diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml index 5bfb7c22..4c09835c 100644 --- a/.pre-commit-config.yaml +++ b/.pre-commit-config.yaml @@ -31,4 +31,3 @@ repos: entry: gitlint stages: [commit-msg] args: [--msg-filename] - diff --git a/dags/aps/aps_params.py b/dags/aps/aps_params.py index 56a37e87..78c304d7 100644 --- a/dags/aps/aps_params.py +++ b/dags/aps/aps_params.py @@ -4,11 +4,11 @@ class APSParams: def __init__( self, - from_date= (date.today() - timedelta(days=1)).strftime("%Y-%m-%d"), - until_date= date.today().strftime("%Y-%m-%d"), - date= "modified", - journals= "", - set= "scoap3", + from_date=(date.today() - timedelta(days=1)).strftime("%Y-%m-%d"), + until_date=date.today().strftime("%Y-%m-%d"), + date="modified", + journals="", + set="scoap3", per_page: int = 100, ): self.from_date = from_date diff --git a/dags/aps/aps_pull_api.py b/dags/aps/aps_pull_api.py index 5e85b928..8b6302e0 100644 --- a/dags/aps/aps_pull_api.py +++ b/dags/aps/aps_pull_api.py @@ -18,11 +18,11 @@ ) def aps_pull_api(): @task() - def set_fetching_intervals(repo = APSRepository(), **kwargs): + def set_fetching_intervals(repo=APSRepository(), **kwargs): return set_harvesting_interval(repo=repo, **kwargs) @task() - def save_json_in_s3(dates: dict, repo = APSRepository(), **kwargs): + def save_json_in_s3(dates: dict, repo=APSRepository(), **kwargs): parameters = APSParams( from_date=dates["from_date"], until_date=dates["until_date"], @@ -40,7 +40,7 @@ def save_json_in_s3(dates: dict, repo = APSRepository(), **kwargs): return None @task() - def trigger_files_processing(key, repo = APSRepository()): + def trigger_files_processing(key, repo=APSRepository()): if key is None: logging.warning("No new files were downloaded to s3") return diff --git a/dags/aps/parser.py b/dags/aps/parser.py index c46d4506..14c8d20a 100644 --- a/dags/aps/parser.py +++ b/dags/aps/parser.py @@ -97,26 +97,27 @@ def _form_authors(self, article): if author["type"] == "Person" ] - def extract_organization_and_ror(self, text): pattern = r'(.*?)' - + ror_url = None - + def replace_and_capture(match): nonlocal ror_url ror_url = match.group(1) return match.group(2) - + modified_text = re.sub(pattern, replace_and_capture, text) - + return modified_text, ror_url def _get_affiliations(self, article, affiliationIds): parsed_affiliations = [ { "value": affiliation["name"], - "organization": self.extract_organization_and_ror(affiliation["name"])[0], + "organization": self.extract_organization_and_ror(affiliation["name"])[ + 0 + ], "ror": self.extract_organization_and_ror(affiliation["name"])[1], } for affiliation in article["affiliations"] diff --git a/dags/aps/repository.py b/dags/aps/repository.py index d07f9f26..cc96e8c9 100644 --- a/dags/aps/repository.py +++ b/dags/aps/repository.py @@ -1,6 +1,5 @@ import io import os -from typing import IO from common.repository import IRepository from common.s3_service import S3Service diff --git a/dags/clean/cleanup_logs.py b/dags/clean/cleanup_logs.py index e8b4db93..babb2219 100644 --- a/dags/clean/cleanup_logs.py +++ b/dags/clean/cleanup_logs.py @@ -2,7 +2,6 @@ import pendulum from airflow.decorators import dag -from airflow.operators.bash import BashOperator from airflow.operators.bash_operator import BashOperator AIRFLOW_HOME = os.getenv("AIRFLOW_HOME") @@ -17,7 +16,7 @@ def cleanup_logs(): bash_command=f""" logs_dir="{logs_dir}" find "$logs_dir" -type d -mtime +30 -exec rm -r {{}} \; - """, + """, # noqa ) diff --git a/dags/common/cleanup.py b/dags/common/cleanup.py index a2b4eb8f..1e795054 100644 --- a/dags/common/cleanup.py +++ b/dags/common/cleanup.py @@ -35,7 +35,7 @@ def replace_cdata_format(text): CDATA_PATTERN = re.compile(r"<\?CDATA(.*)\?>") # pattern = re.compile(r'<\?CDATA\s(.*?)\s\?>', re.DOTALL) - replaced_text = CDATA_PATTERN.sub(r'', text) + replaced_text = CDATA_PATTERN.sub(r"", text) return replaced_text diff --git a/dags/common/constants.py b/dags/common/constants.py index 77e7a2b4..4e9ed959 100644 --- a/dags/common/constants.py +++ b/dags/common/constants.py @@ -91,7 +91,7 @@ ("Brazil", "Brazil"), ("Brasil", "Brazil"), ("Benin", "Benin"), - (u"Bénin", "Benin"), + ("Bénin", "Benin"), ("Bulgaria", "Bulgaria"), ("Bosnia and Herzegovina", "Bosnia and Herzegovina"), ("Canada", "Canada"), @@ -141,7 +141,7 @@ ("Luxembourg", "Luxembourg"), ("Macedonia", "Macedonia"), ("Mexico", "Mexico"), - (u"México", "Mexico"), + ("México", "Mexico"), ("Monaco", "Monaco"), ("Montenegro", "Montenegro"), ("Morocco", "Morocco"), @@ -161,7 +161,7 @@ ("Portugalo", "Portugal"), ("Portugal", "Portugal"), ("P.R.China", "China"), - (u"People’s Republic of China", "China"), + ("People’s Republic of China", "China"), ("Republic of Belarus", "Belarus"), ("Republic of Benin", "Benin"), ("Republic of Korea", "South Korea"), @@ -181,7 +181,7 @@ ("Slovenia", "Slovenia"), ("South Africa", "South Africa"), ("Africa", "South Africa"), - (u"España", "Spain"), + ("España", "Spain"), ("Spain", "Spain"), ("Sudan", "Sudan"), ("Sweden", "Sweden"), @@ -233,19 +233,21 @@ ] ) -INSTITUTIONS_AND_COUNTRIES_MAPPING = OrderedDict([ - ("INFN", "Italy"), - ("European Organization for Nuclear Research", "CERN"), - ("Conseil Européen pour la Recherche Nucléaire", "CERN"), - ("CERN", "CERN"), - ("KEK", "Japan"), - ("DESY", "Germany"), - ("FERMILAB", "USA"), - ("FNAL", "USA"), - ("SLACK", "USA"), - ("Stanford Linear Accelerator Center", "USA"), - ("Joint Institute for Nuclear Research", "JINR"), - ("JINR", "JINR"), - ("ROC", "Taiwan"), - ("R.O.C", "Taiwan"), -]) +INSTITUTIONS_AND_COUNTRIES_MAPPING = OrderedDict( + [ + ("INFN", "Italy"), + ("European Organization for Nuclear Research", "CERN"), + ("Conseil Européen pour la Recherche Nucléaire", "CERN"), + ("CERN", "CERN"), + ("KEK", "Japan"), + ("DESY", "Germany"), + ("FERMILAB", "USA"), + ("FNAL", "USA"), + ("SLACK", "USA"), + ("Stanford Linear Accelerator Center", "USA"), + ("Joint Institute for Nuclear Research", "JINR"), + ("JINR", "JINR"), + ("ROC", "Taiwan"), + ("R.O.C", "Taiwan"), + ] +) diff --git a/dags/common/enhancer.py b/dags/common/enhancer.py index b49ebb61..1e9f6b57 100644 --- a/dags/common/enhancer.py +++ b/dags/common/enhancer.py @@ -2,7 +2,7 @@ import re from common.constants import FN_REGEX -from common.utils import parse_country_from_value, get_country_ISO_name +from common.utils import get_country_ISO_name, parse_country_from_value class Enhancer: @@ -47,7 +47,7 @@ def __construct_titles(self, item, publisher): def __construct_authors(self, item): # add_nations(item) pattern_for_cern_cooperation_agreement = re.compile( - r'cooperation agreement with cern', re.IGNORECASE + r"cooperation agreement with cern", re.IGNORECASE ) for author in item.get("authors", []): for affiliation in author.get("affiliations", []): @@ -65,11 +65,12 @@ def __construct_authors(self, item): affiliation["country"] = _parsed_country if affiliation.get("country"): - affiliation["country"] = get_country_ISO_name(affiliation["country"]) + affiliation["country"] = get_country_ISO_name( + affiliation["country"] + ) return item - def __call__(self, publisher, item): creation_date = datetime.datetime.now().isoformat() item_copy = item.copy() diff --git a/dags/common/parsing/xml_extractors.py b/dags/common/parsing/xml_extractors.py index 4f878f0c..8195424d 100644 --- a/dags/common/parsing/xml_extractors.py +++ b/dags/common/parsing/xml_extractors.py @@ -17,7 +17,7 @@ def __init__( extra_function=lambda s: s, prefixes=None, all_content_between_tags=False, - remove_tags=False + remove_tags=False, ): super().__init__(destination) @@ -92,7 +92,7 @@ def __init__( default_value=None, required=False, extra_function=lambda x: x, - ) : + ): super().__init__(destination) self.destination = destination self.source = source @@ -132,7 +132,7 @@ def extract(self, article): class CustomExtractor(IExtractor): def __init__( self, destination, extraction_function, required=False, default_value=None - ) : + ): super().__init__(destination) self.destination = destination self.extraction_function = extraction_function @@ -154,7 +154,7 @@ def __init__( destination, value, required=False, - ) : + ): super().__init__(destination) self.destination = destination self.required = required diff --git a/dags/common/scoap3_s3.py b/dags/common/scoap3_s3.py index f5c933ea..49f9c4b6 100644 --- a/dags/common/scoap3_s3.py +++ b/dags/common/scoap3_s3.py @@ -8,11 +8,8 @@ logger = get_logger() -FILE_EXTENSIONS = { - "pdf": ".pdf", - "xml": ".xml", - "pdfa": ".pdf" -} +FILE_EXTENSIONS = {"pdf": ".pdf", "xml": ".xml", "pdfa": ".pdf"} + def update_filename_extension(filename, type): extension = FILE_EXTENSIONS.get(type, "") @@ -20,9 +17,10 @@ def update_filename_extension(filename, type): return filename elif extension: if type == "pdfa": - extension = f".a-2b.pdf" + extension = ".a-2b.pdf" return f"{filename}{extension}" + class Scoap3Repository(IRepository): def __init__(self): super().__init__() @@ -55,7 +53,7 @@ def copy_file(self, source_bucket, source_key, prefix=None, type=None): "source_key": source_key, }, "MetadataDirective": "REPLACE", - "ACL": "public-read" + "ACL": "public-read", }, ) logger.info( @@ -67,7 +65,9 @@ def copy_files(self, bucket, files, prefix=None): copied_files = {} for type, path in files.items(): try: - copied_files[type] = self.copy_file(bucket, path, prefix=prefix, type=type) + copied_files[type] = self.copy_file( + bucket, path, prefix=prefix, type=type + ) except Exception as e: logger.error("Failed to copy file.", error=str(e), type=type, path=path) return copied_files diff --git a/dags/common/utils.py b/dags/common/utils.py index 76f36c2c..71847c1f 100644 --- a/dags/common/utils.py +++ b/dags/common/utils.py @@ -1,4 +1,3 @@ -from datetime import date, datetime import io import json import os @@ -6,10 +5,10 @@ import tarfile import xml.etree.ElementTree as ET import zipfile +from datetime import date, datetime from ftplib import error_perm from io import StringIO from stat import S_ISDIR, S_ISREG -from inspire_utils.record import get_value import backoff import country_converter as coco @@ -20,7 +19,6 @@ BY_PATTERN, CDATA_PATTERN, COUNTRIES_DEFAULT_MAPPING, - COUNTRY_PARSING_PATTERN, CREATIVE_COMMONS_PATTERN, INSTITUTIONS_AND_COUNTRIES_MAPPING, LICENSE_PATTERN, @@ -30,11 +28,13 @@ UnknownFileExtension, UnknownLicense, ) +from inspire_utils.record import get_value from structlog import get_logger logger = get_logger() cc = coco.CountryConverter() + def set_harvesting_interval(repo, **kwargs): if ( "params" in kwargs @@ -268,7 +268,7 @@ def iterate_element(item): iterate_element(item) title_part = [i for i in title_parts if i] - full_text = ' '.join(title_part).strip() + full_text = " ".join(title_part).strip() return full_text @@ -311,10 +311,12 @@ def parse_country_from_value(affiliation_value): country_code = cc.convert(country, to="iso2") mapped_countries = [] if country_code != "not found": - mapped_countries = [{ - "code": country_code, - "name": cc.convert(country, to="name_short"), - }] + mapped_countries = [ + { + "code": country_code, + "name": cc.convert(country, to="name_short"), + } + ] if len(mapped_countries) > 1 or len(mapped_countries) == 0: raise FoundMoreThanOneMatchOrNone(affiliation_value) diff --git a/dags/elsevier/elsevier_pull_sftp.py b/dags/elsevier/elsevier_pull_sftp.py index e9b539c1..3a1e4504 100644 --- a/dags/elsevier/elsevier_pull_sftp.py +++ b/dags/elsevier/elsevier_pull_sftp.py @@ -23,9 +23,7 @@ def elsevier_pull_sftp(): @task(executor_config=kubernetes_executor_config) def migrate_from_ftp( - sftp = ElsevierSFTPService(), - repo = ElsevierRepository(), - **kwargs + sftp=ElsevierSFTPService(), repo=ElsevierRepository(), **kwargs ): params = kwargs["params"] specific_files = ( @@ -44,7 +42,7 @@ def migrate_from_ftp( @task(executor_config=kubernetes_executor_config) def trigger_file_processing( - repo = ElsevierRepository(), + repo=ElsevierRepository(), filenames=None, ): return trigger_file_processing_elsevier( diff --git a/dags/executor_config.py b/dags/executor_config.py index db0feb33..971ddecf 100644 --- a/dags/executor_config.py +++ b/dags/executor_config.py @@ -9,9 +9,9 @@ resources=k8s.V1ResourceRequirements( requests={"memory": "1500Mi"}, limits={"memory": "2Gi"}, - ) + ), ) ], ) ), -} \ No newline at end of file +} diff --git a/dags/hindawi/hindawi_api_client.py b/dags/hindawi/hindawi_api_client.py index 0546e11c..a1883da0 100644 --- a/dags/hindawi/hindawi_api_client.py +++ b/dags/hindawi/hindawi_api_client.py @@ -29,7 +29,7 @@ def get_articles_metadata(self, parameters, doi=None): base_url=self.base_url, headers={ "Accept": "application/xml", - "User-Agent": "Mozilla/5.0 (Windows NT 10.0; Win64; x64; rv:89.0) Gecko/20100101 Firefox/89.0" + "User-Agent": "Mozilla/5.0 (Windows NT 10.0; Win64; x64; rv:89.0) Gecko/20100101 Firefox/89.0", }, path_segments=path_segments, parameters=parameters, diff --git a/dags/hindawi/hindawi_params.py b/dags/hindawi/hindawi_params.py index 4d7df7c8..39b0b569 100644 --- a/dags/hindawi/hindawi_params.py +++ b/dags/hindawi/hindawi_params.py @@ -4,12 +4,12 @@ class HindawiParams: def __init__( self, - from_date= (date.today() - timedelta(days=1)).strftime("%Y-%m-%d"), - until_date= date.today().strftime("%Y-%m-%d"), - verb= "listrecords", - set= "HINDAWI.AHEP", - metadataprefix= "marc21", - record= "", + from_date=(date.today() - timedelta(days=1)).strftime("%Y-%m-%d"), + until_date=date.today().strftime("%Y-%m-%d"), + verb="listrecords", + set="HINDAWI.AHEP", + metadataprefix="marc21", + record="", ): self.from_date = from_date self.until_date = until_date diff --git a/dags/hindawi/parser.py b/dags/hindawi/parser.py index a3b706f7..a72b9123 100644 --- a/dags/hindawi/parser.py +++ b/dags/hindawi/parser.py @@ -216,9 +216,7 @@ def _get_publication_info(self, article): "journal_volume": journal.find( "./ns0:subfield/[@code='v']", self.prefixes ).text, - "year": journal.find( - "./ns0:subfield/[@code='y']", self.prefixes - ).text, + "year": journal.find("./ns0:subfield/[@code='y']", self.prefixes).text, } for journal in journals ] diff --git a/dags/iop/iop_process_file.py b/dags/iop/iop_process_file.py index 47a80e3f..62f0084a 100644 --- a/dags/iop/iop_process_file.py +++ b/dags/iop/iop_process_file.py @@ -3,16 +3,16 @@ import pendulum from airflow.decorators import dag, task +from common.cleanup import ( + convert_html_italics_to_latex, + convert_html_subscripts_to_latex, + replace_cdata_format, +) from common.enhancer import Enhancer from common.enricher import Enricher from common.exceptions import EmptyOutputFromPreviousTask from common.scoap3_s3 import Scoap3Repository from common.utils import create_or_update_article, upload_json_to_s3 -from common.cleanup import ( - replace_cdata_format, - convert_html_subscripts_to_latex, - convert_html_italics_to_latex, -) from inspire_utils.record import get_value from iop.parser import IOPParser from iop.repository import IOPRepository @@ -20,12 +20,14 @@ logger = get_logger() + def process_xml(input): input = convert_html_subscripts_to_latex(input) input = convert_html_italics_to_latex(input) input = replace_cdata_format(input) return input + def iop_parse_file(**kwargs): if "params" not in kwargs or "file" not in kwargs["params"]: raise Exception("There was no 'file' parameter. Exiting run.") @@ -33,7 +35,7 @@ def iop_parse_file(**kwargs): file_name = kwargs["params"]["file_name"] xml_bytes = base64.b64decode(encoded_xml) if isinstance(xml_bytes, bytes): - xml_bytes = xml_bytes.decode('utf-8') + xml_bytes = xml_bytes.decode("utf-8") xml_bytes = process_xml(xml_bytes) xml = ET.fromstring(xml_bytes) diff --git a/dags/iop/parser.py b/dags/iop/parser.py index 6161ac2c..57ba642b 100644 --- a/dags/iop/parser.py +++ b/dags/iop/parser.py @@ -17,8 +17,8 @@ extract_text, get_license_type, get_license_type_and_version_from_url, + parse_country_from_value, parse_to_int, - parse_country_from_value ) from idutils import is_arxiv from inspire_utils.date import PartialDate @@ -114,7 +114,7 @@ def __init__(self, file_path=None): required=True, all_content_between_tags=True, source="front/article-meta/title-group/article-title", - remove_tags=True + remove_tags=True, ), TextExtractor( destination="subtitle", @@ -131,7 +131,7 @@ def __init__(self, file_path=None): source="front/article-meta/abstract/p", all_content_between_tags=True, extra_function=lambda x: x, - remove_tags=True + remove_tags=True, ), CustomExtractor( destination="files", diff --git a/dags/oup/oup_pull_ftp.py b/dags/oup/oup_pull_ftp.py index a8fc9444..8b868f2c 100644 --- a/dags/oup/oup_pull_ftp.py +++ b/dags/oup/oup_pull_ftp.py @@ -19,9 +19,7 @@ def oup_pull_ftp(): logger = get_logger().bind(class_name="oup_pull_ftp") @task() - def migrate_from_ftp( - ftp = OUPFTPService(), repo = OUPRepository(), **kwargs - ): + def migrate_from_ftp(ftp=OUPFTPService(), repo=OUPRepository(), **kwargs): params = kwargs["params"] specific_files = ( "filenames_pull" in params @@ -37,7 +35,7 @@ def migrate_from_ftp( @task() def trigger_file_processing( - repo = OUPRepository(), + repo=OUPRepository(), filenames=None, ): return pull_ftp.trigger_file_processing( diff --git a/dags/oup/parser.py b/dags/oup/parser.py index b961dcfb..f0c3f27c 100644 --- a/dags/oup/parser.py +++ b/dags/oup/parser.py @@ -163,7 +163,9 @@ def _get_authors(self, article): ) authors = [] for contribution in contributions: - orcid = get_text_value(contribution.find("contrib-id[@contrib-id-type='orcid']")) + orcid = get_text_value( + contribution.find("contrib-id[@contrib-id-type='orcid']") + ) surname = get_text_value(contribution.find("name/surname")) given_names = get_text_value(contribution.find("name/given-names")) email = get_text_value(contribution.find("email")) diff --git a/dags/springer/parser.py b/dags/springer/parser.py index 9fd62a64..64180b46 100644 --- a/dags/springer/parser.py +++ b/dags/springer/parser.py @@ -9,7 +9,7 @@ CustomExtractor, TextExtractor, ) -from common.utils import construct_license, clean_text +from common.utils import clean_text, construct_license from structlog import get_logger @@ -211,7 +211,11 @@ def _get_affiliations(self, author_group, contrib): affiliations.append(cleaned_aff) mapped_affiliations = [ - {"value": clean_text(aff), "organization": clean_text(org), **({"country": country} if country else {})} + { + "value": clean_text(aff), + "organization": clean_text(org), + **({"country": country} if country else {}), + } for aff, org, country, in affiliations ] diff --git a/documentation/Hindawi/hindawi_fields_mapping.md b/documentation/Hindawi/hindawi_fields_mapping.md index 1824eaaf..a8746b09 100644 --- a/documentation/Hindawi/hindawi_fields_mapping.md +++ b/documentation/Hindawi/hindawi_fields_mapping.md @@ -5,16 +5,16 @@ | dois | generic_parsing : [33] | value | | | arxiv_eprints | enricher : [67] | value | | | | | categories | | -| page_nr | parsing : [6] | | | +| page_nr | parsing : [6] | | | | authors | parsing : [6]
generic_parsing : [22] | surname | | | | | given_names | | | | | full_name | | | | | affiliations | country | | | | | institution | -| collections | parsing [12] | | | +| collections | parsing [12] | | | | license | parsing [11] | url | | | | | license | | -| publication_info | generic_parsing : [40]] | journal_title | | +| publication_info | generic_parsing : [40]] | journal_title | | | | | journal_volume | | | | | year | | | abstracts | enhancer : [46] | value | | @@ -30,8 +30,8 @@ | | | source | | | $schema | enricher : [66] | | | - # [Enricher](#enricher) + | | | | | ------------------------------ | ------------- | ----------------------------------------------------- | | Reference | Field | Enricher | @@ -65,24 +65,24 @@ ### [\_\_construct_abstracts](#__construct_abstracts) -| Reference | Subfield | Value | -| ------------------------------ | -------- | ------------------------------------------------------------------------------ | +| Reference | Subfield | Value | +| ------------------------------ | -------- | ---------------------------------------------------------------------------------------------- | | [53] | value | Take value from generic parsing abstract [23] | -| [54] | source | Constant: Hindawi | +| [54] | source | Constant: Hindawi | ### [\_\_construct_acquisition_source](#__construct_acquisition_source) | Reference | Subfield | Value | | ------------------------------ | -------- | ------------------------------------------------ | -| [55] | source | Constant: Hindawi | -| [56] | method | Constant: Hindawi | +| [55] | source | Constant: Hindawi | +| [56] | method | Constant: Hindawi | | [57] | date | datetime.datetime.now().isoformat() | ### [\_\_construct_copyright](#__construct_copyright) -| Reference | Subfield | Value | -| ------------------------------ | --------- | ----------------------------------------------------------------------------------------- | -| [58] | year | Take value from parsing copyright_year [10] | +| Reference | Subfield | Value | +| ------------------------------ | --------- | --------------------------------------------------------------------------------------- | +| [58] | year | Take value from parsing copyright_year [10] | | [59] | statement | Take value from parsing copyright_statement [9] | ### [\_\_construct_imprints](#__construct_imprints) @@ -107,99 +107,90 @@ ### [\_\_remove_country](#__remove_country) -| | | | | -| ------------------------------ | ---------------------------------------------------------------------------------------- | ----- | -------------------------------------------- | -| Reference | Field | Value | Processing | -| [65] | from parsed affiliation country [55] | | removes county if the value has: | - +| | | | | +| ------------------------------ | ----------------------------------------------------------------------------- | ----- | -------------------------------- | +| Reference | Field | Value | Processing | +| [65] | from parsed affiliation country [55] | | removes county if the value has: | # [Generic parsing](#generic_parsing) -| Reference | Field | Subfield | Processing | Default value | -|-----------|------------------------|----------------------|--------------------------------------------------------------------------------------------------------------------------------------|---------------| +| Reference | Field | Subfield | Processing | Default value | +| ------------------------------ | ---------------------- | -------------------- | -------------------------------------------------------------------------------------------------------------------------------------------- | ------------- | | [22] | authors | surname, given_names | takes authors [2] and splits raw_name: if there is a comma, it means that the surname and given_name are in the second part | | -| [23] | abstract | | takes abstract [3] and cleans white space characters | | -| [24] | collaborations | | NO SUCH A FIELD IN HINDAWI | | -| [25] | title | | takes title [4] and cleans white space characters | | -| [26] | subtitle | | NO SUCH A FIELD IN HINDAWI | | -| [27] | journal_year | | | | -| [28] | preprint_date | | NO SUCH A FIELD IN HINDAWI | | -| [29] | date_published | | takes date_published [5] and forms it f"{tmp_date.year:04d}-{tmp_date.month:02d}-{tmp_date.day:02d}" | | -| [30] | related_article_doi | | NO SUCH A FIELD IN HINDAWI | | -| [31] | free_keywords | | NO SUCH A FIELD IN HINDAWI | | -| [32] | classification_numbers | | NO SUCH A FIELD IN HINDAWI | | -| [33] | dois | | takes dois | | -| [34] | thesis_supervisor | | NO SUCH A FIELD IN HINDAWI | | -| [35] | thesis | | NO SUCH A FIELD IN HINDAWI | | -| [36] | urls | | NO SUCH A FIELD IN HINDAWI | | -| [37] | local_files | | NO SUCH A FIELD IN HINDAWI | | -| [38] | record_creation_date | | NO SUCH A FIELD IN HINDAWI | | -| [39] | control_field | | NO SUCH A FIELD IN HINDAWI | | -| [40] | publication_info | | | | -| [41] | | journal_title | takes journal title [16] | | -| [42] | | journal_volume | takes journal volume [17] | | -| [43] | | journal_year | takes journal year [18] | | -| [44] | | journal_issue | NO SUCH A FIELD IN HINDAWI | | -| [45] | | journal_doctype | NO SUCH A FIELD IN HINDAWI | | - +| [23] | abstract | | takes abstract [3] and cleans white space characters | | +| [24] | collaborations | | NO SUCH A FIELD IN HINDAWI | | +| [25] | title | | takes title [4] and cleans white space characters | | +| [26] | subtitle | | NO SUCH A FIELD IN HINDAWI | | +| [27] | journal_year | | | | +| [28] | preprint_date | | NO SUCH A FIELD IN HINDAWI | | +| [29] | date_published | | takes date_published [5] and forms it f"{tmp_date.year:04d}-{tmp_date.month:02d}-{tmp_date.day:02d}" | | +| [30] | related_article_doi | | NO SUCH A FIELD IN HINDAWI | | +| [31] | free_keywords | | NO SUCH A FIELD IN HINDAWI | | +| [32] | classification_numbers | | NO SUCH A FIELD IN HINDAWI | | +| [33] | dois | | takes dois | | +| [34] | thesis_supervisor | | NO SUCH A FIELD IN HINDAWI | | +| [35] | thesis | | NO SUCH A FIELD IN HINDAWI | | +| [36] | urls | | NO SUCH A FIELD IN HINDAWI | | +| [37] | local_files | | NO SUCH A FIELD IN HINDAWI | | +| [38] | record_creation_date | | NO SUCH A FIELD IN HINDAWI | | +| [39] | control_field | | NO SUCH A FIELD IN HINDAWI | | +| [40] | publication_info | | | | +| [41] | | journal_title | takes journal title [16] | | +| [42] | | journal_volume | takes journal volume [17] | | +| [43] | | journal_year | takes journal year [18] | | +| [44] | | journal_issue | NO SUCH A FIELD IN HINDAWI | | +| [45] | | journal_doctype | NO SUCH A FIELD IN HINDAWI | | # [Parsing](#parsing) - -| Reference | Field | Source | Parsing | -|-----------|---------------------|-----------------------------------------------------------------------------|------------------------------------------------------------------| -| [1] | dois | ns0:metadata/ns1:record/ns0:datafield/[@tag='024']/ns0:subfield/[@code='a'] | lambda x: [x] | -| [2] | authors | | authors_parsing | -| [3] | abstract | ns0:metadata/ns1:record/ns0:datafield/[@tag='520']/ns0:subfield/[@code='a'] | lambda x: " ".join(x.split()) | -| [4] | title | ns0:metadata/ns1:record/ns0:datafield/[@tag='245']/ns0:subfield/[@code='a'] | lambda x: x | -| [5] | date_published | ns0:metadata/ns1:record/ns0:datafield/[@tag='260']/ns0:subfield/[@code='c'] | lambda x: x | -| [6] | page_nr | ns0:metadata/ns1:record/ns0:datafield/[@tag='300']/ns0:subfield/[@code='a'] | lambda x: [int(x)] | -| [7] | publication_info | | _get_publication_info | -| [8] | arxiv_eprints | | _get_arxiv | -| [9] | copyright_statement | ns0:metadata/ns1:record/ns0:datafield/[@tag='542']/ns0:subfield/[@code='f'] | | -| [10] | copyright_year | ns0:metadata/ns1:record/ns0:datafield/[@tag='542']/ns0:subfield/ | re.search(r"[0-9]{4}", value).group(0) | -| [11] | license | | _get_license | -| [12] | collections | | constant: "Advances in High Energy Physics" | - +| Reference | Field | Source | Parsing | +| ------------------------------ | ------------------- | --------------------------------------------------------------------------- | ----------------------------------------------------------- | +| [1] | dois | ns0:metadata/ns1:record/ns0:datafield/[@tag='024']/ns0:subfield/[@code='a'] | lambda x: [x] | +| [2] | authors | | authors_parsing | +| [3] | abstract | ns0:metadata/ns1:record/ns0:datafield/[@tag='520']/ns0:subfield/[@code='a'] | lambda x: " ".join(x.split()) | +| [4] | title | ns0:metadata/ns1:record/ns0:datafield/[@tag='245']/ns0:subfield/[@code='a'] | lambda x: x | +| [5] | date_published | ns0:metadata/ns1:record/ns0:datafield/[@tag='260']/ns0:subfield/[@code='c'] | lambda x: x | +| [6] | page_nr | ns0:metadata/ns1:record/ns0:datafield/[@tag='300']/ns0:subfield/[@code='a'] | lambda x: [int(x)] | +| [7] | publication_info | | \_get_publication_info | +| [8] | arxiv_eprints | | \_get_arxiv | +| [9] | copyright_statement | ns0:metadata/ns1:record/ns0:datafield/[@tag='542']/ns0:subfield/[@code='f'] | | +| [10] | copyright_year | ns0:metadata/ns1:record/ns0:datafield/[@tag='542']/ns0:subfield/ | re.search(r"[0-9]{4}", value).group(0) | +| [11] | license | | \_get_license | +| [12] | collections | | constant: "Advances in High Energy Physics" | ### [authors_parsing](#authors_parsing) -| Reference | Field | Source | Parsing | -|-----------|--------------|-------------------------------------------------|-----------------------------------------------------| +| Reference | Field | Source | Parsing | +| ------------------------------ | ------------ | ----------------------------------------------- | --------------------------------------------------- | | [13] | raw_name | ns0:subfield[@code='a'] | lambda x: [x] | -| [14] | affiliations | | _get_affiliations | +| [14] | affiliations | | \_get_affiliations | | [15] | orcid | ns0:subfield[@code='a']/ns0:subfield[@code='j'] | lambda x: " ".join(x.split()) | +### [\_get_publication_info](#_get_publication_info) - -### [_get_publication_info](#_get_publication_info) - -| Reference | Field | Source | Parsing | -|-----------|----------------|-----------------------------------------------------------------------------|---------| +| Reference | Field | Source | Parsing | +| ------------------------------ | -------------- | --------------------------------------------------------------------------- | ------- | | [16] | journal_title | ns0:metadata/ns1:record/ns0:datafield/[@tag='773']/ns0:subfield/[@code='p'] | | | [17] | journal_volume | ns0:metadata/ns1:record/ns0:datafield/[@tag='773']/ns0:subfield/[@code='v'] | | | [18] | journal_year | ns0:metadata/ns1:record/ns0:datafield/[@tag='773']/ns0:subfield/[@code='y'] | | +### [\_get_arxiv](#_get_arxiv) +| Reference | Field | Source | Parsing | +| ------------------------------ | ----- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------ | +| [19] | value | "ns0:metadata/ns1:record/ns0:datafield/[@tag='037']/ns0:subfield/[@code='a']"
if the field above == 'arxiv'
field above:
ns0:metadata/ns1:record/ns0:datafield/[@tag='037']/ns0:subfield/[@code='9'] | Removing "arxiv" from value, leaving just digits | -### [_get_arxiv](#_get_arxiv) - -| Reference | Field | Source | Parsing | -|-----------|-------|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|--------------------------------------------------| -| [19] | value | "ns0:metadata/ns1:record/ns0:datafield/[@tag='037']/ns0:subfield/[@code='a']"
if the field above == 'arxiv'
field above:
ns0:metadata/ns1:record/ns0:datafield/[@tag='037']/ns0:subfield/[@code='9'] | Removing "arxiv" from value, leaving just digits | - - +### [\_get_license](#_get_arxiv) -### [_get_license](#_get_arxiv) -| Reference | Field | Source | Parsing | -|-----------|---------|---------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| +| Reference | Field | Source | Parsing | +| ------------------------------ | ------- | ------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | [20] | url | License urls: "ns0:metadata/ns1:record/ns0:datafield/[@tag='540']/ns0:subfield/[@code='u']" | | | [21] | license | license text = ns0:metadata/ns1:record/ns0:datafield/[@tag='540']/ns0:subfield/[@code='a'] | url_parts = license_url.text.split("/")
clean_url_parts = list(filter(bool, url_parts))
version = clean_url_parts.pop()
license_type = clean_url_parts.pop()
f"CC-{license_type}-{version}" | -### [_get_affiliations](#_get_affiliations) +### [\_get_affiliations](#_get_affiliations) -| Reference | Field | Source | Parsing | -|-----------|--------------|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------| -| [53] | value | ns0:metadata/ns1:record/ns0:datafield/[@tag='100']/ns0:subfield[@code='u']
ns0:metadata/ns1:record/ns0:datafield/[@tag='700']/ns0:subfield[@code='u'] | | -| [54] | organization | same as value: ns0:metadata/ns1:record/ns0:datafield/[@tag='100']/ns0:subfield[@code='u']
ns0:metadata/ns1:record/ns0:datafield/[@tag='700']/ns0:subfield[@code='u'] | takes the string before the last comma | +| Reference | Field | Source | Parsing | +| ------------------------------ | ------------ | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------- | +| [53] | value | ns0:metadata/ns1:record/ns0:datafield/[@tag='100']/ns0:subfield[@code='u']
ns0:metadata/ns1:record/ns0:datafield/[@tag='700']/ns0:subfield[@code='u'] | | +| [54] | organization | same as value: ns0:metadata/ns1:record/ns0:datafield/[@tag='100']/ns0:subfield[@code='u']
ns0:metadata/ns1:record/ns0:datafield/[@tag='700']/ns0:subfield[@code='u'] | takes the string before the last comma | | [55] | country | same as value: ns0:metadata/ns1:record/ns0:datafield/[@tag='100']/ns0:subfield[@code='u']
ns0:metadata/ns1:record/ns0:datafield/[@tag='700']/ns0:subfield[@code='u'] | takes the last string after comma, which starts with a capital letter | diff --git a/documentation/IOP/iop_fields_mapping.md b/documentation/IOP/iop_fields_mapping.md index f713d35b..c67366c0 100644 --- a/documentation/IOP/iop_fields_mapping.md +++ b/documentation/IOP/iop_fields_mapping.md @@ -80,8 +80,8 @@ | Reference | Subfield | Value | | ------------------------------ | -------- | ------------------------------------------------ | -| [57] | source | Constant: IOP | -| [58] | method | Constant: IOP | +| [57] | source | Constant: IOP | +| [58] | method | Constant: IOP | | [59] | date | datetime.datetime.now().isoformat() | ### [\_\_construct_copyright](#__construct_copyright) diff --git a/requirements-airflow.txt b/requirements-airflow.txt index eceea4bc..88265bbb 100644 --- a/requirements-airflow.txt +++ b/requirements-airflow.txt @@ -1,3 +1,2 @@ -c https://raw.githubusercontent.com/apache/airflow/constraints-2.8.3/constraints-3.10.txt apache-airflow[celery, postgres, redis, cncf.kubernetes, sentry]==2.8.3 - diff --git a/scripts/gitlint_rules/rules.py b/scripts/gitlint_rules/rules.py index 89a7e9fc..5e6790c5 100644 --- a/scripts/gitlint_rules/rules.py +++ b/scripts/gitlint_rules/rules.py @@ -1,6 +1,6 @@ -import re +# import re -from gitlint.rules import CommitRule, RuleViolation +# from gitlint.rules import CommitRule, RuleViolation # EXAMPLE GITLINT CONFIGURATION diff --git a/tests/integration/iop/test_iop_dag_pull_sftp.py b/tests/integration/iop/test_iop_dag_pull_sftp.py index 30236786..71dfde84 100644 --- a/tests/integration/iop/test_iop_dag_pull_sftp.py +++ b/tests/integration/iop/test_iop_dag_pull_sftp.py @@ -1,10 +1,11 @@ +import time + import pytest from airflow.models import DagBag from common.pull_ftp import migrate_from_ftp, trigger_file_processing from iop.repository import IOPRepository from iop.sftp_service import IOPSFTPService from structlog import get_logger -import time DAG_NAME = "iop_pull_sftp" @@ -94,7 +95,7 @@ def test_dag_run(dag, dag_was_paused: bool, iop_empty_repo): def test_dag_migrate_from_FTP(iop_empty_repo): iop_empty_repo.delete_all() assert len(iop_empty_repo.find_all()) == 0 - + with IOPSFTPService() as sftp: migrate_from_ftp( sftp, @@ -165,12 +166,20 @@ def test_dag_migrate_from_FTP(iop_empty_repo): assert len(iop_empty_repo.find_all()) == len(expected_files) - iop_pdf_files = sorted(item["pdf"] for item in iop_empty_repo.find_all() if "pdf" in item) - expected_pdf_files = sorted(item["pdf"] for item in expected_files if "pdf" in item) + iop_pdf_files = sorted( + item["pdf"] for item in iop_empty_repo.find_all() if "pdf" in item + ) + expected_pdf_files = sorted( + item["pdf"] for item in expected_files if "pdf" in item + ) assert iop_pdf_files == expected_pdf_files - iop_xml_files = sorted(item["xml"] for item in iop_empty_repo.find_all() if "xml" in item) - expected_xml_files = sorted(item["xml"] for item in expected_files if "xml" in item) + iop_xml_files = sorted( + item["xml"] for item in iop_empty_repo.find_all() if "xml" in item + ) + expected_xml_files = sorted( + item["xml"] for item in expected_files if "xml" in item + ) assert iop_xml_files == expected_xml_files diff --git a/tests/integration/iop/test_repo.py b/tests/integration/iop/test_repo.py index ae757e72..e9245bc6 100644 --- a/tests/integration/iop/test_repo.py +++ b/tests/integration/iop/test_repo.py @@ -1,9 +1,10 @@ +import time + from common.pull_ftp import migrate_from_ftp from iop.repository import IOPRepository from iop.sftp_service import IOPSFTPService from pytest import fixture from structlog import get_logger -import time @fixture @@ -16,7 +17,7 @@ def iop_empty_repo(): def test_pull_from_sftp(iop_empty_repo): iop_empty_repo.delete_all() assert len(iop_empty_repo.find_all()) == 0 - + with IOPSFTPService() as sftp: migrate_from_ftp( sftp, @@ -49,7 +50,8 @@ def test_pull_from_sftp(iop_empty_repo): { "pdf": "extracted/2022-07-30T03_02_01_content/1674-1137/1674-1137_46/1674-1137_46_8/1674-1137_46_8_085106/cpc_46_8_085106.pdf", "xml": "extracted/2022-07-30T03_02_01_content/1674-1137/1674-1137_46/1674-1137_46_8/1674-1137_46_8_085106/cpc_46_8_085106.xml", - }, { + }, + { "pdf": "extracted/2022-09-01T03_01_40_content/1674-1137/1674-1137_46/1674-1137_46_9/1674-1137_46_9_093111/cpc_46_9_093111.pdf", "xml": "extracted/2022-09-01T03_01_40_content/1674-1137/1674-1137_46/1674-1137_46_9/1674-1137_46_9_093111/cpc_46_9_093111.xml", }, @@ -82,17 +84,24 @@ def test_pull_from_sftp(iop_empty_repo): "xml": "extracted/2022-09-24T03_01_43_content/1674-1137/1674-1137_46/1674-1137_46_10/1674-1137_46_10_103108/cpc_46_10_103108.xml", }, {"xml": "extracted/aca95c/aca95c.xml"}, - ] - + assert len(iop_empty_repo.find_all()) == len(expected_files) - iop_pdf_files = sorted(item["pdf"] for item in iop_empty_repo.find_all() if "pdf" in item) - expected_pdf_files = sorted(item["pdf"] for item in expected_files if "pdf" in item) + iop_pdf_files = sorted( + item["pdf"] for item in iop_empty_repo.find_all() if "pdf" in item + ) + expected_pdf_files = sorted( + item["pdf"] for item in expected_files if "pdf" in item + ) assert iop_pdf_files == expected_pdf_files - iop_xml_files = sorted(item["xml"] for item in iop_empty_repo.find_all() if "xml" in item) - expected_xml_files = sorted(item["xml"] for item in expected_files if "xml" in item) + iop_xml_files = sorted( + item["xml"] for item in iop_empty_repo.find_all() if "xml" in item + ) + expected_xml_files = sorted( + item["xml"] for item in expected_files if "xml" in item + ) assert iop_xml_files == expected_xml_files assert sorted(iop_empty_repo.get_all_raw_filenames()) == sorted( @@ -103,4 +112,4 @@ def test_pull_from_sftp(iop_empty_repo): "2022-09-24T03_01_43_content.zip", "aca95c.zip", ] - ) \ No newline at end of file + ) diff --git a/tests/integration/oup/test_oup_dag_process_file.py b/tests/integration/oup/test_oup_dag_process_file.py index 60c3937c..0e641e93 100644 --- a/tests/integration/oup/test_oup_dag_process_file.py +++ b/tests/integration/oup/test_oup_dag_process_file.py @@ -79,6 +79,7 @@ def test_affiliation_countries_in_enriched(parser, articles): for aff in author.get("affiliations"): assert aff.get("country") is not None + def test_dag_loaded(dag): assert dag assert len(dag.tasks) == 6 diff --git a/tests/units/aps/test_aps_parser.py b/tests/units/aps/test_aps_parser.py index 987ba470..745abdbc 100644 --- a/tests/units/aps/test_aps_parser.py +++ b/tests/units/aps/test_aps_parser.py @@ -61,9 +61,9 @@ def parsed_articles(parser, articles): "surname": "Wu", "affiliations": [ { - "value": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", + "value": 'Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA', "organization": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", - "ror": "https://ror.org/02vm5rt34" + "ror": "https://ror.org/02vm5rt34", } ], }, @@ -73,9 +73,9 @@ def parsed_articles(parser, articles): "surname": "Turner", "affiliations": [ { - "value": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", + "value": 'Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA', "organization": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", - "ror": "https://ror.org/02vm5rt34" + "ror": "https://ror.org/02vm5rt34", } ], }, @@ -85,9 +85,9 @@ def parsed_articles(parser, articles): "surname": "Wang", "affiliations": [ { - "value": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", + "value": 'Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA', "organization": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", - "ror": "https://ror.org/02vm5rt34" + "ror": "https://ror.org/02vm5rt34", } ], }, @@ -97,9 +97,9 @@ def parsed_articles(parser, articles): "surname": "Borel", "affiliations": [ { - "value": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", + "value": 'Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA', "organization": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", - "ror": "https://ror.org/02vm5rt34" + "ror": "https://ror.org/02vm5rt34", } ], }, @@ -111,9 +111,9 @@ def parsed_articles(parser, articles): "surname": "Boudjada", "affiliations": [ { - "value": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", + "value": 'Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA', "organization": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", - "ror": "https://ror.org/02vm5rt34" + "ror": "https://ror.org/02vm5rt34", } ], }, @@ -123,9 +123,9 @@ def parsed_articles(parser, articles): "surname": "Buessen", "affiliations": [ { - "value": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", + "value": 'Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA', "organization": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", - "ror": "https://ror.org/02vm5rt34" + "ror": "https://ror.org/02vm5rt34", } ], }, @@ -135,9 +135,9 @@ def parsed_articles(parser, articles): "surname": "Paramekanti", "affiliations": [ { - "value": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", + "value": 'Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA', "organization": "Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37240, USA", - "ror": "https://ror.org/02vm5rt34" + "ror": "https://ror.org/02vm5rt34", } ], }, diff --git a/tests/units/aps/test_trigger_files_processing.py b/tests/units/aps/test_trigger_files_processing.py index 07192369..e71d7593 100644 --- a/tests/units/aps/test_trigger_files_processing.py +++ b/tests/units/aps/test_trigger_files_processing.py @@ -7,6 +7,7 @@ from airflow.models.dagrun import DagRun from aps.utils import trigger_file_processing_DAG + class S3BucketResultObj: def __init__(self, key): self.key = key diff --git a/tests/units/aps/test_utils.py b/tests/units/aps/test_utils.py index 5b93f4eb..9745f3ff 100644 --- a/tests/units/aps/test_utils.py +++ b/tests/units/aps/test_utils.py @@ -61,6 +61,6 @@ def test_save_file_in_s3(): @freeze_time("2023-12-04 10:00") def test_split_json(): ids_and_articles = split_json(repo=MockedRepo(), key="key/key") - expected_id = f"APS__2023-12-04T10:00:00.000000+0000" + expected_id = "APS__2023-12-04T10:00:00.000000+0000" assert ids_and_articles[0]["id"] == expected_id assert len(ids_and_articles) == 1 diff --git a/tests/units/clean/test_clean.py b/tests/units/clean/test_clean.py index 449b29bc..a24e78eb 100644 --- a/tests/units/clean/test_clean.py +++ b/tests/units/clean/test_clean.py @@ -9,7 +9,7 @@ @fixture def dag(): dagbag = DagBag(dag_folder="dags/", include_examples=False) - assert dagbag.import_errors.get(f"dags/cleanup_logs.py") is None + assert dagbag.import_errors.get("dags/cleanup_logs.py") is None clean_dag = dagbag.get_dag(dag_id="cleanup_logs") return clean_dag diff --git a/tests/units/common/data/file_with_mathML.xml b/tests/units/common/data/file_with_mathML.xml index 106dd94f..dddc4760 100644 --- a/tests/units/common/data/file_with_mathML.xml +++ b/tests/units/common/data/file_with_mathML.xml @@ -1,6 +1,6 @@ <math> - + <!-- Creating Matrix --> <mrow> <mi id=['this will be removed as well']>A</mi> @@ -31,7 +31,7 @@ </mfenced> </mrow> - + <!-- Creating equation --> <br><br> <msub> diff --git a/tests/units/common/test_enhancer.py b/tests/units/common/test_enhancer.py index c5206acc..6c54ec39 100644 --- a/tests/units/common/test_enhancer.py +++ b/tests/units/common/test_enhancer.py @@ -834,6 +834,7 @@ def test_all_contructors(test_input, expected, publisher): enhanced = enhancement(item=test_input, publisher=publisher) assert enhanced == expected + @pytest.mark.skip @pytest.mark.parametrize( "test_input, expected, publisher", @@ -846,7 +847,9 @@ def test_all_contructors(test_input, expected, publisher): ], ) @freeze_time("2022-05-20") -def test_all_contructors_failing_with_wrong_affiliation_value(test_input, expected, publisher): +def test_all_contructors_failing_with_wrong_affiliation_value( + test_input, expected, publisher +): enhancement = Enhancer() enhanced = enhancement(item=test_input, publisher=publisher) assert enhanced == expected diff --git a/tests/units/elsevier/data/CERNQ000000010011/S0370269323000643/main.xml b/tests/units/elsevier/data/CERNQ000000010011/S0370269323000643/main.xml index d6a8d731..206c477b 100644 --- a/tests/units/elsevier/data/CERNQ000000010011/S0370269323000643/main.xml +++ b/tests/units/elsevier/data/CERNQ000000010011/S0370269323000643/main.xml @@ -1 +1 @@ -<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd" [<!ENTITY gr001 SYSTEM "gr001" NDATA IMAGE><!ENTITY gr002 SYSTEM "gr002" NDATA IMAGE><!ENTITY gr003 SYSTEM "gr003" NDATA IMAGE><!ENTITY gr004 SYSTEM "gr004" NDATA IMAGE><!ENTITY gr005 SYSTEM "gr005" NDATA IMAGE><!ENTITY gr006 SYSTEM "gr006" NDATA IMAGE>]><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="sco" xml:lang="en"><item-info><jid>PLB</jid><aid>137730</aid><ce:article-number>137730</ce:article-number><ce:pii>S0370-2693(23)00064-3</ce:pii><ce:doi>10.1016/j.physletb.2023.137730</ce:doi><ce:copyright year="2023" type="other">European Center of Nuclear Research, ALICE experiment</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>Experiments</ce:text></ce:doctopic></ce:doctopics><ce:preprint><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/preprint" xlink:href="arxiv:2204.10210" id="inf0010"/></ce:preprint></item-info><ce:floats><ce:figure id="fg0010"><ce:label>Fig. 1</ce:label><ce:caption id="cp0010"><ce:simple-para id="sp0010">Charged-particle pseudorapidity density in Pb<ce:glyph name="sbnd"/>Pb <ce:cross-ref refid="br0020" id="crf0010">[2]</ce:cross-ref> and p<ce:glyph name="sbnd"/>Pb for the 5% most central collisions, and for pp collisions with INEL>0 trigger class. For symmetric collision systems (Pb<ce:glyph name="sbnd"/>Pb and pp) the data has been symmetrised around <ce:italic>η</ce:italic> = 0 and points for <ce:italic>η</ce:italic> > 3.5 have been reflected around <ce:italic>η</ce:italic> = 0. The boxes around the points and to the right reflect the uncorrelated and correlated, with respect to pseudorapidity, systematic uncertainty, respectively. The relative correlated, normalisation, uncertainties are evaluated at d<ce:italic>N</ce:italic><ce:inf>ch</ce:inf>/d<ce:italic>η</ce:italic>|<ce:inf><ce:italic>η</ce:italic>=0</ce:inf>. The lines show fits of Eq. <ce:cross-ref refid="fm0030" id="crf0020">(1)</ce:cross-ref> (Pb<ce:glyph name="sbnd"/>Pb and pp) and Eq. <ce:cross-ref refid="fm0040" id="crf0030">(2)</ce:cross-ref> (p<ce:glyph name="sbnd"/>Pb) to the data (discussed in Section <ce:cross-ref refid="se0040" id="crf0040">4</ce:cross-ref>). Please note that the ordinate has been cut twice to accommodate for the very different ranges of the charged-particle pseudorapidity densities.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0010">Fig. 1</ce:alt-text><ce:link locator="gr001" xlink:type="simple" xlink:href="pii:S0370269323000643/gr001" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0010"/></ce:figure><ce:figure id="fg0020"><ce:label>Fig. 2</ce:label><ce:caption id="cp0020"><ce:simple-para id="sp0020">Charged-particle pseudorapidity density in p<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> in seven centrality classes based on the V0A and V0C estimators. The lines are obtained using a fit of a scaled, normal distribution in rapidity Eq. <ce:cross-ref refid="fm0040" id="crf0050">(2)</ce:cross-ref> to the data (discussed in Section <ce:cross-ref refid="se0040" id="crf0060">4</ce:cross-ref>).</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0020">Fig. 2</ce:alt-text><ce:link locator="gr002" xlink:type="simple" xlink:href="pii:S0370269323000643/gr002" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0020"/></ce:figure><ce:figure id="fg0030"><ce:label>Fig. 3</ce:label><ce:caption id="cp0030"><ce:simple-para id="sp0030">Ratio <ce:italic>r</ce:italic><ce:inf><ce:italic>X</ce:italic></ce:inf> of the charged-particle pseudorapidity density in Pb<ce:glyph name="sbnd"/>Pb (top) and p<ce:glyph name="sbnd"/>Pb (bottom) in different centrality classes to the charged-particle pseudorapidity density in pp in the INEL>0 event class. Note, for Pb<ce:glyph name="sbnd"/>Pb <ce:italic>η</ce:italic><ce:inf>lab</ce:inf> is the same as the centre-of-mass pseudorapidity.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0030">Fig. 3</ce:alt-text><ce:link locator="gr003" xlink:type="simple" xlink:href="pii:S0370269323000643/gr003" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0030"/></ce:figure><ce:figure id="fg0040"><ce:label>Fig. 4</ce:label><ce:caption id="cp0040"><ce:simple-para id="sp0040">The width (top) and effective <ce:italic>p</ce:italic><ce:inf>T</ce:inf>/<ce:italic>m</ce:italic> (bottom) fit parameters as a function of the mean number of participants in pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>. Vertical uncertainties are the standard error on the best-fit parameter values, while horizontal uncertainties reflect the uncertainty on 〈<ce:italic>N</ce:italic><ce:inf>part</ce:inf>〉 from the Glauber calculations. Also shown are similar fit parameters from the same parameterisation of EPOS-LHC calculations as well as direct calculations of the standard deviation of the d<ce:italic>N</ce:italic><ce:inf>ch</ce:inf>/d<ce:italic>y</ce:italic> distributions and the 〈<ce:italic>p</ce:italic><ce:inf>T</ce:inf>〉/〈<ce:italic>m</ce:italic>〉 ratio from the EPOS-LHC calculations.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0040">Fig. 4</ce:alt-text><ce:link locator="gr004" xlink:type="simple" xlink:href="pii:S0370269323000643/gr004" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0040"/></ce:figure><ce:figure id="fg0050"><ce:label>Fig. 5</ce:label><ce:caption id="cp0050"><ce:simple-para id="sp0050">The transverse area <ce:italic>S</ce:italic><ce:inf>T</ce:inf> as calculated in a numerical Glauber model for two extreme cases: a) only the exclusive overlap of nucleons is considered (∩, open markers) and b) the inclusive area of participating nucleons contribute (∪, closed markers) in both p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0050">Fig. 5</ce:alt-text><ce:link locator="gr005" xlink:type="simple" xlink:href="pii:S0370269323000643/gr005" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0050"/></ce:figure><ce:figure id="fg0060"><ce:label>Fig. 6</ce:label><ce:caption id="cp0060"><ce:simple-para id="sp0060">Estimate of the lower bound on the Bjorken transverse energy density in pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>, considering the exclusive (∩, open markers) and inclusive (∪, full markers) overlap area <ce:italic>S</ce:italic><ce:inf>T</ce:inf> of the nucleons. The expression <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:mi>C</mml:mi><mml:mmultiscripts><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow><mml:none/><mml:none/><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:mmultiscripts></mml:math> is fitted to case ∪, and we find <ce:italic>C</ce:italic> = (0.8 ± 0.3) GeV/(fm<ce:sup>2</ce:sup><ce:italic>c</ce:italic>) and <ce:italic>p</ce:italic> = 0.44 ± 0.08. Also shown is an estimate, via d<ce:italic>E</ce:italic><ce:inf>T</ce:inf>/d<ce:italic>y</ce:italic>, of <ce:italic>ε</ce:italic><ce:inf>Bj</ce:inf> from Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> (stars with uncertainty band) <ce:cross-ref refid="br0310" id="crf0070">[31]</ce:cross-ref>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0060">Fig. 6</ce:alt-text><ce:link locator="gr006" xlink:type="simple" xlink:href="pii:S0370269323000643/gr006" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0060"/></ce:figure></ce:floats><head><ce:title id="ti0010">System-size dependence of the charged-particle pseudorapidity density at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> for pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions</ce:title><ce:author-group id="ag0010"><ce:collaboration id="co0010" collaboration-id="S0370269323000643-3bea72599603117cd9d18494a0279c47"><ce:text>ALICE Collaboration</ce:text><ce:cross-ref refid="fn0080" id="crf0080"><ce:sup>⋆</ce:sup></ce:cross-ref><ce:author-group id="ag0020"><ce:author id="au0010" author-id="S0370269323000643-e60c93a934b81cf9801254193264c6ee"><ce:given-name>S.</ce:given-name><ce:surname>Acharya</ce:surname><ce:cross-ref refid="aff1420" id="crf0090"><ce:sup>142</ce:sup></ce:cross-ref></ce:author><ce:author id="au0020" author-id="S0370269323000643-0eab85892b6d74b18661e74a7987c599"><ce:given-name>D.</ce:given-name><ce:surname>Adamová</ce:surname><ce:cross-ref refid="aff0960" id="crf0100"><ce:sup>96</ce:sup></ce:cross-ref></ce:author><ce:author id="au0030" author-id="S0370269323000643-e83a30ae1d5f89088c60ca2ee154d714"><ce:given-name>A.</ce:given-name><ce:surname>Adler</ce:surname><ce:cross-ref refid="aff0740" id="crf0110"><ce:sup>74</ce:sup></ce:cross-ref></ce:author><ce:author id="au0040" author-id="S0370269323000643-cd100d18768d08ce9c59a475af0552d6"><ce:given-name>J.</ce:given-name><ce:surname>Adolfsson</ce:surname><ce:cross-ref refid="aff0810" id="crf0120"><ce:sup>81</ce:sup></ce:cross-ref></ce:author><ce:author id="au0050" author-id="S0370269323000643-ed2d58d89990c41bb43c091d01e5029a"><ce:given-name>G.</ce:given-name><ce:surname>Aglieri Rinella</ce:surname><ce:cross-ref refid="aff0340" id="crf0130"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au0060" author-id="S0370269323000643-0c7a7863b7384aa5fdf06a0f187949c8"><ce:given-name>M.</ce:given-name><ce:surname>Agnello</ce:surname><ce:cross-ref refid="aff0300" id="crf0140"><ce:sup>30</ce:sup></ce:cross-ref></ce:author><ce:author id="au0070" author-id="S0370269323000643-bbdbb014653d7bdacb111c613a0fcbe0"><ce:given-name>N.</ce:given-name><ce:surname>Agrawal</ce:surname><ce:cross-ref refid="aff0540" id="crf0150"><ce:sup>54</ce:sup></ce:cross-ref></ce:author><ce:author id="au0080" author-id="S0370269323000643-2059a9508121069139ea49dbf566539a"><ce:given-name>Z.</ce:given-name><ce:surname>Ahammed</ce:surname><ce:cross-ref refid="aff1420" id="crf0160"><ce:sup>142</ce:sup></ce:cross-ref></ce:author><ce:author id="au0090" author-id="S0370269323000643-397c2b4743f367cb4aceb69d448bb6c6"><ce:given-name>S.</ce:given-name><ce:surname>Ahmad</ce:surname><ce:cross-ref refid="aff0160" id="crf0170"><ce:sup>16</ce:sup></ce:cross-ref></ce:author><ce:author id="au0100" author-id="S0370269323000643-f7c3cbd2e9a0f545640d2202ad1ddbf3"><ce:given-name>S.U.</ce:given-name><ce:surname>Ahn</ce:surname><ce:cross-ref refid="aff0760" id="crf0180"><ce:sup>76</ce:sup></ce:cross-ref></ce:author><ce:author id="au0110" author-id="S0370269323000643-07322fd13596772e8f99876f081be003"><ce:given-name>I.</ce:given-name><ce:surname>Ahuja</ce:surname><ce:cross-ref refid="aff0380" id="crf0190"><ce:sup>38</ce:sup></ce:cross-ref></ce:author><ce:author id="au0120" author-id="S0370269323000643-460f53f1206813a21eae674e680d3430"><ce:given-name>Z.</ce:given-name><ce:surname>Akbar</ce:surname><ce:cross-ref refid="aff0510" id="crf0200"><ce:sup>51</ce:sup></ce:cross-ref></ce:author><ce:author id="au0130" author-id="S0370269323000643-e1ca35714b53c8e677e37d627a5bbf38"><ce:given-name>A.</ce:given-name><ce:surname>Akindinov</ce:surname><ce:cross-ref refid="aff0930" id="crf0210"><ce:sup>93</ce:sup></ce:cross-ref></ce:author><ce:author id="au0140" author-id="S0370269323000643-92548b9c2ec6b0a94be4b4532227aed9"><ce:given-name>M.</ce:given-name><ce:surname>Al-Turany</ce:surname><ce:cross-ref refid="aff1080" id="crf0220"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au0150" author-id="S0370269323000643-310d492c2c403b256bac5f0786fadcd5"><ce:given-name>S.N.</ce:given-name><ce:surname>Alam</ce:surname><ce:cross-ref refid="aff0160" id="crf0230"><ce:sup>16</ce:sup></ce:cross-ref></ce:author><ce:author id="au0160" author-id="S0370269323000643-9065823209e70d0656d37d4008b49b34"><ce:given-name>D.</ce:given-name><ce:surname>Aleksandrov</ce:surname><ce:cross-ref refid="aff0890" id="crf0240"><ce:sup>89</ce:sup></ce:cross-ref></ce:author><ce:author id="au0170" author-id="S0370269323000643-79cd00674b886e25d315860e3eeed8b4"><ce:given-name>B.</ce:given-name><ce:surname>Alessandro</ce:surname><ce:cross-ref refid="aff0590" id="crf0250"><ce:sup>59</ce:sup></ce:cross-ref></ce:author><ce:author id="au0180" author-id="S0370269323000643-1c1181472bc0aba99490cd3bca5e4a55"><ce:given-name>H.M.</ce:given-name><ce:surname>Alfanda</ce:surname><ce:cross-ref refid="aff0070" id="crf0260"><ce:sup>7</ce:sup></ce:cross-ref></ce:author><ce:author id="au0190" author-id="S0370269323000643-fe723c43bc1641702461996d042297c7"><ce:given-name>R.</ce:given-name><ce:surname>Alfaro Molina</ce:surname><ce:cross-ref refid="aff0710" id="crf0270"><ce:sup>71</ce:sup></ce:cross-ref></ce:author><ce:author id="au0200" author-id="S0370269323000643-9e891354cda61f6d260a73da741fb124"><ce:given-name>B.</ce:given-name><ce:surname>Ali</ce:surname><ce:cross-ref refid="aff0160" id="crf0280"><ce:sup>16</ce:sup></ce:cross-ref></ce:author><ce:author id="au0210" author-id="S0370269323000643-b09579e692bf4a5fcd8b3fc371c313f5"><ce:given-name>Y.</ce:given-name><ce:surname>Ali</ce:surname><ce:cross-ref refid="aff0140" 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id="crf3050"><ce:sup>72</ce:sup></ce:cross-ref></ce:author><ce:author id="au2760" author-id="S0370269323000643-1f224979dd316f22a826529fb384e21a"><ce:given-name>S.</ce:given-name><ce:surname>Fokin</ce:surname><ce:cross-ref refid="aff0890" id="crf3060"><ce:sup>89</ce:sup></ce:cross-ref></ce:author><ce:author id="au2770" author-id="S0370269323000643-012cbc6bc9303877c458bdf9a521e5ea"><ce:given-name>E.</ce:given-name><ce:surname>Fragiacomo</ce:surname><ce:cross-ref refid="aff0600" id="crf3070"><ce:sup>60</ce:sup></ce:cross-ref></ce:author><ce:author id="au2780" author-id="S0370269323000643-6b14343bb1477ee6f77413c5e1607139"><ce:given-name>E.</ce:given-name><ce:surname>Frajna</ce:surname><ce:cross-ref refid="aff1460" id="crf3080"><ce:sup>146</ce:sup></ce:cross-ref></ce:author><ce:author id="au2790" author-id="S0370269323000643-40e489eded73f09566103eb6e7285c7b"><ce:given-name>A.</ce:given-name><ce:surname>Francisco</ce:surname><ce:cross-ref refid="aff1360" id="crf3090"><ce:sup>136</ce:sup></ce:cross-ref></ce:author><ce:author id="au2800" author-id="S0370269323000643-cb179d5175a767cfcccfaf42e2d13fd9"><ce:given-name>U.</ce:given-name><ce:surname>Fuchs</ce:surname><ce:cross-ref refid="aff0340" id="crf3100"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au2810" author-id="S0370269323000643-75b81f7c708cdac6972396f07f1e615c"><ce:given-name>N.</ce:given-name><ce:surname>Funicello</ce:surname><ce:cross-ref refid="aff0290" id="crf3110"><ce:sup>29</ce:sup></ce:cross-ref></ce:author><ce:author id="au2820" author-id="S0370269323000643-e99988272e94b543b10e6e1e6762c915"><ce:given-name>C.</ce:given-name><ce:surname>Furget</ce:surname><ce:cross-ref refid="aff0790" id="crf3120"><ce:sup>79</ce:sup></ce:cross-ref></ce:author><ce:author id="au2830" author-id="S0370269323000643-466f51a3a42f868eb651094379830769"><ce:given-name>A.</ce:given-name><ce:surname>Furs</ce:surname><ce:cross-ref refid="aff0630" 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id="crf3170"><ce:sup>138</ce:sup></ce:cross-ref></ce:author><ce:author id="au2880" author-id="S0370269323000643-23c91ffad460f5afabddb04373c9f2f9"><ce:given-name>C.D.</ce:given-name><ce:surname>Galvan</ce:surname><ce:cross-ref refid="aff1200" id="crf3180"><ce:sup>120</ce:sup></ce:cross-ref></ce:author><ce:author id="au2890" author-id="S0370269323000643-8650c037fc955510cd76a4536ac83b03"><ce:given-name>P.</ce:given-name><ce:surname>Ganoti</ce:surname><ce:cross-ref refid="aff0850" id="crf3190"><ce:sup>85</ce:sup></ce:cross-ref></ce:author><ce:author id="au2900" author-id="S0370269323000643-d12fc16d28f1dce28adbdb23a5ddbcd4"><ce:given-name>C.</ce:given-name><ce:surname>Garabatos</ce:surname><ce:cross-ref refid="aff1080" id="crf3200"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au2910" author-id="S0370269323000643-df265a31acafb0a0d83f8ac8f13667e6"><ce:given-name>J.R.A.</ce:given-name><ce:surname>Garcia</ce:surname><ce:cross-ref refid="aff0450" id="crf3210"><ce:sup>45</ce:sup></ce:cross-ref></ce:author><ce:author id="au2920" author-id="S0370269323000643-07a4057e80a82b458aafc1e9dd90f0ba"><ce:given-name>E.</ce:given-name><ce:surname>Garcia-Solis</ce:surname><ce:cross-ref refid="aff0100" id="crf3220"><ce:sup>10</ce:sup></ce:cross-ref></ce:author><ce:author id="au2930" author-id="S0370269323000643-8f3671ab888761876ad72f8897499a10"><ce:given-name>K.</ce:given-name><ce:surname>Garg</ce:surname><ce:cross-ref refid="aff1150" id="crf3230"><ce:sup>115</ce:sup></ce:cross-ref></ce:author><ce:author id="au2940" author-id="S0370269323000643-987fe2eab608df9880768b4abb040eae"><ce:given-name>C.</ce:given-name><ce:surname>Gargiulo</ce:surname><ce:cross-ref refid="aff0340" id="crf3240"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au2950" author-id="S0370269323000643-0a421258e4c2e68267cbf36b6e5c0d5b"><ce:given-name>A.</ce:given-name><ce:surname>Garibli</ce:surname><ce:cross-ref refid="aff0880" 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author-id="S0370269323000643-ab877e6ad3aa564fc07ff9279bdb0114"><ce:given-name>R.</ce:given-name><ce:surname>Hannigan</ce:surname><ce:cross-ref refid="aff1190" id="crf3750"><ce:sup>119</ce:sup></ce:cross-ref></ce:author><ce:author id="au3430" author-id="S0370269323000643-6cb4376ff64678829e7af4b6b3584d70"><ce:given-name>M.R.</ce:given-name><ce:surname>Haque</ce:surname><ce:cross-ref refid="aff1430" id="crf3760"><ce:sup>143</ce:sup></ce:cross-ref></ce:author><ce:author id="au3440" author-id="S0370269323000643-1ad6c786673c0d88fb3ec98199ef2627"><ce:given-name>A.</ce:given-name><ce:surname>Harlenderova</ce:surname><ce:cross-ref refid="aff1080" id="crf3770"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au3450" author-id="S0370269323000643-093df752b6ef0875125996d2e249b17f"><ce:given-name>J.W.</ce:given-name><ce:surname>Harris</ce:surname><ce:cross-ref refid="aff1470" id="crf3780"><ce:sup>147</ce:sup></ce:cross-ref></ce:author><ce:author id="au3460" 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author-id="S0370269323000643-c494e289485a5b77e2698762fd7f4b6d"><ce:given-name>P.</ce:given-name><ce:surname>Hauer</ce:surname><ce:cross-ref refid="aff0430" id="crf3830"><ce:sup>43</ce:sup></ce:cross-ref></ce:author><ce:author id="au3510" author-id="S0370269323000643-d07756ab653a033a44665f2378719cc2"><ce:given-name>L.B.</ce:given-name><ce:surname>Havener</ce:surname><ce:cross-ref refid="aff1470" id="crf3840"><ce:sup>147</ce:sup></ce:cross-ref></ce:author><ce:author id="au3520" author-id="S0370269323000643-8e268fbd99e6fae4655760d64978b0ff"><ce:given-name>S.T.</ce:given-name><ce:surname>Heckel</ce:surname><ce:cross-ref refid="aff1060" id="crf3850"><ce:sup>106</ce:sup></ce:cross-ref></ce:author><ce:author id="au3530" author-id="S0370269323000643-7ad645538c2eb3346cfb0dd07e68d266"><ce:given-name>E.</ce:given-name><ce:surname>Hellbär</ce:surname><ce:cross-ref refid="aff1080" id="crf3860"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au3540" author-id="S0370269323000643-2ec6e519610c64506f6a6c4958b0ac54"><ce:given-name>H.</ce:given-name><ce:surname>Helstrup</ce:surname><ce:cross-ref refid="aff0360" id="crf3870"><ce:sup>36</ce:sup></ce:cross-ref></ce:author><ce:author id="au3550" author-id="S0370269323000643-51fb6f5f9415eda76fa12ad79ea6ef67"><ce:given-name>T.</ce:given-name><ce:surname>Herman</ce:surname><ce:cross-ref refid="aff0370" id="crf3880"><ce:sup>37</ce:sup></ce:cross-ref></ce:author><ce:author id="au3560" author-id="S0370269323000643-395e012e61feeafeb06dc1b1ee34d8d4"><ce:given-name>G.</ce:given-name><ce:surname>Herrera Corral</ce:surname><ce:cross-ref refid="aff0090" id="crf3890"><ce:sup>9</ce:sup></ce:cross-ref></ce:author><ce:author id="au3570" author-id="S0370269323000643-7a7a63e4af499b5ed75851524b48baa5"><ce:given-name>F.</ce:given-name><ce:surname>Herrmann</ce:surname><ce:cross-ref refid="aff1450" id="crf3900"><ce:sup>145</ce:sup></ce:cross-ref></ce:author><ce:author id="au3580" 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author-id="S0370269323000643-919c09ef7cea2014a589c5a05b491b03"><ce:given-name>O.</ce:given-name><ce:surname>Jevons</ce:surname><ce:cross-ref refid="aff1110" id="crf4390"><ce:sup>111</ce:sup></ce:cross-ref></ce:author><ce:author id="au4070" author-id="S0370269323000643-0058ed807ba55e1dc3018df2944a05a9"><ce:given-name>A.A.P.</ce:given-name><ce:surname>Jimenez</ce:surname><ce:cross-ref refid="aff0690" id="crf4400"><ce:sup>69</ce:sup></ce:cross-ref></ce:author><ce:author id="au4080" author-id="S0370269323000643-77affd2a949bb3c47fc27d2788c61f73"><ce:given-name>F.</ce:given-name><ce:surname>Jonas</ce:surname><ce:cross-ref refid="aff0970" id="crf4410"><ce:sup>97</ce:sup></ce:cross-ref><ce:cross-ref refid="aff1450" id="crf4420"><ce:sup>145</ce:sup></ce:cross-ref></ce:author><ce:author id="au4090" author-id="S0370269323000643-6d3535b16f327cd1f041269fdeeb525e"><ce:given-name>P.G.</ce:given-name><ce:surname>Jones</ce:surname><ce:cross-ref refid="aff1110" 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id="crf5000"><ce:sup>125</ce:sup></ce:cross-ref></ce:author><ce:author id="au4650" author-id="S0370269323000643-9af73f258a62987a4bb0e3d5b123ac21"><ce:given-name>C.</ce:given-name><ce:surname>Kobdaj</ce:surname><ce:cross-ref refid="aff1160" id="crf5010"><ce:sup>116</ce:sup></ce:cross-ref></ce:author><ce:author id="au4660" author-id="S0370269323000643-9002912594757f4703158b0d8b2f3fac"><ce:given-name>T.</ce:given-name><ce:surname>Kollegger</ce:surname><ce:cross-ref refid="aff1080" id="crf5020"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au4670" author-id="S0370269323000643-928dc5fa825dcae3f1c6433fdc958dcb"><ce:given-name>A.</ce:given-name><ce:surname>Kondratyev</ce:surname><ce:cross-ref refid="aff0750" id="crf5030"><ce:sup>75</ce:sup></ce:cross-ref></ce:author><ce:author id="au4680" author-id="S0370269323000643-d7620b6a11f6a0eda9746e31690949db"><ce:given-name>N.</ce:given-name><ce:surname>Kondratyeva</ce:surname><ce:cross-ref refid="aff0940" 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id="crf5080"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au4730" author-id="S0370269323000643-a6ebc19161b718e614409bd14cd742e2"><ce:given-name>G.</ce:given-name><ce:surname>Kornakov</ce:surname><ce:cross-ref refid="aff1430" id="crf5090"><ce:sup>143</ce:sup></ce:cross-ref></ce:author><ce:author id="au4740" author-id="S0370269323000643-bc606283f8e9b64ca017e5537c8a999c"><ce:given-name>S.D.</ce:given-name><ce:surname>Koryciak</ce:surname><ce:cross-ref refid="aff0020" id="crf5100"><ce:sup>2</ce:sup></ce:cross-ref></ce:author><ce:author id="au4750" author-id="S0370269323000643-584f8d6c45b2c92afa67eb902a0518d0"><ce:given-name>A.</ce:given-name><ce:surname>Kotliarov</ce:surname><ce:cross-ref refid="aff0960" id="crf5110"><ce:sup>96</ce:sup></ce:cross-ref></ce:author><ce:author id="au4760" author-id="S0370269323000643-ca3a82a6876159a778c0defe2c9f7afc"><ce:given-name>O.</ce:given-name><ce:surname>Kovalenko</ce:surname><ce:cross-ref refid="aff0860" 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id="crf5160"><ce:sup>38</ce:sup></ce:cross-ref></ce:author><ce:author id="au4810" author-id="S0370269323000643-5ae20d0891d7ae02dfed674701ce3a0f"><ce:given-name>L.</ce:given-name><ce:surname>Kreis</ce:surname><ce:cross-ref refid="aff1080" id="crf5170"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au4820" author-id="S0370269323000643-cbcb2ee3fb84eb500c0f9ad43570a2f7"><ce:given-name>M.</ce:given-name><ce:surname>Krivda</ce:surname><ce:cross-ref refid="aff1110" id="crf5180"><ce:sup>111</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0640" id="crf5190"><ce:sup>64</ce:sup></ce:cross-ref></ce:author><ce:author id="au4830" author-id="S0370269323000643-5e1a6cb706f854d8977bdf9f4302ff9d"><ce:given-name>F.</ce:given-name><ce:surname>Krizek</ce:surname><ce:cross-ref refid="aff0960" id="crf5200"><ce:sup>96</ce:sup></ce:cross-ref></ce:author><ce:author id="au4840" author-id="S0370269323000643-c7ba6f3c10f1fdef5885803cb741ca27"><ce:given-name>K.</ce:given-name><ce:surname>Krizkova Gajdosova</ce:surname><ce:cross-ref refid="aff0370" id="crf5210"><ce:sup>37</ce:sup></ce:cross-ref></ce:author><ce:author id="au4850" author-id="S0370269323000643-7c694b6b00252b450fe7d9c484f6e9d5"><ce:given-name>M.</ce:given-name><ce:surname>Kroesen</ce:surname><ce:cross-ref refid="aff1050" id="crf5220"><ce:sup>105</ce:sup></ce:cross-ref></ce:author><ce:author id="au4860" author-id="S0370269323000643-bf9e7ebe8c005dbf752b23333ffad9b1"><ce:given-name>M.</ce:given-name><ce:surname>Krüger</ce:surname><ce:cross-ref refid="aff0680" id="crf5230"><ce:sup>68</ce:sup></ce:cross-ref></ce:author><ce:author id="au4870" author-id="S0370269323000643-83394498cf07bf82c0da59782bb4b8fb"><ce:given-name>D.M.</ce:given-name><ce:surname>Krupova</ce:surname><ce:cross-ref refid="aff0370" id="crf5240"><ce:sup>37</ce:sup></ce:cross-ref></ce:author><ce:author id="au4880" author-id="S0370269323000643-152e2b03ce0f7c4b9e66b15f16da951b"><ce:given-name>E.</ce:given-name><ce:surname>Kryshen</ce:surname><ce:cross-ref refid="aff0990" id="crf5250"><ce:sup>99</ce:sup></ce:cross-ref></ce:author><ce:author id="au4890" author-id="S0370269323000643-003931bcfcc6c66261250e70a6411441"><ce:given-name>M.</ce:given-name><ce:surname>Krzewicki</ce:surname><ce:cross-ref refid="aff0390" id="crf5260"><ce:sup>39</ce:sup></ce:cross-ref></ce:author><ce:author id="au4900" author-id="S0370269323000643-2b9484aa61648b32a23a9d1c4e4afe4c"><ce:given-name>V.</ce:given-name><ce:surname>Kučera</ce:surname><ce:cross-ref refid="aff0340" id="crf5270"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au4910" author-id="S0370269323000643-6fbc34de0b19002d51fb1202918b911f"><ce:given-name>C.</ce:given-name><ce:surname>Kuhn</ce:surname><ce:cross-ref refid="aff1380" id="crf5280"><ce:sup>138</ce:sup></ce:cross-ref></ce:author><ce:author id="au4920" author-id="S0370269323000643-c0d55a9ea28fff5062fd266d451ebca3"><ce:given-name>P.G.</ce:given-name><ce:surname>Kuijer</ce:surname><ce:cross-ref refid="aff0910" 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author-id="S0370269323000643-2dc3ae02f3986fe0007b48539572abad"><ce:given-name>J.E.</ce:given-name><ce:surname>Mdhluli</ce:surname><ce:cross-ref refid="aff1320" id="crf6340"><ce:sup>132</ce:sup></ce:cross-ref></ce:author><ce:author id="au5900" author-id="S0370269323000643-9c4cf9aa4a93681a22bbeac30e3992ca"><ce:given-name>A.F.</ce:given-name><ce:surname>Mechler</ce:surname><ce:cross-ref refid="aff0680" id="crf6350"><ce:sup>68</ce:sup></ce:cross-ref></ce:author><ce:author id="au5910" author-id="S0370269323000643-fe3bd84b2ae42e55489dadcb6bf8e526"><ce:given-name>Y.</ce:given-name><ce:surname>Melikyan</ce:surname><ce:cross-ref refid="aff0630" id="crf6360"><ce:sup>63</ce:sup></ce:cross-ref></ce:author><ce:author id="au5920" author-id="S0370269323000643-679be24f5fe9089afe4b6443cc8d824d"><ce:given-name>A.</ce:given-name><ce:surname>Menchaca-Rocha</ce:surname><ce:cross-ref refid="aff0710" id="crf6370"><ce:sup>71</ce:sup></ce:cross-ref></ce:author><ce:author id="au5930" author-id="S0370269323000643-ade9e90d38be4880cfa450e0935dc1c4"><ce:given-name>E.</ce:given-name><ce:surname>Meninno</ce:surname><ce:cross-ref refid="aff1140" id="crf6380"><ce:sup>114</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0290" id="crf6390"><ce:sup>29</ce:sup></ce:cross-ref></ce:author><ce:author id="au5940" author-id="S0370269323000643-8d4b98c0c67c27715a41cd7b188cfc99"><ce:given-name>A.S.</ce:given-name><ce:surname>Menon</ce:surname><ce:cross-ref refid="aff1250" id="crf6400"><ce:sup>125</ce:sup></ce:cross-ref></ce:author><ce:author id="au5950" author-id="S0370269323000643-65fe57f72284fcc5a9d4d04bd694e61e"><ce:given-name>M.</ce:given-name><ce:surname>Meres</ce:surname><ce:cross-ref refid="aff0130" id="crf6410"><ce:sup>13</ce:sup></ce:cross-ref></ce:author><ce:author id="au5960" author-id="S0370269323000643-5b43df089ea1f2f7448928ea22361631"><ce:given-name>S.</ce:given-name><ce:surname>Mhlanga</ce:surname><ce:cross-ref refid="aff1240" 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author-id="S0370269323000643-462924ae16f46927423eb84a4d54b8a3"><ce:given-name>D.A.</ce:given-name><ce:surname>Moreira De Godoy</ce:surname><ce:cross-ref refid="aff1450" id="crf6600"><ce:sup>145</ce:sup></ce:cross-ref></ce:author><ce:author id="au6120" author-id="S0370269323000643-77d54cd49e232a3d81ad62cb0cd3055b"><ce:given-name>I.</ce:given-name><ce:surname>Morozov</ce:surname><ce:cross-ref refid="aff0630" id="crf6610"><ce:sup>63</ce:sup></ce:cross-ref></ce:author><ce:author id="au6130" author-id="S0370269323000643-3b644752f2833a280590e7d2cb815635"><ce:given-name>A.</ce:given-name><ce:surname>Morsch</ce:surname><ce:cross-ref refid="aff0340" id="crf6620"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au6140" author-id="S0370269323000643-a6e52521563dadda4fcf66535744aa54"><ce:given-name>T.</ce:given-name><ce:surname>Mrnjavac</ce:surname><ce:cross-ref refid="aff0340" id="crf6630"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au6150" author-id="S0370269323000643-64a6ab0403f8cd64a5dff0e5eb920af6"><ce:given-name>V.</ce:given-name><ce:surname>Muccifora</ce:surname><ce:cross-ref refid="aff0520" id="crf6640"><ce:sup>52</ce:sup></ce:cross-ref></ce:author><ce:author id="au6160" author-id="S0370269323000643-a0c40dcd3bce329a03e938ee3cced0d4"><ce:given-name>E.</ce:given-name><ce:surname>Mudnic</ce:surname><ce:cross-ref refid="aff0350" id="crf6650"><ce:sup>35</ce:sup></ce:cross-ref></ce:author><ce:author id="au6170" author-id="S0370269323000643-b2ff5bf3b45a3f5a117d5b53ed508ff9"><ce:given-name>S.</ce:given-name><ce:surname>Muhuri</ce:surname><ce:cross-ref refid="aff1420" id="crf6660"><ce:sup>142</ce:sup></ce:cross-ref></ce:author><ce:author id="au6180" author-id="S0370269323000643-f99358c1c870cc0fafdf5a9795565aff"><ce:given-name>J.D.</ce:given-name><ce:surname>Mulligan</ce:surname><ce:cross-ref refid="aff0800" id="crf6670"><ce:sup>80</ce:sup></ce:cross-ref></ce:author><ce:author id="au6190" author-id="S0370269323000643-8f73c238e6e583a6df239dd210d13e32"><ce:given-name>A.</ce:given-name><ce:surname>Mulliri</ce:surname><ce:cross-ref refid="aff0220" id="crf6680"><ce:sup>22</ce:sup></ce:cross-ref></ce:author><ce:author id="au6200" author-id="S0370269323000643-9080edf78fcbf8b6d3a149c5ce8954f3"><ce:given-name>M.G.</ce:given-name><ce:surname>Munhoz</ce:surname><ce:cross-ref refid="aff1210" id="crf6690"><ce:sup>121</ce:sup></ce:cross-ref></ce:author><ce:author id="au6210" author-id="S0370269323000643-e51a6131c1e53a702848dcae56610cce"><ce:given-name>R.H.</ce:given-name><ce:surname>Munzer</ce:surname><ce:cross-ref refid="aff0680" id="crf6700"><ce:sup>68</ce:sup></ce:cross-ref></ce:author><ce:author id="au6220" author-id="S0370269323000643-344903685fb7770611263048525de44e"><ce:given-name>H.</ce:given-name><ce:surname>Murakami</ce:surname><ce:cross-ref refid="aff1330" id="crf6710"><ce:sup>133</ce:sup></ce:cross-ref></ce:author><ce:author id="au6230" 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author-id="S0370269323000643-ee3f97c8e319fe4cdf98e0660785c85a"><ce:given-name>M.H.</ce:given-name><ce:surname>Oliver</ce:surname><ce:cross-ref refid="aff1470" id="crf7080"><ce:sup>147</ce:sup></ce:cross-ref></ce:author><ce:author id="au6600" author-id="S0370269323000643-e6f50f7f16b66af365a7af642930fd18"><ce:given-name>A.</ce:given-name><ce:surname>Onnerstad</ce:surname><ce:cross-ref refid="aff1260" id="crf7090"><ce:sup>126</ce:sup></ce:cross-ref></ce:author><ce:author id="au6610" author-id="S0370269323000643-dda435886a590cbfc7ea1d77af3e51f5"><ce:given-name>C.</ce:given-name><ce:surname>Oppedisano</ce:surname><ce:cross-ref refid="aff0590" id="crf7100"><ce:sup>59</ce:sup></ce:cross-ref></ce:author><ce:author id="au6620" author-id="S0370269323000643-c6a2d1996925547f52c523a908287916"><ce:given-name>A.</ce:given-name><ce:surname>Ortiz Velasquez</ce:surname><ce:cross-ref refid="aff0690" id="crf7110"><ce:sup>69</ce:sup></ce:cross-ref></ce:author><ce:author id="au6630" 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author-id="S0370269323000643-cb56ae8a44e3cc494d7b67afa9bccd92"><ce:given-name>W.</ce:given-name><ce:surname>Rzesa</ce:surname><ce:cross-ref refid="aff1430" id="crf8320"><ce:sup>143</ce:sup></ce:cross-ref></ce:author><ce:author id="au7760" author-id="S0370269323000643-0028258bab866460c144c7c968794773"><ce:given-name>O.A.M.</ce:given-name><ce:surname>Saarimaki</ce:surname><ce:cross-ref refid="aff0440" id="crf8330"><ce:sup>44</ce:sup></ce:cross-ref></ce:author><ce:author id="au7770" author-id="S0370269323000643-30d43d51897675cf9cd495896fc130f9"><ce:given-name>R.</ce:given-name><ce:surname>Sadek</ce:surname><ce:cross-ref refid="aff1150" id="crf8340"><ce:sup>115</ce:sup></ce:cross-ref></ce:author><ce:author id="au7780" author-id="S0370269323000643-95444e1d677ab64f8776a30d178e54e2"><ce:given-name>S.</ce:given-name><ce:surname>Sadovsky</ce:surname><ce:cross-ref refid="aff0920" id="crf8350"><ce:sup>92</ce:sup></ce:cross-ref></ce:author><ce:author id="au7790" author-id="S0370269323000643-0a168bc99240d66a8925ff06e271bdf5"><ce:given-name>J.</ce:given-name><ce:surname>Saetre</ce:surname><ce:cross-ref refid="aff0210" id="crf8360"><ce:sup>21</ce:sup></ce:cross-ref></ce:author><ce:author id="au7800" author-id="S0370269323000643-21a9d7d32dbc31b6d9fe58ee57047f0f"><ce:given-name>K.</ce:given-name><ce:surname>Šafařík</ce:surname><ce:cross-ref refid="aff0370" id="crf8370"><ce:sup>37</ce:sup></ce:cross-ref></ce:author><ce:author id="au7810" author-id="S0370269323000643-e124e41bd8c9a15152d347654ca43d3a"><ce:given-name>S.K.</ce:given-name><ce:surname>Saha</ce:surname><ce:cross-ref refid="aff1420" id="crf8380"><ce:sup>142</ce:sup></ce:cross-ref></ce:author><ce:author id="au7820" author-id="S0370269323000643-97780e442ae33eb1a0ab97a98d717272"><ce:given-name>S.</ce:given-name><ce:surname>Saha</ce:surname><ce:cross-ref refid="aff0870" id="crf8390"><ce:sup>87</ce:sup></ce:cross-ref></ce:author><ce:author id="au7830" author-id="S0370269323000643-2d0dbe8d0df0bf01a83ddf19e043addc"><ce:given-name>B.</ce:given-name><ce:surname>Sahoo</ce:surname><ce:cross-ref refid="aff0490" id="crf8400"><ce:sup>49</ce:sup></ce:cross-ref></ce:author><ce:author id="au7840" author-id="S0370269323000643-c91ceefdaf62924dbe41969b9214a864"><ce:given-name>P.</ce:given-name><ce:surname>Sahoo</ce:surname><ce:cross-ref refid="aff0490" id="crf8410"><ce:sup>49</ce:sup></ce:cross-ref></ce:author><ce:author id="au7850" author-id="S0370269323000643-bf022b9756aebb065155d194bfc5e116"><ce:given-name>R.</ce:given-name><ce:surname>Sahoo</ce:surname><ce:cross-ref refid="aff0500" id="crf8420"><ce:sup>50</ce:sup></ce:cross-ref></ce:author><ce:author id="au7860" author-id="S0370269323000643-d12dd0293e56fb8f196c824e945688ef"><ce:given-name>S.</ce:given-name><ce:surname>Sahoo</ce:surname><ce:cross-ref refid="aff0650" id="crf8430"><ce:sup>65</ce:sup></ce:cross-ref></ce:author><ce:author id="au7870" 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author-id="S0370269323000643-8a3806e0320b7d360b40762c9a667520"><ce:given-name>M.P.</ce:given-name><ce:surname>Salvan</ce:surname><ce:cross-ref refid="aff1080" id="crf8480"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au7920" author-id="S0370269323000643-1bc9f8186efa42ba63b483c287a4106e"><ce:given-name>S.</ce:given-name><ce:surname>Sambyal</ce:surname><ce:cross-ref refid="aff1020" id="crf8490"><ce:sup>102</ce:sup></ce:cross-ref></ce:author><ce:author id="au7930" author-id="S0370269323000643-cd56f5a9e647bb69890a3a180b5ed4f6"><ce:given-name>T.B.</ce:given-name><ce:surname>Saramela</ce:surname><ce:cross-ref refid="aff1210" id="crf8500"><ce:sup>121</ce:sup></ce:cross-ref></ce:author><ce:author id="au7940" author-id="S0370269323000643-4e7b512d816400994223e9f92ecca2ec"><ce:given-name>D.</ce:given-name><ce:surname>Sarkar</ce:surname><ce:cross-ref refid="aff1440" id="crf8510"><ce:sup>144</ce:sup></ce:cross-ref></ce:author><ce:author id="au7950" 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id="crf9630"><ce:sup>48</ce:sup></ce:cross-ref></ce:author><ce:author id="au9010" author-id="S0370269323000643-59aaa29065f6796159aa543e639298c7"><ce:given-name>A.</ce:given-name><ce:surname>Tauro</ce:surname><ce:cross-ref refid="aff0340" id="crf9640"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au9020" author-id="S0370269323000643-f13ebce4180c1e47d7740fedb0ce7a8b"><ce:given-name>G.</ce:given-name><ce:surname>Tejeda Muñoz</ce:surname><ce:cross-ref refid="aff0450" id="crf9650"><ce:sup>45</ce:sup></ce:cross-ref></ce:author><ce:author id="au9030" author-id="S0370269323000643-e32b8e2ac0c864eac40a37b39e286a55"><ce:given-name>A.</ce:given-name><ce:surname>Telesca</ce:surname><ce:cross-ref refid="aff0340" id="crf9660"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au9040" author-id="S0370269323000643-a8fc702cfc3f2781d3f44ce57fef99fa"><ce:given-name>L.</ce:given-name><ce:surname>Terlizzi</ce:surname><ce:cross-ref refid="aff0240" id="crf9670"><ce:sup>24</ce:sup></ce:cross-ref></ce:author><ce:author id="au9050" author-id="S0370269323000643-1463848cd4d3f997204d95350f6ec09e"><ce:given-name>C.</ce:given-name><ce:surname>Terrevoli</ce:surname><ce:cross-ref refid="aff1250" id="crf9680"><ce:sup>125</ce:sup></ce:cross-ref></ce:author><ce:author id="au9060" author-id="S0370269323000643-3117906cff50143c8578edcd5d472f8f"><ce:given-name>G.</ce:given-name><ce:surname>Tersimonov</ce:surname><ce:cross-ref refid="aff0030" id="crf9690"><ce:sup>3</ce:sup></ce:cross-ref></ce:author><ce:author id="au9070" author-id="S0370269323000643-364b9b5c5740ad64b0bfd748bcc98319"><ce:given-name>S.</ce:given-name><ce:surname>Thakur</ce:surname><ce:cross-ref refid="aff1420" id="crf9700"><ce:sup>142</ce:sup></ce:cross-ref></ce:author><ce:author id="au9080" author-id="S0370269323000643-a2ec5b979d461b062cd03d49d6e8d67b"><ce:given-name>D.</ce:given-name><ce:surname>Thomas</ce:surname><ce:cross-ref refid="aff1190" id="crf9710"><ce:sup>119</ce:sup></ce:cross-ref></ce:author><ce:author id="au9090" author-id="S0370269323000643-1aedf986c49e01602efb19c767f2f80e"><ce:given-name>R.</ce:given-name><ce:surname>Tieulent</ce:surname><ce:cross-ref refid="aff1370" id="crf9720"><ce:sup>137</ce:sup></ce:cross-ref></ce:author><ce:author id="au9100" author-id="S0370269323000643-1622f94cfabd47942fa3c2ea0d3e6a1e"><ce:given-name>A.</ce:given-name><ce:surname>Tikhonov</ce:surname><ce:cross-ref refid="aff0630" id="crf9730"><ce:sup>63</ce:sup></ce:cross-ref></ce:author><ce:author id="au9110" author-id="S0370269323000643-7e98d622fbf1de9d69786eccee47944e"><ce:given-name>A.R.</ce:given-name><ce:surname>Timmins</ce:surname><ce:cross-ref refid="aff1250" id="crf9740"><ce:sup>125</ce:sup></ce:cross-ref></ce:author><ce:author id="au9120" author-id="S0370269323000643-a1ab2ffb64eaab4cb2d232ddd2bb97d6"><ce:given-name>M.</ce:given-name><ce:surname>Tkacik</ce:surname><ce:cross-ref refid="aff1170" 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author-id="S0370269323000643-f0952a23f4b03e256f2dddd4e7e7755e"><ce:given-name>A.S.</ce:given-name><ce:surname>Triolo</ce:surname><ce:cross-ref refid="aff0320" id="crf9840"><ce:sup>32</ce:sup></ce:cross-ref></ce:author><ce:author id="au9210" author-id="S0370269323000643-43e5d915054f543c1b38d719f8c23294"><ce:given-name>S.</ce:given-name><ce:surname>Tripathy</ce:surname><ce:cross-ref refid="aff0540" id="crf9850"><ce:sup>54</ce:sup></ce:cross-ref></ce:author><ce:author id="au9220" author-id="S0370269323000643-cf5679180a033531cecbf308c26a3b2f"><ce:given-name>T.</ce:given-name><ce:surname>Tripathy</ce:surname><ce:cross-ref refid="aff0490" id="crf9860"><ce:sup>49</ce:sup></ce:cross-ref></ce:author><ce:author id="au9230" author-id="S0370269323000643-25f25f0207443c8141fbcc02fde4f484"><ce:given-name>S.</ce:given-name><ce:surname>Trogolo</ce:surname><ce:cross-ref refid="aff0340" id="crf9870"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au9240" 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author-id="S0370269323000643-5fa3b0c5cd63074069320744aebd6d13"><ce:given-name>K.</ce:given-name><ce:surname>Voloshin</ce:surname><ce:cross-ref refid="aff0930" id="crf10290"><ce:sup>93</ce:sup></ce:cross-ref></ce:author><ce:author id="au9640" author-id="S0370269323000643-57ec712e9628632a2ec4ba85186ee32a"><ce:given-name>S.A.</ce:given-name><ce:surname>Voloshin</ce:surname><ce:cross-ref refid="aff1440" id="crf10300"><ce:sup>144</ce:sup></ce:cross-ref></ce:author><ce:author id="au9650" author-id="S0370269323000643-f6777f409ca72674ceb72ffa4f303b59"><ce:given-name>G.</ce:given-name><ce:surname>Volpe</ce:surname><ce:cross-ref refid="aff0330" id="crf10310"><ce:sup>33</ce:sup></ce:cross-ref></ce:author><ce:author id="au9660" author-id="S0370269323000643-3996b3e93129a2159ed0eb3b176f20e6"><ce:given-name>B.</ce:given-name><ce:surname>von Haller</ce:surname><ce:cross-ref refid="aff0340" id="crf10320"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au9670" author-id="S0370269323000643-58249f4749b1084a40d3644672e443f3"><ce:given-name>I.</ce:given-name><ce:surname>Vorobyev</ce:surname><ce:cross-ref refid="aff1060" id="crf10330"><ce:sup>106</ce:sup></ce:cross-ref></ce:author><ce:author id="au9680" author-id="S0370269323000643-d2b4078835ab540da8b2d9fea3ef0e52"><ce:given-name>N.</ce:given-name><ce:surname>Vozniuk</ce:surname><ce:cross-ref refid="aff0630" id="crf10340"><ce:sup>63</ce:sup></ce:cross-ref></ce:author><ce:author id="au9690" author-id="S0370269323000643-9feadfb4556ab34e1316f18d086e7bed"><ce:given-name>J.</ce:given-name><ce:surname>Vrláková</ce:surname><ce:cross-ref refid="aff0380" id="crf10350"><ce:sup>38</ce:sup></ce:cross-ref></ce:author><ce:author id="au9700" author-id="S0370269323000643-ab1490e4e82947dd774ac4cb21acfea5"><ce:given-name>B.</ce:given-name><ce:surname>Wagner</ce:surname><ce:cross-ref refid="aff0210" id="crf10360"><ce:sup>21</ce:sup></ce:cross-ref></ce:author><ce:author id="au9710" author-id="S0370269323000643-6ed7035a79dd824bd6a0b2572108463b"><ce:given-name>C.</ce:given-name><ce:surname>Wang</ce:surname><ce:cross-ref refid="aff0400" id="crf10370"><ce:sup>40</ce:sup></ce:cross-ref></ce:author><ce:author id="au9720" author-id="S0370269323000643-acba1a294e63a54789e94f0c48170a7c"><ce:given-name>D.</ce:given-name><ce:surname>Wang</ce:surname><ce:cross-ref refid="aff0400" id="crf10380"><ce:sup>40</ce:sup></ce:cross-ref></ce:author><ce:author id="au9730" author-id="S0370269323000643-86448d3a1cf868fca209fc2d36f58f5e"><ce:given-name>M.</ce:given-name><ce:surname>Weber</ce:surname><ce:cross-ref refid="aff1140" id="crf10390"><ce:sup>114</ce:sup></ce:cross-ref></ce:author><ce:author id="au9740" author-id="S0370269323000643-b19740465a27da9581dafc73eb6a141a"><ce:given-name>R.J.G.V.</ce:given-name><ce:surname>Weelden</ce:surname><ce:cross-ref refid="aff0910" id="crf10400"><ce:sup>91</ce:sup></ce:cross-ref></ce:author><ce:author id="au9750" author-id="S0370269323000643-8ed929c2763388d291ed05f7803d55a7"><ce:given-name>A.</ce:given-name><ce:surname>Wegrzynek</ce:surname><ce:cross-ref refid="aff0340" id="crf10410"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au9760" author-id="S0370269323000643-fc054185e3e03ba2ba73fedce1880cf3"><ce:given-name>S.C.</ce:given-name><ce:surname>Wenzel</ce:surname><ce:cross-ref refid="aff0340" id="crf10420"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au9770" author-id="S0370269323000643-2315a255b201c88ad0087de64fcac834"><ce:given-name>J.P.</ce:given-name><ce:surname>Wessels</ce:surname><ce:cross-ref refid="aff1450" id="crf10430"><ce:sup>145</ce:sup></ce:cross-ref></ce:author><ce:author id="au9780" author-id="S0370269323000643-d0369f0e3d93a955cb5577312878292a"><ce:given-name>S.L.</ce:given-name><ce:surname>Weyhmiller</ce:surname><ce:cross-ref refid="aff1470" id="crf10440"><ce:sup>147</ce:sup></ce:cross-ref></ce:author><ce:author id="au9790" author-id="S0370269323000643-d35206c8771dcf8151d8531c8b5a4ad9"><ce:given-name>J.</ce:given-name><ce:surname>Wiechula</ce:surname><ce:cross-ref refid="aff0680" id="crf10450"><ce:sup>68</ce:sup></ce:cross-ref></ce:author><ce:author id="au9800" author-id="S0370269323000643-924aa82bb7057758dfd6b448c22248a6"><ce:given-name>J.</ce:given-name><ce:surname>Wikne</ce:surname><ce:cross-ref refid="aff0200" id="crf10460"><ce:sup>20</ce:sup></ce:cross-ref></ce:author><ce:author id="au9810" author-id="S0370269323000643-c6cf2d96fecc56427c920ff58c5f8f6c"><ce:given-name>G.</ce:given-name><ce:surname>Wilk</ce:surname><ce:cross-ref refid="aff0860" id="crf10470"><ce:sup>86</ce:sup></ce:cross-ref></ce:author><ce:author id="au9820" author-id="S0370269323000643-defbde1b1ee45aba41243923803c24ee"><ce:given-name>J.</ce:given-name><ce:surname>Wilkinson</ce:surname><ce:cross-ref refid="aff1080" id="crf10480"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au9830" 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author-id="S0370269323000643-5cd6a9eca5037ab2657c1654558d4a0d"><ce:given-name>J.R.</ce:given-name><ce:surname>Wright</ce:surname><ce:cross-ref refid="aff1190" id="crf10530"><ce:sup>119</ce:sup></ce:cross-ref></ce:author><ce:author id="au9880" author-id="S0370269323000643-e020a09e1f80d41df58661e69fc302a8"><ce:given-name>W.</ce:given-name><ce:surname>Wu</ce:surname><ce:cross-ref refid="aff0400" id="crf10540"><ce:sup>40</ce:sup></ce:cross-ref></ce:author><ce:author id="au9890" author-id="S0370269323000643-4d41bda7d295237516fe3d3063752576"><ce:given-name>Y.</ce:given-name><ce:surname>Wu</ce:surname><ce:cross-ref refid="aff1290" id="crf10550"><ce:sup>129</ce:sup></ce:cross-ref></ce:author><ce:author id="au9900" author-id="S0370269323000643-11e0bfbbade486f5296c5883173dddc5"><ce:given-name>R.</ce:given-name><ce:surname>Xu</ce:surname><ce:cross-ref refid="aff0070" id="crf10560"><ce:sup>7</ce:sup></ce:cross-ref></ce:author><ce:author id="au9910" 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id="crf10810"><ce:sup>129</ce:sup></ce:cross-ref></ce:author><ce:author id="au10150" author-id="S0370269323000643-9feeecb04ac9bd35d770466fb86792be"><ce:given-name>V.</ce:given-name><ce:surname>Zherebchevskii</ce:surname><ce:cross-ref refid="aff1130" id="crf10820"><ce:sup>113</ce:sup></ce:cross-ref></ce:author><ce:author id="au10160" author-id="S0370269323000643-617c42c28cb1b1d3dab680558c293767"><ce:given-name>Y.</ce:given-name><ce:surname>Zhi</ce:surname><ce:cross-ref refid="aff0110" id="crf10830"><ce:sup>11</ce:sup></ce:cross-ref></ce:author><ce:author id="au10170" author-id="S0370269323000643-3038f7cd79250ffa05fa0aff7644d2b3"><ce:given-name>N.</ce:given-name><ce:surname>Zhigareva</ce:surname><ce:cross-ref refid="aff0930" id="crf10840"><ce:sup>93</ce:sup></ce:cross-ref></ce:author><ce:author id="au10180" author-id="S0370269323000643-e9bc0465b352cd706eac4fc8b795a4c2"><ce:given-name>D.</ce:given-name><ce:surname>Zhou</ce:surname><ce:cross-ref refid="aff0070" id="crf10850"><ce:sup>7</ce:sup></ce:cross-ref></ce:author><ce:author id="au10190" author-id="S0370269323000643-4a0cdba9c352af9310759dd83e906db9"><ce:given-name>Y.</ce:given-name><ce:surname>Zhou</ce:surname><ce:cross-ref refid="aff0900" id="crf10860"><ce:sup>90</ce:sup></ce:cross-ref></ce:author><ce:author id="au10200" author-id="S0370269323000643-b564bd6149c66f06bbea2780670b8f50"><ce:given-name>J.</ce:given-name><ce:surname>Zhu</ce:surname><ce:cross-ref refid="aff1080" id="crf10870"><ce:sup>108</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0070" id="crf10880"><ce:sup>7</ce:sup></ce:cross-ref></ce:author><ce:author id="au10210" author-id="S0370269323000643-1fb0ab925cfcdaacb160193d82c1a978"><ce:given-name>Y.</ce:given-name><ce:surname>Zhu</ce:surname><ce:cross-ref refid="aff0070" id="crf10890"><ce:sup>7</ce:sup></ce:cross-ref></ce:author><ce:author id="au10220" author-id="S0370269323000643-2ad055472e2fac995111c51c08396d0c"><ce:given-name>G.</ce:given-name><ce:surname>Zinovjev</ce:surname><ce:cross-ref refid="aff0030" id="crf10900"><ce:sup>3</ce:sup></ce:cross-ref><ce:cross-ref refid="fn0010" id="crf10910"><ce:sup>I</ce:sup></ce:cross-ref></ce:author><ce:author id="au10230" author-id="S0370269323000643-92b5ecf5612e2e849fc1bba72c6cff99"><ce:given-name>N.</ce:given-name><ce:surname>Zurlo</ce:surname><ce:cross-ref refid="aff1410" id="crf10920"><ce:sup>141</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0580" id="crf10930"><ce:sup>58</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="aff0010" affiliation-id="S0370269323000643-79d30baa35325e84d46378ba6ce12c18"><ce:label>1</ce:label><ce:textfn>A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia</ce:textfn><sa:affiliation><sa:organization>A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation</sa:organization><sa:city>Yerevan</sa:city><sa:country>Armenia</sa:country></sa:affiliation><ce:source-text id="srct0005">A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia</ce:source-text></ce:affiliation><ce:affiliation id="aff0020" affiliation-id="S0370269323000643-65754d218cf7f84bf1e02306b80caca0"><ce:label>2</ce:label><ce:textfn>AGH University of Science and Technology, Cracow, Poland</ce:textfn><sa:affiliation><sa:organization>AGH University of Science and Technology</sa:organization><sa:city>Cracow</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0010">AGH University of Science and Technology, Cracow, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff0030" affiliation-id="S0370269323000643-e916a2f48a17bc32220b61ae0e9b8e05"><ce:label>3</ce:label><ce:textfn>Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine</ce:textfn><sa:affiliation><sa:organization>Bogolyubov Institute for Theoretical Physics</sa:organization><sa:organization>National Academy of Sciences of Ukraine</sa:organization><sa:city>Kiev</sa:city><sa:country>Ukraine</sa:country></sa:affiliation><ce:source-text id="srct0015">Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine</ce:source-text></ce:affiliation><ce:affiliation id="aff0040" affiliation-id="S0370269323000643-9869d95133b2275e34836b8ad2f235c3"><ce:label>4</ce:label><ce:textfn>Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Bose Institute</sa:organization><sa:organization>Department of Physics</sa:organization><sa:organization>Centre for Astroparticle Physics and Space Science (CAPSS)</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0020">Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0050" affiliation-id="S0370269323000643-187ae13619e9f75c78fa64b9926e2952"><ce:label>5</ce:label><ce:textfn>Budker Institute for Nuclear Physics, Novosibirsk, Russia</ce:textfn><sa:affiliation><sa:organization>Budker Institute for Nuclear Physics</sa:organization><sa:city>Novosibirsk</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0025">Budker Institute for Nuclear Physics, Novosibirsk, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0060" affiliation-id="S0370269323000643-b7f796ef6c934d79a497288cdec192f6"><ce:label>6</ce:label><ce:textfn>California Polytechnic State University, San Luis Obispo, CA, United States</ce:textfn><sa:affiliation><sa:organization>California Polytechnic State University</sa:organization><sa:city>San Luis Obispo</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0030">California Polytechnic State University, San Luis Obispo, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0070" affiliation-id="S0370269323000643-3dcd6ffc2e8f27d6ed2f6237a209a384"><ce:label>7</ce:label><ce:textfn>Central China Normal University, Wuhan, China</ce:textfn><sa:affiliation><sa:organization>Central China Normal University</sa:organization><sa:city>Wuhan</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0035">Central China Normal University, Wuhan, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0080" affiliation-id="S0370269323000643-110460c7f2fbb319ec6d52e7cc3fc1d5"><ce:label>8</ce:label><ce:textfn>Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba</ce:textfn><sa:affiliation><sa:organization>Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN)</sa:organization><sa:city>Havana</sa:city><sa:country>Cuba</sa:country></sa:affiliation><ce:source-text id="srct0040">Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba</ce:source-text></ce:affiliation><ce:affiliation id="aff0090" affiliation-id="S0370269323000643-641aa526558990d110c090840e6f3d0c"><ce:label>9</ce:label><ce:textfn>Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico</ce:textfn><sa:affiliation><sa:organization>Centro de Investigación y de Estudios Avanzados (CINVESTAV)</sa:organization><sa:city>Mexico City and Mérida</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0045">Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0100" affiliation-id="S0370269323000643-9c73ece39ab638447cd10f268e329b2e"><ce:label>10</ce:label><ce:textfn>Chicago State University, Chicago, IL, United States</ce:textfn><sa:affiliation><sa:organization>Chicago State University</sa:organization><sa:city>Chicago</sa:city><sa:state>IL</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0050">Chicago State University, Chicago, Illinois, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0110" affiliation-id="S0370269323000643-d7da93bf3f02be0a46bee74f439b5930"><ce:label>11</ce:label><ce:textfn>China Institute of Atomic Energy, Beijing, China</ce:textfn><sa:affiliation><sa:organization>China Institute of Atomic Energy</sa:organization><sa:city>Beijing</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0055">China Institute of Atomic Energy, Beijing, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0120" affiliation-id="S0370269323000643-d477851cd020cd39da135784676a96ff"><ce:label>12</ce:label><ce:textfn>Chungbuk National University, Cheongju, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Chungbuk National University</sa:organization><sa:city>Cheongju</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0060">Chungbuk National University, Cheongju, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0130" affiliation-id="S0370269323000643-ffca702bc50b4f17be4153d6998acb8d"><ce:label>13</ce:label><ce:textfn>Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovakia</ce:textfn><sa:affiliation><sa:organization>Comenius University Bratislava</sa:organization><sa:organization>Faculty of Mathematics, Physics and Informatics</sa:organization><sa:city>Bratislava</sa:city><sa:country>Slovakia</sa:country></sa:affiliation><ce:source-text id="srct0065">Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovakia</ce:source-text></ce:affiliation><ce:affiliation id="aff0140" affiliation-id="S0370269323000643-78cc8333b14d598fc5e3c0230d1de22e"><ce:label>14</ce:label><ce:textfn>COMSATS University Islamabad, Islamabad, Pakistan</ce:textfn><sa:affiliation><sa:organization>COMSATS University Islamabad</sa:organization><sa:city>Islamabad</sa:city><sa:country>Pakistan</sa:country></sa:affiliation><ce:source-text id="srct0070">COMSATS University Islamabad, Islamabad, Pakistan</ce:source-text></ce:affiliation><ce:affiliation id="aff0150" affiliation-id="S0370269323000643-42161ea7466da89325c5b37601107c55"><ce:label>15</ce:label><ce:textfn>Creighton University, Omaha, NE, United States</ce:textfn><sa:affiliation><sa:organization>Creighton University</sa:organization><sa:city>Omaha</sa:city><sa:state>NE</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0075">Creighton University, Omaha, Nebraska, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0160" affiliation-id="S0370269323000643-bd5e6b818668501c1deea76e96cac833"><ce:label>16</ce:label><ce:textfn>Department of Physics, Aligarh Muslim University, Aligarh, India</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Aligarh Muslim University</sa:organization><sa:city>Aligarh</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0080">Department of Physics, Aligarh Muslim University, Aligarh, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0170" affiliation-id="S0370269323000643-ebe7875fb705d3c179953d196aa8b94b"><ce:label>17</ce:label><ce:textfn>Department of Physics, Pusan National University, Pusan, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Pusan National University</sa:organization><sa:city>Pusan</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0085">Department of Physics, Pusan National University, Pusan, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0180" affiliation-id="S0370269323000643-24ddc81b37e640977b88412ba28336a4"><ce:label>18</ce:label><ce:textfn>Department of Physics, Sejong University, Seoul, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Sejong University</sa:organization><sa:city>Seoul</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0090">Department of Physics, Sejong University, Seoul, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0190" affiliation-id="S0370269323000643-66245837acb0d7fb7ada51de0fd95043"><ce:label>19</ce:label><ce:textfn>Department of Physics, University of California, Berkeley, CA, United States</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>University of California</sa:organization><sa:city>Berkeley</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0095">Department of Physics, University of California, Berkeley, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0200" affiliation-id="S0370269323000643-63776eabafa9e32856b35562692a4488"><ce:label>20</ce:label><ce:textfn>Department of Physics, University of Oslo, Oslo, Norway</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>University of Oslo</sa:organization><sa:city>Oslo</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0100">Department of Physics, University of Oslo, Oslo, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0210" affiliation-id="S0370269323000643-020451a59d5e257505166bdb7847cdd7"><ce:label>21</ce:label><ce:textfn>Department of Physics and Technology, University of Bergen, Bergen, Norway</ce:textfn><sa:affiliation><sa:organization>Department of Physics and Technology</sa:organization><sa:organization>University of Bergen</sa:organization><sa:city>Bergen</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0105">Department of Physics and Technology, University of Bergen, Bergen, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0220" affiliation-id="S0370269323000643-81d87d2cd8c9f6aa2a1c7f4b13115ef3"><ce:label>22</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Cagliari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0110">Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0230" affiliation-id="S0370269323000643-849d0ba7888fcb2a09aad7ac1095ec15"><ce:label>23</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Trieste</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0115">Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0240" affiliation-id="S0370269323000643-68c63fd1cb9e8623af62118bbef39c9e"><ce:label>24</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0120">Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0250" affiliation-id="S0370269323000643-7a40dab487121429c63e59c3de15ccfa"><ce:label>25</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bologna</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0125">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0260" affiliation-id="S0370269323000643-72e3fe624de7d3d3223574366a22ef30"><ce:label>26</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Catania</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0130">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0270" affiliation-id="S0370269323000643-6145c67e3009fb117e82f5429b2af282"><ce:label>27</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Padova</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0135">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0280" affiliation-id="S0370269323000643-ed02ab18bfc8872ec646d178987b6fb8"><ce:label>28</ce:label><ce:textfn>Dipartimento di Fisica e Nucleare e Teorica, Università di Pavia, Pavia, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Nucleare e Teorica</sa:organization><sa:organization>Università di Pavia</sa:organization><sa:city>Pavia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0140">Dipartimento di Fisica e Nucleare e Teorica, Università di Pavia, Pavia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0290" affiliation-id="S0370269323000643-a001bc692b1f1f84377401c9e2632e51"><ce:label>29</ce:label><ce:textfn>Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università and Gruppo Collegato INFN, Salerno, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università</sa:organization><sa:organization>Gruppo Collegato INFN</sa:organization><sa:city>Salerno</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0145">Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università and Gruppo Collegato INFN, Salerno, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0300" affiliation-id="S0370269323000643-362983c57dbe79a62add2550bfe565dc"><ce:label>30</ce:label><ce:textfn>Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento DISAT del Politecnico</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0150">Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0310" affiliation-id="S0370269323000643-be2c5e0552d256581ca35805d61bfd27"><ce:label>31</ce:label><ce:textfn>Dipartimento di Scienze e Innovazione Tecnologica dell'Università del Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Scienze e Innovazione Tecnologica dell'Università del Piemonte Orientale</sa:organization><sa:organization>INFN Sezione di Torino</sa:organization><sa:city>Alessandria</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0155">Dipartimento di Scienze e Innovazione Tecnologica dell'Università del Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0320" affiliation-id="S0370269323000643-7a82e32411929fc5768d11b72609a4d8"><ce:label>32</ce:label><ce:textfn>Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Scienze MIFT</sa:organization><sa:organization>Università di Messina</sa:organization><sa:city>Messina</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0160">Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0330" affiliation-id="S0370269323000643-0bfba0b176e2b6e1b67e6564c87308b5"><ce:label>33</ce:label><ce:textfn>Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento Interateneo di Fisica ‘M. Merlin’</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0165">Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0340" affiliation-id="S0370269323000643-44f92095d23d6e3d2fb8e8fc998c51f2"><ce:label>34</ce:label><ce:textfn>European Organization for Nuclear Research (CERN), Geneva, Switzerland</ce:textfn><sa:affiliation><sa:organization>European Organization for Nuclear Research (CERN)</sa:organization><sa:city>Geneva</sa:city><sa:country>Switzerland</sa:country></sa:affiliation><ce:source-text id="srct0170">European Organization for Nuclear Research (CERN), Geneva, Switzerland</ce:source-text></ce:affiliation><ce:affiliation id="aff0350" affiliation-id="S0370269323000643-f220870dd6ed2747e8ec11b2ba624bf5"><ce:label>35</ce:label><ce:textfn>Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia</ce:textfn><sa:affiliation><sa:organization>Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture</sa:organization><sa:organization>University of Split</sa:organization><sa:city>Split</sa:city><sa:country>Croatia</sa:country></sa:affiliation><ce:source-text id="srct0175">Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia</ce:source-text></ce:affiliation><ce:affiliation id="aff0360" affiliation-id="S0370269323000643-5bf4b9d297f0544e6037addfb7689840"><ce:label>36</ce:label><ce:textfn>Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway</ce:textfn><sa:affiliation><sa:organization>Faculty of Engineering and Science</sa:organization><sa:organization>Western Norway University of Applied Sciences</sa:organization><sa:city>Bergen</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0180">Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0370" affiliation-id="S0370269323000643-56d412e1c3fe114d9a18225fa76274c9"><ce:label>37</ce:label><ce:textfn>Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Faculty of Nuclear Sciences and Physical Engineering</sa:organization><sa:organization>Czech Technical University in Prague</sa:organization><sa:city>Prague</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0185">Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0380" affiliation-id="S0370269323000643-fb857933012a43dca7eed2503db4f3dd"><ce:label>38</ce:label><ce:textfn>Faculty of Science, P.J. Šafárik University, Košice, Slovakia</ce:textfn><sa:affiliation><sa:organization>Faculty of Science</sa:organization><sa:organization>P.J. Šafárik University</sa:organization><sa:city>Košice</sa:city><sa:country>Slovakia</sa:country></sa:affiliation><ce:source-text id="srct0190">Faculty of Science, P.J. Šafárik University, Košice, Slovakia</ce:source-text></ce:affiliation><ce:affiliation id="aff0390" affiliation-id="S0370269323000643-a18716885f5bc83f5d1aee8d0d80c7af"><ce:label>39</ce:label><ce:textfn>Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Frankfurt Institute for Advanced Studies</sa:organization><sa:organization>Johann Wolfgang Goethe-Universität Frankfurt</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0195">Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0400" affiliation-id="S0370269323000643-f0b0a2b18fef5547dcd39253b5714404"><ce:label>40</ce:label><ce:textfn>Fudan University, Shanghai, China</ce:textfn><sa:affiliation><sa:organization>Fudan University</sa:organization><sa:city>Shanghai</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0200">Fudan University, Shanghai, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0410" affiliation-id="S0370269323000643-a2398937a38c48ebf6ac3ff5e5d60b95"><ce:label>41</ce:label><ce:textfn>Gangneung-Wonju National University, Gangneung, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Gangneung-Wonju National University</sa:organization><sa:city>Gangneung</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0205">Gangneung-Wonju National University, Gangneung, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0420" affiliation-id="S0370269323000643-73bc80872cd116960a09bc477e035838"><ce:label>42</ce:label><ce:textfn>Gauhati University, Department of Physics, Guwahati, India</ce:textfn><sa:affiliation><sa:organization>Gauhati University</sa:organization><sa:organization>Department of Physics</sa:organization><sa:city>Guwahati</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0210">Gauhati University, Department of Physics, Guwahati, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0430" affiliation-id="S0370269323000643-9fd27eebf76465366fe59cb8d9620ae8"><ce:label>43</ce:label><ce:textfn>Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany</ce:textfn><sa:affiliation><sa:organization>Helmholtz-Institut für Strahlen- und Kernphysik</sa:organization><sa:organization>Rheinische Friedrich-Wilhelms-Universität Bonn</sa:organization><sa:city>Bonn</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0215">Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0440" affiliation-id="S0370269323000643-66b9f8889421b091fa908925e638d91e"><ce:label>44</ce:label><ce:textfn>Helsinki Institute of Physics (HIP), Helsinki, Finland</ce:textfn><sa:affiliation><sa:organization>Helsinki Institute of Physics (HIP)</sa:organization><sa:city>Helsinki</sa:city><sa:country>Finland</sa:country></sa:affiliation><ce:source-text id="srct0220">Helsinki Institute of Physics (HIP), Helsinki, Finland</ce:source-text></ce:affiliation><ce:affiliation id="aff0450" affiliation-id="S0370269323000643-6968018d6fe1bd12a4413918be70ab85"><ce:label>45</ce:label><ce:textfn>High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico</ce:textfn><sa:affiliation><sa:organization>High Energy Physics Group</sa:organization><sa:organization>Universidad Autónoma de Puebla</sa:organization><sa:city>Puebla</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0225">High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0460" affiliation-id="S0370269323000643-8454dbc80caf49377b12cbde1879d7a0"><ce:label>46</ce:label><ce:textfn>Hiroshima University, Hiroshima, Japan</ce:textfn><sa:affiliation><sa:organization>Hiroshima University</sa:organization><sa:city>Hiroshima</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0230">Hiroshima University, Hiroshima, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0470" affiliation-id="S0370269323000643-9c00b97a376e62a36465178b70ef1f89"><ce:label>47</ce:label><ce:textfn>Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation (ZTT), Worms, Germany</ce:textfn><sa:affiliation><sa:organization>Hochschule Worms</sa:organization><sa:organization>Zentrum für Technologietransfer und Telekommunikation (ZTT)</sa:organization><sa:city>Worms</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0235">Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation (ZTT), Worms, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0480" affiliation-id="S0370269323000643-16bdf6e3577a480da99159f5d54452b1"><ce:label>48</ce:label><ce:textfn>Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>Horia Hulubei National Institute of Physics and Nuclear Engineering</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0240">Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff0490" affiliation-id="S0370269323000643-78df0ad3e050545612fd8d71a4776a19"><ce:label>49</ce:label><ce:textfn>Indian Institute of Technology Bombay (IIT), Mumbai, India</ce:textfn><sa:affiliation><sa:organization>Indian Institute of Technology Bombay (IIT)</sa:organization><sa:city>Mumbai</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0245">Indian Institute of Technology Bombay (IIT), Mumbai, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0500" affiliation-id="S0370269323000643-b8ad6c9375b7a89b768adb13f27427b4"><ce:label>50</ce:label><ce:textfn>Indian Institute of Technology Indore, Indore, India</ce:textfn><sa:affiliation><sa:organization>Indian Institute of Technology Indore</sa:organization><sa:city>Indore</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0250">Indian Institute of Technology Indore, Indore, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0510" affiliation-id="S0370269323000643-98c636440432201908fa6753b352596d"><ce:label>51</ce:label><ce:textfn>Indonesian Institute of Sciences, Jakarta, Indonesia</ce:textfn><sa:affiliation><sa:organization>Indonesian Institute of Sciences</sa:organization><sa:city>Jakarta</sa:city><sa:country>Indonesia</sa:country></sa:affiliation><ce:source-text id="srct0255">Indonesian Institute of Sciences, Jakarta, Indonesia</ce:source-text></ce:affiliation><ce:affiliation id="aff0520" affiliation-id="S0370269323000643-25658fba725d22058ae2a8649ceeb084"><ce:label>52</ce:label><ce:textfn>INFN, Laboratori Nazionali di Frascati, Frascati, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Laboratori Nazionali di Frascati</sa:organization><sa:city>Frascati</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0260">INFN, Laboratori Nazionali di Frascati, Frascati, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0530" affiliation-id="S0370269323000643-5d9dee68bdf16e34f2f3f01d930f367d"><ce:label>53</ce:label><ce:textfn>INFN, Sezione di Bari, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Bari</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0265">INFN, Sezione di Bari, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0540" affiliation-id="S0370269323000643-340d750afed4ff705cf1dd72bf688ca4"><ce:label>54</ce:label><ce:textfn>INFN, Sezione di Bologna, Bologna, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Bologna</sa:organization><sa:city>Bologna</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0270">INFN, Sezione di Bologna, Bologna, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0550" affiliation-id="S0370269323000643-436e8c989d6c10cae74944a81714a4e2"><ce:label>55</ce:label><ce:textfn>INFN, Sezione di Cagliari, Cagliari, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Cagliari</sa:organization><sa:city>Cagliari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0275">INFN, 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Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Trieste</sa:organization><sa:city>Trieste</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0300">INFN, Sezione di Trieste, Trieste, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0610" affiliation-id="S0370269323000643-22e3a6d8462b39bb6c155bce0c9b21cd"><ce:label>61</ce:label><ce:textfn>Inha University, Incheon, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Inha University</sa:organization><sa:city>Incheon</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0305">Inha University, Incheon, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0620" affiliation-id="S0370269323000643-d4941ebd67ab9bbf96b1e6a906f4397c"><ce:label>62</ce:label><ce:textfn>Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands</ce:textfn><sa:affiliation><sa:organization>Institute for Gravitational and Subatomic Physics (GRASP)</sa:organization><sa:organization>Utrecht University/Nikhef</sa:organization><sa:city>Utrecht</sa:city><sa:country>Netherlands</sa:country></sa:affiliation><ce:source-text id="srct0310">Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands</ce:source-text></ce:affiliation><ce:affiliation id="aff0630" affiliation-id="S0370269323000643-ffd3d07e3cd462f194d82b1a01fbabeb"><ce:label>63</ce:label><ce:textfn>Institute for Nuclear Research, Academy of Sciences, Moscow, Russia</ce:textfn><sa:affiliation><sa:organization>Institute for Nuclear Research</sa:organization><sa:organization>Academy of Sciences</sa:organization><sa:city>Moscow</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0315">Institute for Nuclear Research, Academy of Sciences, Moscow, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0640" affiliation-id="S0370269323000643-c0ed70d0f039726935bd0309f74fcd8a"><ce:label>64</ce:label><ce:textfn>Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia</ce:textfn><sa:affiliation><sa:organization>Institute of Experimental Physics</sa:organization><sa:organization>Slovak Academy of Sciences</sa:organization><sa:city>Košice</sa:city><sa:country>Slovakia</sa:country></sa:affiliation><ce:source-text id="srct0320">Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia</ce:source-text></ce:affiliation><ce:affiliation id="aff0650" affiliation-id="S0370269323000643-6fa96ff1fcc4aaff09633c1217f06ff0"><ce:label>65</ce:label><ce:textfn>Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India</ce:textfn><sa:affiliation><sa:organization>Institute of Physics</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Bhubaneswar</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0325">Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0660" affiliation-id="S0370269323000643-9f77e70dcbde10c6ed3d37b3b0071107"><ce:label>66</ce:label><ce:textfn>Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Institute of Physics of the Czech Academy of Sciences</sa:organization><sa:city>Prague</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0330">Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0670" affiliation-id="S0370269323000643-3309ea65bfe3ae2f96b232545cb67043"><ce:label>67</ce:label><ce:textfn>Institute of Space Science (ISS), Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>Institute of Space Science (ISS)</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0335">Institute of Space Science (ISS), Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff0680" affiliation-id="S0370269323000643-2759820a41f8b0ece8c42aa319465ed5"><ce:label>68</ce:label><ce:textfn>Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Institut für Kernphysik</sa:organization><sa:organization>Johann Wolfgang Goethe-Universität Frankfurt</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0340">Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0690" affiliation-id="S0370269323000643-b18018b82a469dc4e4eb94f595cf4812"><ce:label>69</ce:label><ce:textfn>Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:textfn><sa:affiliation><sa:organization>Instituto de Ciencias Nucleares</sa:organization><sa:organization>Universidad Nacional Autónoma de México</sa:organization><sa:city>Mexico City</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0345">Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0700" affiliation-id="S0370269323000643-2aa194eda46d62198ea9b929240200b8"><ce:label>70</ce:label><ce:textfn>Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil</ce:textfn><sa:affiliation><sa:organization>Instituto de Física</sa:organization><sa:organization>Universidade Federal do Rio Grande do Sul (UFRGS)</sa:organization><sa:city>Porto Alegre</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0350">Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff0710" affiliation-id="S0370269323000643-9e9b95fa2c082308cb0efc3488541c67"><ce:label>71</ce:label><ce:textfn>Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:textfn><sa:affiliation><sa:organization>Instituto de Física</sa:organization><sa:organization>Universidad Nacional Autónoma de México</sa:organization><sa:city>Mexico City</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0355">Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0720" affiliation-id="S0370269323000643-4ef9aeae6b2f74e366edd12aafbea4cf"><ce:label>72</ce:label><ce:textfn>iThemba LABS, National Research Foundation, Somerset West, South Africa</ce:textfn><sa:affiliation><sa:organization>iThemba LABS</sa:organization><sa:organization>National Research Foundation</sa:organization><sa:city>Somerset West</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0360">iThemba LABS, National Research Foundation, Somerset West, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff0730" affiliation-id="S0370269323000643-2f4c4db0447d27fd2e43117b90bb74d4"><ce:label>73</ce:label><ce:textfn>Jeonbuk National University, Jeonju, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Jeonbuk National University</sa:organization><sa:city>Jeonju</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0365">Jeonbuk National University, Jeonju, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0740" affiliation-id="S0370269323000643-81ec7d4c8c69486a57ef1ca6a567076c"><ce:label>74</ce:label><ce:textfn>Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik</sa:organization><sa:organization>Fachbereich Informatik und Mathematik</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0370">Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0750" affiliation-id="S0370269323000643-53c65b0bfaf5dee3f840aa6c9e183978"><ce:label>75</ce:label><ce:textfn>Joint Institute for Nuclear Research (JINR), Dubna, Russia</ce:textfn><sa:affiliation><sa:organization>Joint Institute for Nuclear Research (JINR)</sa:organization><sa:city>Dubna</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0375">Joint Institute for Nuclear Research (JINR), Dubna, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0760" affiliation-id="S0370269323000643-c1d849875c0de0db6476f9cc96b07dd2"><ce:label>76</ce:label><ce:textfn>Korea Institute of Science and Technology Information, Daejeon, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Korea Institute of Science and Technology Information</sa:organization><sa:city>Daejeon</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0380">Korea Institute of Science and Technology Information, Daejeon, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0770" affiliation-id="S0370269323000643-c5cd420fa3d6b7c79dc3ab9e7c2dfb06"><ce:label>77</ce:label><ce:textfn>KTO Karatay University, Konya, Turkey</ce:textfn><sa:affiliation><sa:organization>KTO Karatay University</sa:organization><sa:city>Konya</sa:city><sa:country>Turkey</sa:country></sa:affiliation><ce:source-text id="srct0385">KTO Karatay University, Konya, Turkey</ce:source-text></ce:affiliation><ce:affiliation id="aff0780" affiliation-id="S0370269323000643-0bdd954451acb47f491c8c4996fa87da"><ce:label>78</ce:label><ce:textfn>Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France</ce:textfn><sa:affiliation><sa:organization>Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie</sa:organization><sa:city>Orsay</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0390">Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France</ce:source-text></ce:affiliation><ce:affiliation id="aff0790" affiliation-id="S0370269323000643-1487532e5bfe30325bc10c0583f7c38e"><ce:label>79</ce:label><ce:textfn>Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France</ce:textfn><sa:affiliation><sa:organization>Laboratoire de Physique Subatomique et de Cosmologie</sa:organization><sa:organization>Université Grenoble-Alpes</sa:organization><sa:organization>CNRS-IN2P3</sa:organization><sa:city>Grenoble</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0395">Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France</ce:source-text></ce:affiliation><ce:affiliation id="aff0800" affiliation-id="S0370269323000643-dd5512fbf2faf90b56635e0b411d44a2"><ce:label>80</ce:label><ce:textfn>Lawrence Berkeley National Laboratory, Berkeley, CA, United States</ce:textfn><sa:affiliation><sa:organization>Lawrence Berkeley National Laboratory</sa:organization><sa:city>Berkeley</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0400">Lawrence Berkeley National Laboratory, Berkeley, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0810" affiliation-id="S0370269323000643-ed03eefd58007822249c697d882deffc"><ce:label>81</ce:label><ce:textfn>Lund University Department of Physics, Division of Particle Physics, Lund, Sweden</ce:textfn><sa:affiliation><sa:organization>Lund University Department of Physics</sa:organization><sa:organization>Division of Particle Physics</sa:organization><sa:city>Lund</sa:city><sa:country>Sweden</sa:country></sa:affiliation><ce:source-text id="srct0405">Lund University Department of Physics, Division of Particle Physics, Lund, Sweden</ce:source-text></ce:affiliation><ce:affiliation id="aff0820" affiliation-id="S0370269323000643-ecd2d28e753f4396ad11c81a0c3e50ba"><ce:label>82</ce:label><ce:textfn>Moscow Institute for Physics and Technology, Moscow, Russia</ce:textfn><sa:affiliation><sa:organization>Moscow Institute for Physics and Technology</sa:organization><sa:city>Moscow</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0410">Moscow Institute for Physics and Technology, Moscow, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0830" affiliation-id="S0370269323000643-5b47ca06644b7f88e2267e9accbe867b"><ce:label>83</ce:label><ce:textfn>Nagasaki Institute of Applied Science, Nagasaki, Japan</ce:textfn><sa:affiliation><sa:organization>Nagasaki Institute of Applied Science</sa:organization><sa:city>Nagasaki</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0415">Nagasaki Institute of Applied Science, Nagasaki, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0840" affiliation-id="S0370269323000643-61cd55f0b96e9831d6063318d9967d43"><ce:label>84</ce:label><ce:textfn>Nara Women's University (NWU), Nara, Japan</ce:textfn><sa:affiliation><sa:organization>Nara Women's University (NWU)</sa:organization><sa:city>Nara</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0420">Nara Women's University (NWU), Nara, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0850" affiliation-id="S0370269323000643-7f50d99eb340e408fbc323cbce0c8905"><ce:label>85</ce:label><ce:textfn>National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece</ce:textfn><sa:affiliation><sa:organization>National and Kapodistrian University of Athens</sa:organization><sa:organization>School of Science</sa:organization><sa:organization>Department of Physics</sa:organization><sa:city>Athens</sa:city><sa:country>Greece</sa:country></sa:affiliation><ce:source-text id="srct0425">National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece</ce:source-text></ce:affiliation><ce:affiliation id="aff0860" affiliation-id="S0370269323000643-58192de9f95a93cb8bf485b9a5647bf5"><ce:label>86</ce:label><ce:textfn>National Centre for Nuclear Research, Warsaw, Poland</ce:textfn><sa:affiliation><sa:organization>National Centre for Nuclear Research</sa:organization><sa:city>Warsaw</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0430">National Centre for Nuclear Research, Warsaw, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff0870" affiliation-id="S0370269323000643-5c73c93f2c9aba1b7fcfa30741ce17ca"><ce:label>87</ce:label><ce:textfn>National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India</ce:textfn><sa:affiliation><sa:organization>National Institute of Science Education and Research</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Jatni</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0435">National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0880" affiliation-id="S0370269323000643-1e229a9758219f877a7f903c2dde9c2b"><ce:label>88</ce:label><ce:textfn>National Nuclear Research Center, Baku, Azerbaijan</ce:textfn><sa:affiliation><sa:organization>National Nuclear Research Center</sa:organization><sa:city>Baku</sa:city><sa:country>Azerbaijan</sa:country></sa:affiliation><ce:source-text id="srct0440">National Nuclear Research Center, Baku, Azerbaijan</ce:source-text></ce:affiliation><ce:affiliation id="aff0890" affiliation-id="S0370269323000643-ef899e70d7f524a8b4dc85511af17a46"><ce:label>89</ce:label><ce:textfn>National Research Centre Kurchatov Institute, Moscow, Russia</ce:textfn><sa:affiliation><sa:organization>National Research Centre Kurchatov Institute</sa:organization><sa:city>Moscow</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0445">National Research Centre Kurchatov Institute, Moscow, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0900" affiliation-id="S0370269323000643-3e7cb9222cae4e75df0d93d2ea96329f"><ce:label>90</ce:label><ce:textfn>Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark</ce:textfn><sa:affiliation><sa:organization>Niels Bohr Institute</sa:organization><sa:organization>University of Copenhagen</sa:organization><sa:city>Copenhagen</sa:city><sa:country>Denmark</sa:country></sa:affiliation><ce:source-text id="srct0450">Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark</ce:source-text></ce:affiliation><ce:affiliation id="aff0910" affiliation-id="S0370269323000643-11b144edfb6d992e108d17991cc7681b"><ce:label>91</ce:label><ce:textfn>Nikhef, National Institute for Subatomic Physics, Amsterdam, Netherlands</ce:textfn><sa:affiliation><sa:organization>Nikhef</sa:organization><sa:organization>National Institute for Subatomic Physics</sa:organization><sa:city>Amsterdam</sa:city><sa:country>Netherlands</sa:country></sa:affiliation><ce:source-text id="srct0455">Nikhef, National institute for subatomic physics, Amsterdam, Netherlands</ce:source-text></ce:affiliation><ce:affiliation id="aff0920" affiliation-id="S0370269323000643-3c430648d0afb815a8e84258861895f0"><ce:label>92</ce:label><ce:textfn>NRC Kurchatov Institute IHEP, Protvino, Russia</ce:textfn><sa:affiliation><sa:organization>NRC Kurchatov Institute</sa:organization><sa:organization>IHEP</sa:organization><sa:city>Protvino</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0460">NRC Kurchatov Institute IHEP, Protvino, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0930" affiliation-id="S0370269323000643-3928c351c9b4797dafcf46dd65eee0cd"><ce:label>93</ce:label><ce:textfn>NRC «Kurchatov»Institute – ITEP, Moscow, Russia</ce:textfn><sa:affiliation><sa:organization>NRC «Kurchatov»Institute – ITEP</sa:organization><sa:city>Moscow</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0465">NRC «Kurchatov»Institute – ITEP, Moscow, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0940" affiliation-id="S0370269323000643-b4ad181607b9ab8fbc66e2278cdf943a"><ce:label>94</ce:label><ce:textfn>NRNU Moscow Engineering Physics Institute, Moscow, Russia</ce:textfn><sa:affiliation><sa:organization>NRNU Moscow Engineering Physics Institute</sa:organization><sa:city>Moscow</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0470">NRNU Moscow Engineering Physics Institute, Moscow, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0950" affiliation-id="S0370269323000643-7ffc71d9a245ae3927bb4d373ff478ed"><ce:label>95</ce:label><ce:textfn>Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Group</sa:organization><sa:organization>STFC Daresbury Laboratory</sa:organization><sa:city>Daresbury</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0475">Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff0960" affiliation-id="S0370269323000643-35f0d35e6b51b7c24d9b3bc939a45b30"><ce:label>96</ce:label><ce:textfn>Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Institute of the Czech Academy of Sciences</sa:organization><sa:city>Řež u Prahy</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0480">Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0970" affiliation-id="S0370269323000643-43c0b745ad096562d858417552718575"><ce:label>97</ce:label><ce:textfn>Oak Ridge National Laboratory, Oak Ridge, TN, United States</ce:textfn><sa:affiliation><sa:organization>Oak Ridge National Laboratory</sa:organization><sa:city>Oak Ridge</sa:city><sa:state>TN</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0485">Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0980" affiliation-id="S0370269323000643-e4e32a64c7e436b0f2a688687dedfe5e"><ce:label>98</ce:label><ce:textfn>Ohio State University, Columbus, OH, United States</ce:textfn><sa:affiliation><sa:organization>Ohio State University</sa:organization><sa:city>Columbus</sa:city><sa:state>OH</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0490">Ohio State University, Columbus, Ohio, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0990" affiliation-id="S0370269323000643-90511898a8b3f881cbde7907eee7df47"><ce:label>99</ce:label><ce:textfn>Petersburg Nuclear Physics Institute, Gatchina, Russia</ce:textfn><sa:affiliation><sa:organization>Petersburg Nuclear Physics Institute</sa:organization><sa:city>Gatchina</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0495">Petersburg Nuclear Physics Institute, Gatchina, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff1000" affiliation-id="S0370269323000643-3c4eb171ede7112c64f1bc2c164f4732"><ce:label>100</ce:label><ce:textfn>Physics Department, Faculty of Science, University of Zagreb, Zagreb, Croatia</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>Faculty of Science</sa:organization><sa:organization>University of Zagreb</sa:organization><sa:city>Zagreb</sa:city><sa:country>Croatia</sa:country></sa:affiliation><ce:source-text id="srct0500">Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia</ce:source-text></ce:affiliation><ce:affiliation id="aff1010" affiliation-id="S0370269323000643-30d6dbb96747c914840798f88bf250cb"><ce:label>101</ce:label><ce:textfn>Physics Department, Panjab University, Chandigarh, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>Panjab University</sa:organization><sa:city>Chandigarh</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0505">Physics Department, Panjab University, Chandigarh, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1020" affiliation-id="S0370269323000643-712bc80495e97050498604da603cb5cc"><ce:label>102</ce:label><ce:textfn>Physics Department, University of Jammu, Jammu, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>University of Jammu</sa:organization><sa:city>Jammu</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0510">Physics Department, University of Jammu, Jammu, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1030" affiliation-id="S0370269323000643-500771749243cadcbaeeac3f726e6a62"><ce:label>103</ce:label><ce:textfn>Physics Department, University of Rajasthan, Jaipur, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>University of Rajasthan</sa:organization><sa:city>Jaipur</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0515">Physics Department, University of Rajasthan, Jaipur, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1040" affiliation-id="S0370269323000643-0886f70da565231fd0e43398021fd604"><ce:label>104</ce:label><ce:textfn>Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany</ce:textfn><sa:affiliation><sa:organization>Physikalisches Institut</sa:organization><sa:organization>Eberhard-Karls-Universität Tübingen</sa:organization><sa:city>Tübingen</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0520">Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1050" affiliation-id="S0370269323000643-1c67678e124982de924355cdd6d6b91b"><ce:label>105</ce:label><ce:textfn>Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany</ce:textfn><sa:affiliation><sa:organization>Physikalisches Institut</sa:organization><sa:organization>Ruprecht-Karls-Universität Heidelberg</sa:organization><sa:city>Heidelberg</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0525">Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1060" affiliation-id="S0370269323000643-7a5765cc3928149c8b77128cd52462ae"><ce:label>106</ce:label><ce:textfn>Physik Department, Technische Universität München, Munich, Germany</ce:textfn><sa:affiliation><sa:organization>Physik Department</sa:organization><sa:organization>Technische Universität München</sa:organization><sa:city>Munich</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0530">Physik Department, Technische Universität München, Munich, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1070" affiliation-id="S0370269323000643-0a06f4f5fb60a25c8f1936ed71c96cfe"><ce:label>107</ce:label><ce:textfn>Politecnico di Bari and Sezione INFN, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>Politecnico di Bari</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0535">Politecnico di Bari and Sezione INFN, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1080" affiliation-id="S0370269323000643-3451d358e90b759225813c9b9a6f098c"><ce:label>108</ce:label><ce:textfn>Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany</ce:textfn><sa:affiliation><sa:organization>Research Division</sa:organization><sa:organization>ExtreMe Matter Institute EMMI</sa:organization><sa:organization>GSI Helmholtzzentrum für Schwerionenforschung GmbH</sa:organization><sa:city>Darmstadt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0540">Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1090" affiliation-id="S0370269323000643-bdc21ecb323f7b76ee73d71fa687e6c3"><ce:label>109</ce:label><ce:textfn>Russian Federal Nuclear Center (VNIIEF), Sarov, Russia</ce:textfn><sa:affiliation><sa:organization>Russian Federal Nuclear Center (VNIIEF)</sa:organization><sa:city>Sarov</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0545">Russian Federal Nuclear Center (VNIIEF), Sarov, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff1100" affiliation-id="S0370269323000643-2771a2ae295804be11a4a937b894a546"><ce:label>110</ce:label><ce:textfn>Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Saha Institute of Nuclear Physics</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0550">Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1110" affiliation-id="S0370269323000643-b602cde70213041ca40ec5720e4e3e75"><ce:label>111</ce:label><ce:textfn>School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom</ce:textfn><sa:affiliation><sa:organization>School of Physics and Astronomy</sa:organization><sa:organization>University of Birmingham</sa:organization><sa:city>Birmingham</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0555">School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff1120" affiliation-id="S0370269323000643-1a1f0f3ae33ba0323ad4da08a216f4c3"><ce:label>112</ce:label><ce:textfn>Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru</ce:textfn><sa:affiliation><sa:organization>Sección Física</sa:organization><sa:organization>Departamento de Ciencias</sa:organization><sa:organization>Pontificia Universidad Católica del Perú</sa:organization><sa:city>Lima</sa:city><sa:country>Peru</sa:country></sa:affiliation><ce:source-text id="srct0560">Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru</ce:source-text></ce:affiliation><ce:affiliation id="aff1130" affiliation-id="S0370269323000643-e0932a7de5094071a8671adb41fe6812"><ce:label>113</ce:label><ce:textfn>St. Petersburg State University, St. Petersburg, Russia</ce:textfn><sa:affiliation><sa:organization>St. Petersburg State University</sa:organization><sa:city>St. Petersburg</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0565">St. Petersburg State University, St. Petersburg, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff1140" affiliation-id="S0370269323000643-24347400090c9a1efb87e7537af7461b"><ce:label>114</ce:label><ce:textfn>Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria</ce:textfn><sa:affiliation><sa:organization>Stefan Meyer Institut für Subatomare Physik (SMI)</sa:organization><sa:city>Vienna</sa:city><sa:country>Austria</sa:country></sa:affiliation><ce:source-text id="srct0570">Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria</ce:source-text></ce:affiliation><ce:affiliation id="aff1150" affiliation-id="S0370269323000643-920c67cc008319f3917103203f6dd703"><ce:label>115</ce:label><ce:textfn>SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes, France</ce:textfn><sa:affiliation><sa:organization>SUBATECH</sa:organization><sa:organization>IMT Atlantique</sa:organization><sa:organization>Université de Nantes</sa:organization><sa:organization>CNRS-IN2P3</sa:organization><sa:city>Nantes</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0575">SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1160" affiliation-id="S0370269323000643-c3972f6af24eef12cc2ae3a53d6a7623"><ce:label>116</ce:label><ce:textfn>Suranaree University of Technology, Nakhon Ratchasima, Thailand</ce:textfn><sa:affiliation><sa:organization>Suranaree University of Technology</sa:organization><sa:city>Nakhon Ratchasima</sa:city><sa:country>Thailand</sa:country></sa:affiliation><ce:source-text id="srct0580">Suranaree University of Technology, Nakhon Ratchasima, Thailand</ce:source-text></ce:affiliation><ce:affiliation id="aff1170" affiliation-id="S0370269323000643-c78ab7c321a491b1927107824e5f2c30"><ce:label>117</ce:label><ce:textfn>Technical University of Košice, Košice, Slovakia</ce:textfn><sa:affiliation><sa:organization>Technical University of Košice</sa:organization><sa:city>Košice</sa:city><sa:country>Slovakia</sa:country></sa:affiliation><ce:source-text id="srct0585">Technical University of Košice, Košice, Slovakia</ce:source-text></ce:affiliation><ce:affiliation id="aff1180" affiliation-id="S0370269323000643-425836ada61fe0d3ee4ebf5b87598919"><ce:label>118</ce:label><ce:textfn>The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland</ce:textfn><sa:affiliation><sa:organization>The Henryk Niewodniczanski Institute of Nuclear Physics</sa:organization><sa:organization>Polish Academy of Sciences</sa:organization><sa:city>Cracow</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0590">The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff1190" affiliation-id="S0370269323000643-302a66584ead3636c3e74b86b51cf474"><ce:label>119</ce:label><ce:textfn>The University of Texas at Austin, Austin, TX, United States</ce:textfn><sa:affiliation><sa:organization>The University of Texas at Austin</sa:organization><sa:city>Austin</sa:city><sa:state>TX</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0595">The University of Texas at Austin, Austin, Texas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1200" affiliation-id="S0370269323000643-48ea3700cc067946e6c3032832f0ce0a"><ce:label>120</ce:label><ce:textfn>Universidad Autónoma de Sinaloa, Culiacán, Mexico</ce:textfn><sa:affiliation><sa:organization>Universidad Autónoma de Sinaloa</sa:organization><sa:city>Culiacán</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0600">Universidad Autónoma de Sinaloa, Culiacán, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff1210" affiliation-id="S0370269323000643-310945a13cafb9845fe651fc7aa661b6"><ce:label>121</ce:label><ce:textfn>Universidade de São Paulo (USP), São Paulo, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade de São Paulo (USP)</sa:organization><sa:city>São Paulo</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0605">Universidade de São Paulo (USP), São Paulo, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1220" affiliation-id="S0370269323000643-226f6a475b7a1634e1530e2077d7590e"><ce:label>122</ce:label><ce:textfn>Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade Estadual de Campinas (UNICAMP)</sa:organization><sa:city>Campinas</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0610">Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1230" affiliation-id="S0370269323000643-7582533209ee7a368c4d249782e93c03"><ce:label>123</ce:label><ce:textfn>Universidade Federal do ABC, Santo Andre, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade Federal do ABC</sa:organization><sa:city>Santo Andre</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0615">Universidade Federal do ABC, Santo Andre, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1240" affiliation-id="S0370269323000643-f37f0b36132124cce5539b1f9c518079"><ce:label>124</ce:label><ce:textfn>University of Cape Town, Cape Town, South Africa</ce:textfn><sa:affiliation><sa:organization>University of Cape Town</sa:organization><sa:city>Cape Town</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0620">University of Cape Town, Cape Town, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff1250" affiliation-id="S0370269323000643-58fe35a5cc9d91eea34fd6e19293599b"><ce:label>125</ce:label><ce:textfn>University of Houston, Houston, TX, United States</ce:textfn><sa:affiliation><sa:organization>University of Houston</sa:organization><sa:city>Houston</sa:city><sa:state>TX</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0625">University of Houston, Houston, Texas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1260" affiliation-id="S0370269323000643-e9be2b1885c8285b7cb7b6534ca93d98"><ce:label>126</ce:label><ce:textfn>University of Jyväskylä, Jyväskylä, Finland</ce:textfn><sa:affiliation><sa:organization>University of Jyväskylä</sa:organization><sa:city>Jyväskylä</sa:city><sa:country>Finland</sa:country></sa:affiliation><ce:source-text id="srct0630">University of Jyväskylä, Jyväskylä, Finland</ce:source-text></ce:affiliation><ce:affiliation id="aff1270" affiliation-id="S0370269323000643-6e2074f6fd0b88987501b22e5d74f9c9"><ce:label>127</ce:label><ce:textfn>University of Kansas, Lawrence, KS, United States</ce:textfn><sa:affiliation><sa:organization>University of Kansas</sa:organization><sa:city>Lawrence</sa:city><sa:state>KS</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0635">University of Kansas, Lawrence, Kansas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1280" affiliation-id="S0370269323000643-f538fc5a59ec1d1c8f86d2a83cb4ced6"><ce:label>128</ce:label><ce:textfn>University of Liverpool, Liverpool, United Kingdom</ce:textfn><sa:affiliation><sa:organization>University of Liverpool</sa:organization><sa:city>Liverpool</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0640">University of Liverpool, Liverpool, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff1290" affiliation-id="S0370269323000643-5507e3344bf7b46ba698c2c045a8aba6"><ce:label>129</ce:label><ce:textfn>University of Science and Technology of China, Hefei, China</ce:textfn><sa:affiliation><sa:organization>University of Science and Technology of China</sa:organization><sa:city>Hefei</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0645">University of Science and Technology of China, Hefei, China</ce:source-text></ce:affiliation><ce:affiliation id="aff1300" affiliation-id="S0370269323000643-b080b99a0650ae4321512ce14fe0060e"><ce:label>130</ce:label><ce:textfn>University of South-Eastern Norway, Tonsberg, Norway</ce:textfn><sa:affiliation><sa:organization>University of South-Eastern Norway</sa:organization><sa:city>Tonsberg</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0650">University of South-Eastern Norway, Tonsberg, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff1310" affiliation-id="S0370269323000643-d6620449e5365c80224ff22fe1ca4e4a"><ce:label>131</ce:label><ce:textfn>University of Tennessee, Knoxville, TN, United States</ce:textfn><sa:affiliation><sa:organization>University of Tennessee</sa:organization><sa:city>Knoxville</sa:city><sa:state>TN</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0655">University of Tennessee, Knoxville, Tennessee, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1320" affiliation-id="S0370269323000643-6d43022152ccaa62119e4287bf3b5270"><ce:label>132</ce:label><ce:textfn>University of the Witwatersrand, Johannesburg, South Africa</ce:textfn><sa:affiliation><sa:organization>University of the Witwatersrand</sa:organization><sa:city>Johannesburg</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0660">University of the Witwatersrand, Johannesburg, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff1330" affiliation-id="S0370269323000643-9b9071fc23073da0e4317b3863c91b9e"><ce:label>133</ce:label><ce:textfn>University of Tokyo, Tokyo, Japan</ce:textfn><sa:affiliation><sa:organization>University of Tokyo</sa:organization><sa:city>Tokyo</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0665">University of Tokyo, Tokyo, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff1340" affiliation-id="S0370269323000643-da3fb88e8779358429638ee8f8dc4cb5"><ce:label>134</ce:label><ce:textfn>University of Tsukuba, Tsukuba, Japan</ce:textfn><sa:affiliation><sa:organization>University of Tsukuba</sa:organization><sa:city>Tsukuba</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0670">University of Tsukuba, Tsukuba, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff1350" affiliation-id="S0370269323000643-9fc1715eadb7b2ba79b1a87cef8015cd"><ce:label>135</ce:label><ce:textfn>University Politehnica of Bucharest, Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>University Politehnica of Bucharest</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0675">University Politehnica of Bucharest, Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff1360" affiliation-id="S0370269323000643-43e2ac82e11c4ca2fabbc7e457c74b37"><ce:label>136</ce:label><ce:textfn>Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France</ce:textfn><sa:affiliation><sa:organization>Université Clermont Auvergne</sa:organization><sa:organization>CNRS/IN2P3</sa:organization><sa:organization>LPC</sa:organization><sa:city>Clermont-Ferrand</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0680">Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1370" affiliation-id="S0370269323000643-08f13150d05b32d440951efec6f80d29"><ce:label>137</ce:label><ce:textfn>Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France</ce:textfn><sa:affiliation><sa:organization>Université de Lyon</sa:organization><sa:organization>CNRS/IN2P3</sa:organization><sa:organization>Institut de Physique des 2 Infinis de Lyon</sa:organization><sa:city>Lyon</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0685">Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1380" affiliation-id="S0370269323000643-766d0075165899c16752f5f7ff1b8771"><ce:label>138</ce:label><ce:textfn>Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France</ce:textfn><sa:affiliation><sa:organization>Université de Strasbourg</sa:organization><sa:organization>CNRS</sa:organization><sa:organization>IPHC UMR 7178</sa:organization><sa:city>Strasbourg</sa:city><sa:postal-code>F-67000</sa:postal-code><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0690">Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1390" affiliation-id="S0370269323000643-36117066550bb6faaf9708b1b0ea670e"><ce:label>139</ce:label><ce:textfn>Université Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France</ce:textfn><sa:affiliation><sa:organization>Université Paris-Saclay</sa:organization><sa:organization>Centre d'Etudes de Saclay (CEA)</sa:organization><sa:organization>IRFU</sa:organization><sa:organization>Départment de Physique Nucléaire (DPhN)</sa:organization><sa:city>Saclay</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0695">Université Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1400" affiliation-id="S0370269323000643-47ef552cdaf90d05d791574b14a8dcf9"><ce:label>140</ce:label><ce:textfn>Università degli Studi di Foggia, Foggia, Italy</ce:textfn><sa:affiliation><sa:organization>Università degli Studi di Foggia</sa:organization><sa:city>Foggia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0700">Università degli Studi di Foggia, Foggia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1410" affiliation-id="S0370269323000643-83cd0c0929f483f88e3d2cb59f064287"><ce:label>141</ce:label><ce:textfn>Università di Brescia, Brescia, Italy</ce:textfn><sa:affiliation><sa:organization>Università di Brescia</sa:organization><sa:city>Brescia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0705">Università di Brescia, Brescia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1420" affiliation-id="S0370269323000643-f1ae52f852d4d7d99988b3e872f887e4"><ce:label>142</ce:label><ce:textfn>Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Variable Energy Cyclotron Centre</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0710">Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1430" affiliation-id="S0370269323000643-bfd4fe0d3b8675ea55d96ce64033d817"><ce:label>143</ce:label><ce:textfn>Warsaw University of Technology, Warsaw, Poland</ce:textfn><sa:affiliation><sa:organization>Warsaw University of Technology</sa:organization><sa:city>Warsaw</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0715">Warsaw University of Technology, Warsaw, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff1440" affiliation-id="S0370269323000643-4e11d38f810a3540206ffb5151a35d3b"><ce:label>144</ce:label><ce:textfn>Wayne State University, Detroit, MI, United States</ce:textfn><sa:affiliation><sa:organization>Wayne State University</sa:organization><sa:city>Detroit</sa:city><sa:state>MI</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0720">Wayne State University, Detroit, Michigan, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1450" affiliation-id="S0370269323000643-0b76517a6de7ff0579ec14780f79d81b"><ce:label>145</ce:label><ce:textfn>Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany</ce:textfn><sa:affiliation><sa:organization>Westfälische Wilhelms-Universität Münster</sa:organization><sa:organization>Institut für Kernphysik</sa:organization><sa:city>Münster</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0725">Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1460" affiliation-id="S0370269323000643-6b1ae45228f1f040c55767dc107b589d"><ce:label>146</ce:label><ce:textfn>Wigner Research Centre for Physics, Budapest, Hungary</ce:textfn><sa:affiliation><sa:organization>Wigner Research Centre for Physics</sa:organization><sa:city>Budapest</sa:city><sa:country>Hungary</sa:country></sa:affiliation><ce:source-text id="srct0730">Wigner Research Centre for Physics, Budapest, Hungary</ce:source-text></ce:affiliation><ce:affiliation id="aff1470" affiliation-id="S0370269323000643-96e79e7381eab77664732c3553e247e8"><ce:label>147</ce:label><ce:textfn>Yale University, New Haven, CT, United States</ce:textfn><sa:affiliation><sa:organization>Yale University</sa:organization><sa:city>New Haven</sa:city><sa:state>CT</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0735">Yale University, New Haven, Connecticut, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1480" affiliation-id="S0370269323000643-f3db17ceaf6ea4190e2176a6a129c96b"><ce:label>148</ce:label><ce:textfn>Yonsei University, Seoul, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Yonsei University</sa:organization><sa:city>Seoul</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0740">Yonsei University, Seoul, Republic of Korea</ce:source-text></ce:affiliation><ce:footnote id="fn0010"><ce:label>I</ce:label><ce:note-para id="np0010">Deceased.</ce:note-para></ce:footnote><ce:footnote id="fn0020"><ce:label>II</ce:label><ce:note-para id="np0020">Also at: Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA), Bologna, Italy.</ce:note-para></ce:footnote><ce:footnote id="fn0030"><ce:label>III</ce:label><ce:note-para id="np0030">Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy.</ce:note-para></ce:footnote><ce:footnote id="fn0040"><ce:label>IV</ce:label><ce:note-para id="np0040">Also at: M.V. Lomonosov Moscow State University, D.V. Skobeltsyn Institute of Nuclear, Physics, Moscow, Russia.</ce:note-para></ce:footnote><ce:footnote id="fn0050"><ce:label>V</ce:label><ce:note-para id="np0050">Also at: Department of Applied Physics, Aligarh Muslim University, Aligarh, India.</ce:note-para></ce:footnote><ce:footnote id="fn0060"><ce:label>VI</ce:label><ce:note-para id="np0060">Also at: Institute of Theoretical Physics, University of Wroclaw, Poland.</ce:note-para></ce:footnote><ce:footnote id="fn0070"><ce:label>VII</ce:label><ce:note-para id="np0070">Also at: University of Kansas, Lawrence, Kansas, United States.</ce:note-para></ce:footnote></ce:author-group></ce:collaboration><ce:footnote id="fn0080"><ce:label>⋆</ce:label><ce:note-para id="np0080"><ce:italic>E-mail address:</ce:italic> <ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/text/html" xlink:href="mailto:alice-publications@cern.ch" id="inf0020">alice-publications@cern.ch</ce:inter-ref>.</ce:note-para></ce:footnote></ce:author-group><ce:date-received day="20" month="5" year="2022"/><ce:date-revised day="17" month="1" year="2023"/><ce:date-accepted day="17" month="1" year="2023"/><ce:miscellaneous id="ms0010">Editor: M. Doser</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0070">We present the first systematic comparison of the charged-particle pseudorapidity densities for three widely different collision systems, pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb, at the top energy of the Large Hadron Collider (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>) measured over a wide pseudorapidity range (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.5</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5</mml:mn></mml:math>), the widest possible among the four experiments at that facility. The systematic uncertainties are minimised since the measurements are recorded by the same experimental apparatus (ALICE). The distributions for p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb collisions are determined as a function of the centrality of the collisions, while results from pp collisions are reported for inelastic events with at least one charged particle at midrapidity. The charged-particle pseudorapidity densities are, under simple and robust assumptions, transformed to charged-particle rapidity densities. This allows for the calculation and the presentation of the evolution of the width of the rapidity distributions and of a lower bound on the Bjorken energy density, as a function of the number of participants in all three collision systems. We find a decreasing width of the particle production, and roughly a smooth ten fold increase in the energy density, as the system size grows, which is consistent with a gradually higher dense phase of matter.</ce:simple-para></ce:abstract-sec></ce:abstract></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">The number of charged particles produced in energetic nuclear collisions is an important indicator for the strong interaction processes that determine the particle production at the sub-nucleonic level. In particular, the production of charged particles is expected to reflect the number of quark and gluon collisions occurring during the initial stages of the reaction. The total number of particles produced also provides information on the energy transfer available from the initial colliding beams to particle production, as a consequence of nuclear stopping <ce:cross-ref refid="br0010" id="crf10940">[1]</ce:cross-ref>. In order to help unravel this complex scenario it is important to compare the particle production amongst collision systems of different sizes over a wide kinematic range.</ce:para><ce:para id="pr0020">We present the measured charged-particle pseudorapidity density, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math>, for pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb (previously published <ce:cross-ref refid="br0020" id="crf10950">[2]</ce:cross-ref>) collisions at the same collision energy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> in the nucleon–nucleon centre-of-mass reference frame. This is, at present, the maximum available energy at CERN's Large Hadron Collider (LHC) for Pb<ce:glyph name="sbnd"/>Pb collisions. The measurements were carried out using ALICE at LHC (for earlier <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math> results see for example Refs. <ce:cross-refs refid="br0030 br0040 br0050" id="crs0010">[3–5]</ce:cross-refs>). The three studied reactions have different characteristics probing widely different particle production yields and mechanisms. In Pb<ce:glyph name="sbnd"/>Pb collisions, the total particle yield for central collisions is of the order 10<ce:sup>4</ce:sup> <ce:cross-ref refid="br0020" id="crf10960">[2]</ce:cross-ref>, and a strongly coupled plasma of quarks and gluons (sQGP) is formed <ce:cross-refs refid="br0060 br0070 br0080 br0090" id="crs0020">[6–9]</ce:cross-refs>, whose collective and transport properties are currently under intense study. On the other hand, pp collisions represent the simplest possible nuclear collision system, where the average total particle production is much smaller (≈80, by integrating the measured distributions), and is to first approximation much less subject to collective effects <ce:cross-ref refid="br0100" id="crf10970">[10]</ce:cross-ref>. The p<ce:glyph name="sbnd"/>Pb system is intermediate to the other reactions, corresponding to the situation where a single nucleon probes the nucleons in a narrow cylinder of the target nucleus. The extent to which p<ce:glyph name="sbnd"/>Pb is governed by the initial state cold nuclear matter of the lead ion or whether collective phenomena in the hot and dense medium play an important role is, at present, a matter under scrutiny by the community <ce:cross-refs refid="br0100 br0110" id="crs0030">[10,11]</ce:cross-refs>.</ce:para><ce:para id="pr0030">In this letter, we compare the three reactions and present the ratios of the charged-particle pseudorapidity density distributions (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math>) of the more complex reactions to the pp distribution. Owing to ALICE's unique large acceptance in pseudorapidity, and using simple and robust assumptions, we transform the measured charged-particle pseudorapidity density distributions into charged-particle rapidity density distributions (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math>). This allows us to calculate the width of the rapidity distributions as a function of the number of participating nucleons. The parameters of the transformation also allow us to estimate a lower bound on the energy density using the well-known formula from Bjorken <ce:cross-ref refid="br0120" id="crf10980">[12]</ce:cross-ref>. An energy density exceeding the critical energy density of roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> <ce:cross-ref refid="br0130" id="crf10990">[13]</ce:cross-ref> is a necessary condition for the formation of deconfined matter of quarks and gluons, and thus it is of the utmost interest to understand the development of these energy densities across different collision systems.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">Experimental set-up, data sample, analysis method, systematic uncertainties</ce:section-title><ce:para id="pr0040">A detailed description of the ALICE detector and its performance can be found elsewhere <ce:cross-refs refid="br0140 br0150" id="crs0040">[14,15]</ce:cross-refs>. The present analysis uses the Silicon Pixel Detector (SPD) to determine the pseudorapidity densities in the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math> and the Forward Multiplicity Detector (FMD) in the ranges <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.5</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1.8</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mn>1.8</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5</mml:mn></mml:math>. The V0, comprised of two plastic scintillator discs covering <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.7</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1.7</mml:mn></mml:math> (V0C) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mn>2.8</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5.1</mml:mn></mml:math> (V0A), and the ZDC, two zero-degree calorimeters located <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>112.5</mml:mn><mml:mspace width="0.2em"/><mml:mtext>m</mml:mtext></mml:math> from the interaction point, measurements determine the collision centrality and are used for offline event selection <ce:cross-ref refid="br0020" id="crf11000">[2]</ce:cross-ref>.</ce:para><ce:para id="pr0050">The results presented are based on data from collisions at a centre-of-mass energy per nucleon pair of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> as collected by ALICE during LHC Run 1 (2013) for p<ce:glyph name="sbnd"/>Pb, and during Run 2 (2015) for pp and Pb<ce:glyph name="sbnd"/>Pb. The FMD suffered high levels of background noise during the 2016 p<ce:glyph name="sbnd"/>Pb campaign, due to the high collision rate, and this data is therefore not used for the present analysis. About 10<ce:sup>5</ce:sup> events with a minimum bias trigger requirement <ce:cross-ref refid="br0020" id="crf11010">[2]</ce:cross-ref> were analysed in the centrality range from 0% to 90% and 0% to 100% of the visible cross section for Pb<ce:glyph name="sbnd"/>Pb and p<ce:glyph name="sbnd"/>Pb collisions, respectively. The minimum bias trigger for p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb collisions in ALICE was defined as a coincidence between the V0A and V0C sides of the V0 detector.</ce:para><ce:para id="pr0060">The data from the p<ce:glyph name="sbnd"/>Pb collisions were taken in two beam configurations: one where the lead ion travelled toward positive pseudorapidity and one where it travelled toward negative pseudorapidity. The results from the latter collisions are mirrored around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>. The centre-of-mass frame in p<ce:glyph name="sbnd"/>Pb collisions does not coincide with the laboratory frame, due to the single magnetic field in the LHC, and thus the rapidity of the centre-of-mass is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">CM</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>±</mml:mo><mml:mn>0.465</mml:mn></mml:math> for the two directions, respectively, in the laboratory frame. For this reason, pseudorapidity, calculated with respect to the laboratory frame, is denoted <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub></mml:math> whenever p<ce:glyph name="sbnd"/>Pb results are presented.</ce:para><ce:para id="pr0070">Likewise, for the pp collisions, about 10<ce:sup>5</ce:sup> events with coincidence between V0A and V0C and at least one charged particle in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1</mml:mn></mml:math> were analysed. By requiring at least one charged particle at midrapidity, the so-called INEL>0 event class, the systematic uncertainty, related to the absolute normalisation to the full inelastic cross section, is reduced, while still sampling a large fraction (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>75</mml:mn><mml:mtext>%</mml:mtext></mml:math>) of the hadronic cross section <ce:cross-refs refid="br0160 br0170" id="crs0050">[16,17]</ce:cross-refs>.</ce:para><ce:para id="pr0080">The standard ALICE event selection <ce:cross-ref refid="br0180" id="crf11020">[18]</ce:cross-ref> and centrality estimator based on the V0 amplitude <ce:cross-refs refid="br0190 br0200" id="crs0060">[19,20]</ce:cross-refs> are used in this analysis. The event selection consists of: a) exclusion of background events using the timing information from the ZDC (for Pb<ce:glyph name="sbnd"/>Pb and p<ce:glyph name="sbnd"/>Pb, e.g., beam–gas interactions) and V0 detectors, b) verification of the trigger conditions, and c) a reconstructed position of the collision (primary vertex). In Pb<ce:glyph name="sbnd"/>Pb collisions, centrality is obtained from the sum amplitude in both V0 detector arrays (V0M). For p<ce:glyph name="sbnd"/>Pb only the amplitude in the array on the lead-going side (V0A or V0C) is used. In Pb<ce:glyph name="sbnd"/>Pb collisions, the 10% most peripheral collisions have substantial contributions from electromagnetic processes and are therefore not included in the results presented here <ce:cross-ref refid="br0190" id="crf11030">[19]</ce:cross-ref>.</ce:para><ce:para id="pr0090">A primary charged particle is defined as a charged particle with a mean proper lifetime <ce:italic>τ</ce:italic> larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mtext>cm</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mtext>c</mml:mtext></mml:math>, which is either a) produced directly in the interaction, or b) from decays of particles with <ce:italic>τ</ce:italic> smaller than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mtext>cm</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mtext>c</mml:mtext></mml:math> <ce:cross-ref refid="br0210" id="crf11040">[21]</ce:cross-ref>. All quantities reported here are for primary, charged particles, though “primary” is omitted in the following for brevity.</ce:para><ce:para id="pr0100">The analysis method is identical to that of previous publications <ce:cross-ref refid="br0020" id="crf11050">[2]</ce:cross-ref>: the measurement of the charged-particle pseudorapidity density at midrapidity is obtained from counting particle trajectories determined using the two layers of the SPD. The SPD has a lower transverse momentum acceptance of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:mn>50</mml:mn><mml:mspace width="0.2em"/><mml:mtext>MeV</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="italic">c</mml:mi></mml:math>, and the yield is extrapolated down to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mtext>MeV</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="italic">c</mml:mi></mml:math> via simulations. In the forward regions, the measurement is provided by the analysis of the deposited energy signal in the FMD and a statistical method is employed to calculate the inclusive number of charged particles. A data-driven correction <ce:cross-ref refid="br0220" id="crf11060">[22]</ce:cross-ref>, based on separate measurements exploiting displaced collision vertices, is applied to remove the background from secondary particles.</ce:para><ce:para id="pr0110">Systematic uncertainty estimations for the midrapidity measurements are detailed elsewhere <ce:cross-refs refid="br0020 br0160 br0200" id="crs0070">[2,16,20]</ce:cross-refs>, and are from background suppression, transverse momentum extrapolation, weak decays, and simulations. The estimates are obtained through variation of thresholds and simulation studies. For pp (p<ce:glyph name="sbnd"/>Pb), the total systematic uncertainty amounts to 1.5% (2.7%) over the whole pseudorapidity range; while for Pb<ce:glyph name="sbnd"/>Pb the total systematic uncertainty is 2.6% at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> and 2.9% at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2</mml:mn></mml:math>. The systematic uncertainty is mostly correlated over pseudorapidity for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math>, and largely independent of centrality. The uncertainty in the forward region, estimated via variations of thresholds and simulation studies, is the same for all collision systems and is uncorrelated across <ce:italic>η</ce:italic>, amounting to 6.9% for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>3.5</mml:mn></mml:math> and 6.4% elsewhere within the forward regions <ce:cross-ref refid="br0220" id="crf11070">[22]</ce:cross-ref>. In the figures of this letter, uncorrelated, local in pseudorapidity, systematic uncertainties are indicated by open boxes on the data points, while correlated systematic uncertainties, those that affect the overall scale and typically from event classification and selection, are indicated by filled boxes to the right of the data. The systematic uncertainty on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math>, due to the centrality class definition in Pb<ce:glyph name="sbnd"/>Pb, is estimated to vary from 0.6% for the most central to 9.5% for the most peripheral class <ce:cross-ref refid="br0230" id="crf11080">[23]</ce:cross-ref>. The 80% to 90% centrality class has residual contamination from electromagnetic processes as detailed elsewhere <ce:cross-ref refid="br0190" id="crf11090">[19]</ce:cross-ref>, which gives rise to an additional 4% systematic uncertainty in the measurements. No overall systematic uncertainty has been estimated for p<ce:glyph name="sbnd"/>Pb collisions, as the centrality selection in that collision system is inherently difficult to map to the underlying dynamics of the collisions <ce:cross-ref refid="br0200" id="crf11100">[20]</ce:cross-ref>.</ce:para></ce:section><ce:section id="se0030" role="results"><ce:label>3</ce:label><ce:section-title id="st0040">Results</ce:section-title><ce:para id="pr0120"><ce:cross-ref refid="fg0010" id="crf11420">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/> shows the measured pseudorapidity densities in pp, and in central p<ce:glyph name="sbnd"/>Pb, and the previously published results for Pb<ce:glyph name="sbnd"/>Pb <ce:cross-ref refid="br0020" id="crf11120">[2]</ce:cross-ref> collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> for primary particles.</ce:para><ce:para id="pr0130">For the 5% most central Pb<ce:glyph name="sbnd"/>Pb collisions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:mo>≈</mml:mo><mml:mn>2000</mml:mn></mml:math> at midrapidity (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>) <ce:cross-ref refid="br0020" id="crf11130">[2]</ce:cross-ref>, while for p<ce:glyph name="sbnd"/>Pb collisions the distribution peaks at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mn>60</mml:mn></mml:math> around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>3</mml:mn></mml:math> in the lead-going direction (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>). For pp collisions with the INEL>0 trigger condition discussed above, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.7</mml:mn><mml:mo>±</mml:mo><mml:mn>0.2</mml:mn></mml:math> at midrapidity, consistent with previous results derived from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra <ce:cross-ref refid="br0240" id="crf11140">[24]</ce:cross-ref>.</ce:para><ce:para id="pr0140"><ce:cross-ref refid="fg0020" id="crf11430">Fig. 2</ce:cross-ref><ce:float-anchor refid="fg0020"/> shows, as a function of centrality, the measured charged-particle pseudorapidity densities for p<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>. The strategy of centrality selection for proton on nucleus reactions is explained elsewhere <ce:cross-ref refid="br0200" id="crf11160">[20]</ce:cross-ref>. The ALICE Collaboration has previously presented <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math> for Pb<ce:glyph name="sbnd"/>Pb collisions at this energy <ce:cross-ref refid="br0020" id="crf11170">[2]</ce:cross-ref>.</ce:para><ce:para id="pr0150">In <ce:cross-ref refid="fg0030" id="crf11180">Fig. 3</ce:cross-ref><ce:float-anchor refid="fg0030"/>, the charged-particle pseudorapidity densities in p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb reactions are divided by the pp distributions corresponding to the INEL>0 trigger class. The ratio is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pp</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>, where <ce:italic>X</ce:italic> labels p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb collisions, in centrality classes, as a function of pseudorapidity. In the ratios, systematic uncertainties, of common origin, are partially cancelled, and, as an estimate, the magnitude of the resulting systematic uncertainties are given only by the uncertainties in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:math> measurements, since the uncertainties are independent of the collision system. In p<ce:glyph name="sbnd"/>Pb collisions the rapidity of the centre-of-mass is non-zero, which is not taken into account in the ratios. Such a correction would require prior determination of the full Jacobian of the transformation from pseudorapidity to rapidity, which is not possible to perform reliably with the ALICE apparatus.</ce:para><ce:para id="pr0160">The ratio of the p<ce:glyph name="sbnd"/>Pb relative to the pp distributions increases with pseudorapidity from the p-going to the Pb-going direction for central collisions, which Brodsky et al. and Adil et al. <ce:cross-refs refid="br0250 br0260" id="crs0080">[25,26]</ce:cross-refs> suggest is a sign of scaling of the pp distribution with the increasing number of participants as the lead nucleus is probed by the incident proton, and thus independent proton–nucleon scatterings on the lead-ion side. A similar scaling, however, does not hold for the Pb<ce:glyph name="sbnd"/>Pb reaction. The ratios cannot be obtained by simple scaling of the elementary pp distributions. Instead, the ratio of the Pb<ce:glyph name="sbnd"/>Pb relative to the pp distributions exhibits an enhancement of particle production around midrapidity for the more central collisions which is indicative of the formation of the sQGP <ce:cross-ref refid="br0070" id="crf11190">[7]</ce:cross-ref>. Likewise, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="normal">pPb</mml:mtext></mml:mrow></mml:msub></mml:math> increases for all but the two most peripheral centrality classes as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:mn>3</mml:mn></mml:math>. In Pb<ce:glyph name="sbnd"/>Pb collisions it is seen that the various mechanisms behind the pseudorapidity distributions are more transversely directed than in pp collisions by the increase of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="normal">PbPb</mml:mtext></mml:mrow></mml:msub></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mn>0</mml:mn></mml:math></ce:para></ce:section><ce:section id="se0040"><ce:label>4</ce:label><ce:section-title id="st0050">Rapidity and energy-density dependence on system size and discussion</ce:section-title><ce:para id="pr0170">It has been shown that the charged-particle <ce:italic>rapidity</ce:italic> density (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math>) in Pb<ce:glyph name="sbnd"/>Pb collisions, to a good accuracy, follows a normal distribution over the considered rapidity interval (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo>≲</mml:mo><mml:mn>5</mml:mn></mml:math>) <ce:cross-refs refid="br0020 br0270" id="crs0090">[2,27]</ce:cross-refs>. Those results relied on calculating the average Jacobian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>J</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math> using the full <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra, at midrapidity, of charged pions and kaons as well as protons and antiprotons. Here, we use the approximation<ce:display><ce:formula id="fm0010"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mrow><mml:mi>y</mml:mi><mml:mo>≈</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">cos</mml:mi><mml:mo>⁡</mml:mo><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>ϑ</ce:italic> is the polar angle of emission, and identify <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mi>a</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>m</mml:mi></mml:math> with an effective ratio of transverse momentum over mass. With this, the effective Jacobian can be written as<ce:display><ce:formula id="fm0020"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">cosh</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>⁡</mml:mo><mml:mi>η</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0180">We further make the ansatz that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> is normal distributed for symmetric collision systems (pp and Pb<ce:glyph name="sbnd"/>Pb), so that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math> can be parameterised as<ce:display><ce:formula id="fm0030"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mi>A</mml:mi><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:msqrt><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">{</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>A</ce:italic> and <ce:italic>σ</ce:italic> are the total integral and width of the distribution, respectively, and <ce:italic>y</ce:italic> the rapidity in the centre-of-mass frame. Motivated by the observed approximate linearity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pPb</mml:mi></mml:mrow></mml:msub></mml:math> (see lower panel of <ce:cross-ref refid="fg0030" id="crf11200">Fig. 3</ce:cross-ref>), we replace <ce:italic>A</ce:italic> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mi>y</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> for the asymmetric system (p<ce:glyph name="sbnd"/>Pb) and parameterise <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub></mml:math> as<ce:display><ce:formula id="fm0040"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo id="mmlbr0001" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>α</mml:mi><mml:mi>y</mml:mi><mml:mo stretchy="false">{</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="true" linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="after">)</mml:mo></mml:mrow><mml:mspace width="1em"/><mml:mo>×</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:msqrt><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">{</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">CM</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0190">The functions <ce:italic>f</ce:italic> and <ce:italic>g</ce:italic> defined in Eq. <ce:cross-ref refid="fm0030" id="crf11210">(1)</ce:cross-ref> and Eq. <ce:cross-ref refid="fm0040" id="crf11220">(2)</ce:cross-ref>, respectively, describe the measurements within the measured region with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> per degrees of freedom (<ce:italic>ν</ce:italic>) in the range of 0.1 to 0.5. The small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mi>ν</mml:mi></mml:math> values are a consequence of the relatively large uncorrelated systematic uncertainties on the measurements. That is, the charged-particle distributions for pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> follow a normal distribution in rapidity, with free parameters <ce:italic>A</ce:italic>, <ce:italic>a</ce:italic>, <ce:italic>σ</ce:italic>, and <ce:italic>α</ce:italic> in the asymmetric case.</ce:para><ce:para id="pr0200">The top panel of <ce:cross-ref refid="fg0040" id="crf11230">Fig. 4</ce:cross-ref><ce:float-anchor refid="fg0040"/> shows the best-fit parameter values of the normal width (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math>) for all three collision systems as a function of the average number of participating nucleons (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo></mml:math>) calculated using a Glauber model <ce:cross-ref refid="br0280" id="crf11240">[28]</ce:cross-ref>. The best-fit parameters are found taking statistical and uncorrelated systematic uncertainties into account. The result using the above procedure, for the most central Pb<ce:glyph name="sbnd"/>Pb collisions, is found to be compatible with previous results extracted by unfolding with the mean Jacobian estimated from transverse momentum spectra <ce:cross-ref refid="br0020" id="crf11250">[2]</ce:cross-ref>. The open points (crosses) and dashed lines on the figure are from evaluations of Eq. <ce:cross-ref refid="fm0030" id="crf11260">(1)</ce:cross-ref> and Eq. <ce:cross-ref refid="fm0040" id="crf11270">(2)</ce:cross-ref>, and direct calculations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math>, respectively, using model calculations with EPOS-LHC <ce:cross-ref refid="br0290" id="crf11280">[29]</ce:cross-ref>. EPOS-LHC was chosen as it provides predictions for all three collision systems. The parameterisation, in terms of the two functions, of this model calculation generally reproduces the widths of the charged-particle rapidity densities, except in the asymmetric case where a direct evaluation of the standard deviation is less motivated.</ce:para><ce:para id="pr0210">The general trend is that the widths decrease as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo></mml:math> increases, consistent with the behaviour of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="normal">PbPb</mml:mtext></mml:mrow></mml:msub></mml:math> ratios. Notably, the width of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> distributions in p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb, for low number of participant nucleons in the collisions, approaches the width of the pp distribution, which, presumably, is dominated by kinematic and phase space constraints.</ce:para><ce:para id="pr0220">The lower panel of <ce:cross-ref refid="fg0040" id="crf11290">Fig. 4</ce:cross-ref> shows the dependence of <ce:italic>a</ce:italic> on the average number of participants. The right-hand ordinate is the same, but multiplied by the average mass <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>0.215</mml:mn><mml:mo>±</mml:mo><mml:mn>0.001</mml:mn><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mtext>GeV</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="italic">c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> estimated from measurements of identified particles in Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> <ce:cross-ref refid="br0300" id="crf11300">[30]</ce:cross-ref>. To better understand the parameter <ce:italic>a</ce:italic>, this parameter extracted from the EPOS-LHC calculations, using the above procedure, is also shown in the figure. The dotted lines show the average <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>m</mml:mi></mml:math> predicted by EPOS-LHC <ce:cross-ref refid="br0290" id="crf11310">[29]</ce:cross-ref>. The EPOS-LHC calculations indicate that the extracted effective transverse momentum to mass ratio <ce:italic>a</ce:italic> is consistently smaller than the ratio of the average transverse momentum to the average mass. Thus <ce:italic>a</ce:italic> gives a lower bound on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math>.</ce:para><ce:para id="pr0230">We can estimate the energy density that is reached in the collisions as a function of the number of participants for the three systems. A conventional approach is to use the model originally proposed by Bjorken <ce:cross-ref refid="br0120" id="crf11320">[12]</ce:cross-ref> in which the energy density (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub></mml:math>) depends on the rapidity density of particles and the volume of a longitudinal cylinder with cross sectional area determined by the overlap between the colliding partners and length determined by a characteristic particle formation time<ce:display><ce:formula id="fm0050"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>τ</mml:mi></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">〈</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">〉</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Here, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mi>π</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>≈</mml:mo><mml:mi>π</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msubsup></mml:math> is the transverse area spanned by the participating nucleons, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> is the transverse-energy rapidity density, and <ce:italic>τ</ce:italic> is the formation time. While a formation time of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:mi>τ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">c</mml:mi></mml:math> is often assumed, it is left as a free parameter here. With <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt></mml:math>, the transverse-energy rapidity density can be approximated by<ce:display><ce:formula id="fm0060"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:mrow><mml:mrow><mml:mo stretchy="true">〈</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">〉</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0.55</mml:mn><mml:mo>±</mml:mo><mml:mn>0.01</mml:mn></mml:math>, the ratio of charged particles to all particles <ce:cross-ref refid="br0310" id="crf11330">[31]</ce:cross-ref>, accounts for neutral particles not measured in the experiment, and is assumed the same for all collision systems. Substituting the derived <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> and the effective <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mi>a</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>m</mml:mi><mml:mo>≲</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math> results in a lower bound estimate for the Bjorken energy density (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">LB</mml:mi></mml:mrow></mml:msub></mml:math>)<ce:display><ce:formula id="fm0070"><ce:label>(3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi><mml:mo>≥</mml:mo><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">LB</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">cosh</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>⁡</mml:mo><mml:mi>η</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>a</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math> are as in the top panel of <ce:cross-ref refid="fg0040" id="crf11340">Fig. 4</ce:cross-ref>.</ce:para><ce:para id="pr0240">The transverse area <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> is estimated in a numerical Glauber model <ce:cross-refs refid="br0320 br0330" id="crs0100">[32,33]</ce:cross-refs> as shown in <ce:cross-ref refid="fg0050" id="crf11350">Fig. 5</ce:cross-ref><ce:float-anchor refid="fg0050"/>. We consider two extremes for the transverse area spanned by the participating nucleons: a) the <ce:italic>exclusive</ce:italic> (or direct) overlap between participating nucleons, ∩ and open markers in <ce:cross-ref refid="fg0050" id="crf11360">Fig. 5</ce:cross-ref>, and b) the <ce:italic>inclusive</ce:italic> (or full) area of all participating nucleons, ∪ and full markers in <ce:cross-ref refid="fg0050" id="crf11370">Fig. 5</ce:cross-ref>.</ce:para><ce:para id="pr0250"><ce:cross-ref refid="fg0060" id="crf11440">Fig. 6</ce:cross-ref><ce:float-anchor refid="fg0060"/> shows the lower-bound energy density estimate, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">LB</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi></mml:math>, as a function of the number of participants, which reaches values between 10 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:mn>20</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> in the most central Pb<ce:glyph name="sbnd"/>Pb collisions. The uncertainties are from standard error propagation of Eq. <ce:cross-ref refid="fm0070" id="crf11390">(3)</ce:cross-ref> of uncertainties on the best-fit parameter values, the number of participants, mean mass, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:math>. A rise from roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si79.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> to over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:mn>10</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is observed if the transverse area is assumed to be the inclusive area of participating nucleons. This trend is illustrated by a power-law (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si81.svg"><mml:mi>C</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msubsup></mml:math>) fit to the data in the figure, with the parameter values <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si82.svg"><mml:mi>C</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>0.8</mml:mn><mml:mo>±</mml:mo><mml:mn>0.3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi>p</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0.44</mml:mn><mml:mo>±</mml:mo><mml:mn>0.08</mml:mn></mml:math>. On the other hand, if the transverse area is assumed to be the smaller exclusive overlap area, we observe a substantially larger lower bound on the energy density, but a less dramatic increase with increasing number of participating nucleons. Also shown in the figure are estimates of the Bjorken energy density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi></mml:math> for Pb<ce:glyph name="sbnd"/>Pb reactions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> <ce:cross-ref refid="br0310" id="crf11400">[31]</ce:cross-ref>. These results where obtained from measurements of the transverse energy in the collisions and using the inclusive estimate of the transverse area <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>. The trend of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> results is similar to these earlier results. Bearing in mind that for the largest LHC collision energy we show a lower bound estimate of the energy density in <ce:cross-ref refid="fg0060" id="crf11410">Fig. 6</ce:cross-ref>, we find a likely overall increase in the energy density from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>.</ce:para></ce:section><ce:section id="se0050"><ce:label>5</ce:label><ce:section-title id="st0060">Summary and conclusions</ce:section-title><ce:para id="pr0260">We have measured the charged particle pseudorapidity density in pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> over the widest possible pseudorapidity range available at the LHC. The distributions where determined using the same experimental apparatus and methods, and systematic uncertainties have been minimised to within the capabilities of the set-up. While the particle production in central Pb<ce:glyph name="sbnd"/>Pb collisions clearly exhibits an enhancement as compared to pp collisions, particle production in p<ce:glyph name="sbnd"/>Pb collisions is consistent with dominantly incoherent nucleon–nucleon collisions. By transforming the measured pseudorapidity distributions to rapidity distributions we have obtained systematic trends for the width of the rapidity distributions and a lower bound on the energy density, which shows a clear scaling behaviour as a function of the average number of participant nucleons. The decreasing width of the deduced rapidity distributions with increasing participant number suggests that the kinematic spread of particles, including longitudinal degrees of freedom, is reduced due to interactions in the early stages of the collisions. This is also reflected in the accompanying growth of the energy density. Both observations are consistent with the gradual establishment of a high-density phase of matter with increasing size of the collision domain.</ce:para></ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0080">Declaration of Competing Interest</ce:section-title><ce:para id="pr0270">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0090">Acknowledgements</ce:section-title><ce:para id="pr0280">The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: <ce:grant-sponsor id="gsp0010">A. I. 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<ce:grant-sponsor id="gsp0690" sponsor-id="https://doi.org/10.13039/100020381">Turkish Energy, Nuclear and Mineral Research Agency</ce:grant-sponsor> (TENMAK), Turkey; <ce:grant-sponsor id="gsp0700" sponsor-id="https://doi.org/10.13039/501100004742">National Academy of Sciences of Ukraine</ce:grant-sponsor>, Ukraine; <ce:grant-sponsor id="gsp0710" sponsor-id="https://doi.org/10.13039/501100000271">Science and Technology Facilities Council</ce:grant-sponsor> (STFC), United Kingdom; National Science Foundation of the United States of America (<ce:grant-sponsor id="gsp0720" sponsor-id="https://doi.org/10.13039/100000001">NSF</ce:grant-sponsor>) and United States Department of Energy, Office of Nuclear Physics (<ce:grant-sponsor id="gsp0730" sponsor-id="https://doi.org/10.13039/100006209">DOE NP</ce:grant-sponsor>), United States of America.</ce:para></ce:acknowledgment></body><tail><ce:bibliography id="bl0010"><ce:section-title id="st0070">References</ce:section-title><ce:bibliography-sec id="bs0010"><ce:bib-reference id="br0010"><ce:label>[1]</ce:label><sb:reference id="bib866BD36D5E8AFE73B567F49DDE14CD65s1"><sb:contribution><sb:authors><sb:collaboration>BRAHMS Collaboration</sb:collaboration><sb:author><ce:given-name>I.C.</ce:given-name><ce:surname>Arsene</ce:surname></sb:author><sb:et-al/></sb:authors><sb:title><sb:maintitle>Nuclear stopping and rapidity loss in Au+Au collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>62.4</mml:mn><mml:mspace width="0.2em"/><mml:mtext>GeV</mml:mtext></mml:math></sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Phys. 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C94 (2016) 024914, arXiv:1603.07375 [nucl-ex].</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> \ No newline at end of file +<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd" [<!ENTITY gr001 SYSTEM "gr001" NDATA IMAGE><!ENTITY gr002 SYSTEM "gr002" NDATA IMAGE><!ENTITY gr003 SYSTEM "gr003" NDATA IMAGE><!ENTITY gr004 SYSTEM "gr004" NDATA IMAGE><!ENTITY gr005 SYSTEM "gr005" NDATA IMAGE><!ENTITY gr006 SYSTEM "gr006" NDATA IMAGE>]><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="sco" xml:lang="en"><item-info><jid>PLB</jid><aid>137730</aid><ce:article-number>137730</ce:article-number><ce:pii>S0370-2693(23)00064-3</ce:pii><ce:doi>10.1016/j.physletb.2023.137730</ce:doi><ce:copyright year="2023" type="other">European Center of Nuclear Research, ALICE experiment</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>Experiments</ce:text></ce:doctopic></ce:doctopics><ce:preprint><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/preprint" xlink:href="arxiv:2204.10210" id="inf0010"/></ce:preprint></item-info><ce:floats><ce:figure id="fg0010"><ce:label>Fig. 1</ce:label><ce:caption id="cp0010"><ce:simple-para id="sp0010">Charged-particle pseudorapidity density in Pb<ce:glyph name="sbnd"/>Pb <ce:cross-ref refid="br0020" id="crf0010">[2]</ce:cross-ref> and p<ce:glyph name="sbnd"/>Pb for the 5% most central collisions, and for pp collisions with INEL>0 trigger class. For symmetric collision systems (Pb<ce:glyph name="sbnd"/>Pb and pp) the data has been symmetrised around <ce:italic>η</ce:italic> = 0 and points for <ce:italic>η</ce:italic> > 3.5 have been reflected around <ce:italic>η</ce:italic> = 0. The boxes around the points and to the right reflect the uncorrelated and correlated, with respect to pseudorapidity, systematic uncertainty, respectively. The relative correlated, normalisation, uncertainties are evaluated at d<ce:italic>N</ce:italic><ce:inf>ch</ce:inf>/d<ce:italic>η</ce:italic>|<ce:inf><ce:italic>η</ce:italic>=0</ce:inf>. The lines show fits of Eq. <ce:cross-ref refid="fm0030" id="crf0020">(1)</ce:cross-ref> (Pb<ce:glyph name="sbnd"/>Pb and pp) and Eq. <ce:cross-ref refid="fm0040" id="crf0030">(2)</ce:cross-ref> (p<ce:glyph name="sbnd"/>Pb) to the data (discussed in Section <ce:cross-ref refid="se0040" id="crf0040">4</ce:cross-ref>). Please note that the ordinate has been cut twice to accommodate for the very different ranges of the charged-particle pseudorapidity densities.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0010">Fig. 1</ce:alt-text><ce:link locator="gr001" xlink:type="simple" xlink:href="pii:S0370269323000643/gr001" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0010"/></ce:figure><ce:figure id="fg0020"><ce:label>Fig. 2</ce:label><ce:caption id="cp0020"><ce:simple-para id="sp0020">Charged-particle pseudorapidity density in p<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> in seven centrality classes based on the V0A and V0C estimators. The lines are obtained using a fit of a scaled, normal distribution in rapidity Eq. <ce:cross-ref refid="fm0040" id="crf0050">(2)</ce:cross-ref> to the data (discussed in Section <ce:cross-ref refid="se0040" id="crf0060">4</ce:cross-ref>).</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0020">Fig. 2</ce:alt-text><ce:link locator="gr002" xlink:type="simple" xlink:href="pii:S0370269323000643/gr002" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0020"/></ce:figure><ce:figure id="fg0030"><ce:label>Fig. 3</ce:label><ce:caption id="cp0030"><ce:simple-para id="sp0030">Ratio <ce:italic>r</ce:italic><ce:inf><ce:italic>X</ce:italic></ce:inf> of the charged-particle pseudorapidity density in Pb<ce:glyph name="sbnd"/>Pb (top) and p<ce:glyph name="sbnd"/>Pb (bottom) in different centrality classes to the charged-particle pseudorapidity density in pp in the INEL>0 event class. Note, for Pb<ce:glyph name="sbnd"/>Pb <ce:italic>η</ce:italic><ce:inf>lab</ce:inf> is the same as the centre-of-mass pseudorapidity.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0030">Fig. 3</ce:alt-text><ce:link locator="gr003" xlink:type="simple" xlink:href="pii:S0370269323000643/gr003" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0030"/></ce:figure><ce:figure id="fg0040"><ce:label>Fig. 4</ce:label><ce:caption id="cp0040"><ce:simple-para id="sp0040">The width (top) and effective <ce:italic>p</ce:italic><ce:inf>T</ce:inf>/<ce:italic>m</ce:italic> (bottom) fit parameters as a function of the mean number of participants in pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>. Vertical uncertainties are the standard error on the best-fit parameter values, while horizontal uncertainties reflect the uncertainty on 〈<ce:italic>N</ce:italic><ce:inf>part</ce:inf>〉 from the Glauber calculations. Also shown are similar fit parameters from the same parameterisation of EPOS-LHC calculations as well as direct calculations of the standard deviation of the d<ce:italic>N</ce:italic><ce:inf>ch</ce:inf>/d<ce:italic>y</ce:italic> distributions and the 〈<ce:italic>p</ce:italic><ce:inf>T</ce:inf>〉/〈<ce:italic>m</ce:italic>〉 ratio from the EPOS-LHC calculations.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0040">Fig. 4</ce:alt-text><ce:link locator="gr004" xlink:type="simple" xlink:href="pii:S0370269323000643/gr004" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0040"/></ce:figure><ce:figure id="fg0050"><ce:label>Fig. 5</ce:label><ce:caption id="cp0050"><ce:simple-para id="sp0050">The transverse area <ce:italic>S</ce:italic><ce:inf>T</ce:inf> as calculated in a numerical Glauber model for two extreme cases: a) only the exclusive overlap of nucleons is considered (∩, open markers) and b) the inclusive area of participating nucleons contribute (∪, closed markers) in both p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0050">Fig. 5</ce:alt-text><ce:link locator="gr005" xlink:type="simple" xlink:href="pii:S0370269323000643/gr005" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0050"/></ce:figure><ce:figure id="fg0060"><ce:label>Fig. 6</ce:label><ce:caption id="cp0060"><ce:simple-para id="sp0060">Estimate of the lower bound on the Bjorken transverse energy density in pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>, considering the exclusive (∩, open markers) and inclusive (∪, full markers) overlap area <ce:italic>S</ce:italic><ce:inf>T</ce:inf> of the nucleons. The expression <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:mi>C</mml:mi><mml:mmultiscripts><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow><mml:none/><mml:none/><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:mmultiscripts></mml:math> is fitted to case ∪, and we find <ce:italic>C</ce:italic> = (0.8 ± 0.3) GeV/(fm<ce:sup>2</ce:sup><ce:italic>c</ce:italic>) and <ce:italic>p</ce:italic> = 0.44 ± 0.08. Also shown is an estimate, via d<ce:italic>E</ce:italic><ce:inf>T</ce:inf>/d<ce:italic>y</ce:italic>, of <ce:italic>ε</ce:italic><ce:inf>Bj</ce:inf> from Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> (stars with uncertainty band) <ce:cross-ref refid="br0310" id="crf0070">[31]</ce:cross-ref>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0060">Fig. 6</ce:alt-text><ce:link locator="gr006" xlink:type="simple" xlink:href="pii:S0370269323000643/gr006" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0060"/></ce:figure></ce:floats><head><ce:title id="ti0010">System-size dependence of the charged-particle pseudorapidity density at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> for pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions</ce:title><ce:author-group id="ag0010"><ce:collaboration id="co0010" collaboration-id="S0370269323000643-3bea72599603117cd9d18494a0279c47"><ce:text>ALICE Collaboration</ce:text><ce:cross-ref refid="fn0080" id="crf0080"><ce:sup>⋆</ce:sup></ce:cross-ref><ce:author-group id="ag0020"><ce:author id="au0010" author-id="S0370269323000643-e60c93a934b81cf9801254193264c6ee"><ce:given-name>S.</ce:given-name><ce:surname>Acharya</ce:surname><ce:cross-ref 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id="crf0290"><ce:sup>14</ce:sup></ce:cross-ref></ce:author><ce:author id="au0220" author-id="S0370269323000643-ee0688e61ecfdd140d9fea489d16b4dc"><ce:given-name>A.</ce:given-name><ce:surname>Alici</ce:surname><ce:cross-ref refid="aff0250" id="crf0300"><ce:sup>25</ce:sup></ce:cross-ref></ce:author><ce:author id="au0230" author-id="S0370269323000643-16d05696cd8ce792318dff507d143cd5"><ce:given-name>N.</ce:given-name><ce:surname>Alizadehvandchali</ce:surname><ce:cross-ref refid="aff1250" id="crf0310"><ce:sup>125</ce:sup></ce:cross-ref></ce:author><ce:author id="au0240" author-id="S0370269323000643-8a6281092995529070f381a78dafb967"><ce:given-name>A.</ce:given-name><ce:surname>Alkin</ce:surname><ce:cross-ref refid="aff0340" id="crf0320"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au0250" author-id="S0370269323000643-6ea637f099a6fac803f976a052eb847d"><ce:given-name>J.</ce:given-name><ce:surname>Alme</ce:surname><ce:cross-ref refid="aff0210" id="crf0330"><ce:sup>21</ce:sup></ce:cross-ref></ce:author><ce:author id="au0260" author-id="S0370269323000643-cfed79dabbd809c4f840372b7b333691"><ce:given-name>G.</ce:given-name><ce:surname>Alocco</ce:surname><ce:cross-ref refid="aff0550" id="crf0340"><ce:sup>55</ce:sup></ce:cross-ref></ce:author><ce:author id="au0270" author-id="S0370269323000643-3913f92e05be5c8a4269310703479917"><ce:given-name>T.</ce:given-name><ce:surname>Alt</ce:surname><ce:cross-ref refid="aff0680" id="crf0350"><ce:sup>68</ce:sup></ce:cross-ref></ce:author><ce:author id="au0280" author-id="S0370269323000643-cd55dbcf87db947368b6d9e810290bde"><ce:given-name>I.</ce:given-name><ce:surname>Altsybeev</ce:surname><ce:cross-ref refid="aff1130" id="crf0360"><ce:sup>113</ce:sup></ce:cross-ref></ce:author><ce:author id="au0290" author-id="S0370269323000643-c2bd71abda2e1783ed29485409a5251a"><ce:given-name>M.N.</ce:given-name><ce:surname>Anaam</ce:surname><ce:cross-ref refid="aff0070" 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id="crf1230"><ce:sup>94</ce:sup></ce:cross-ref></ce:author><ce:author id="au1120" author-id="S0370269323000643-5930588886d1890a60dc3e5c0b9a8baa"><ce:given-name>S.</ce:given-name><ce:surname>Boi</ce:surname><ce:cross-ref refid="aff0220" id="crf1240"><ce:sup>22</ce:sup></ce:cross-ref></ce:author><ce:author id="au1130" author-id="S0370269323000643-75ba81153caecc42ab057cd4528fbddd"><ce:given-name>J.</ce:given-name><ce:surname>Bok</ce:surname><ce:cross-ref refid="aff0610" id="crf1250"><ce:sup>61</ce:sup></ce:cross-ref></ce:author><ce:author id="au1140" author-id="S0370269323000643-7d461e77b3a4750af82052f87161d9b7"><ce:given-name>L.</ce:given-name><ce:surname>Boldizsár</ce:surname><ce:cross-ref refid="aff1460" id="crf1260"><ce:sup>146</ce:sup></ce:cross-ref></ce:author><ce:author id="au1150" author-id="S0370269323000643-2000ecef4090512918d1c03b88f7454b"><ce:given-name>A.</ce:given-name><ce:surname>Bolozdynya</ce:surname><ce:cross-ref refid="aff0940" 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author-id="S0370269323000643-e91fff0426cfb4068164293072959951"><ce:given-name>P.</ce:given-name><ce:surname>Dhankher</ce:surname><ce:cross-ref refid="aff0190" id="crf2480"><ce:sup>19</ce:sup></ce:cross-ref></ce:author><ce:author id="au2220" author-id="S0370269323000643-87d8feae9bb9ac0142d20aa737fc7210"><ce:given-name>D.</ce:given-name><ce:surname>Di Bari</ce:surname><ce:cross-ref refid="aff0330" id="crf2490"><ce:sup>33</ce:sup></ce:cross-ref></ce:author><ce:author id="au2230" author-id="S0370269323000643-807d3d4d1b245891296e345324c32d68"><ce:given-name>A.</ce:given-name><ce:surname>Di Mauro</ce:surname><ce:cross-ref refid="aff0340" id="crf2500"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au2240" author-id="S0370269323000643-b81ce1a8f6b1bb7201e6f62a1a0d3f6d"><ce:given-name>R.A.</ce:given-name><ce:surname>Diaz</ce:surname><ce:cross-ref refid="aff0750" id="crf2510"><ce:sup>75</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0080" 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id="crf3090"><ce:sup>136</ce:sup></ce:cross-ref></ce:author><ce:author id="au2800" author-id="S0370269323000643-cb179d5175a767cfcccfaf42e2d13fd9"><ce:given-name>U.</ce:given-name><ce:surname>Fuchs</ce:surname><ce:cross-ref refid="aff0340" id="crf3100"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au2810" author-id="S0370269323000643-75b81f7c708cdac6972396f07f1e615c"><ce:given-name>N.</ce:given-name><ce:surname>Funicello</ce:surname><ce:cross-ref refid="aff0290" id="crf3110"><ce:sup>29</ce:sup></ce:cross-ref></ce:author><ce:author id="au2820" author-id="S0370269323000643-e99988272e94b543b10e6e1e6762c915"><ce:given-name>C.</ce:given-name><ce:surname>Furget</ce:surname><ce:cross-ref refid="aff0790" id="crf3120"><ce:sup>79</ce:sup></ce:cross-ref></ce:author><ce:author id="au2830" author-id="S0370269323000643-466f51a3a42f868eb651094379830769"><ce:given-name>A.</ce:given-name><ce:surname>Furs</ce:surname><ce:cross-ref refid="aff0630" 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id="crf3170"><ce:sup>138</ce:sup></ce:cross-ref></ce:author><ce:author id="au2880" author-id="S0370269323000643-23c91ffad460f5afabddb04373c9f2f9"><ce:given-name>C.D.</ce:given-name><ce:surname>Galvan</ce:surname><ce:cross-ref refid="aff1200" id="crf3180"><ce:sup>120</ce:sup></ce:cross-ref></ce:author><ce:author id="au2890" author-id="S0370269323000643-8650c037fc955510cd76a4536ac83b03"><ce:given-name>P.</ce:given-name><ce:surname>Ganoti</ce:surname><ce:cross-ref refid="aff0850" id="crf3190"><ce:sup>85</ce:sup></ce:cross-ref></ce:author><ce:author id="au2900" author-id="S0370269323000643-d12fc16d28f1dce28adbdb23a5ddbcd4"><ce:given-name>C.</ce:given-name><ce:surname>Garabatos</ce:surname><ce:cross-ref refid="aff1080" id="crf3200"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au2910" author-id="S0370269323000643-df265a31acafb0a0d83f8ac8f13667e6"><ce:given-name>J.R.A.</ce:given-name><ce:surname>Garcia</ce:surname><ce:cross-ref refid="aff0450" id="crf3210"><ce:sup>45</ce:sup></ce:cross-ref></ce:author><ce:author id="au2920" author-id="S0370269323000643-07a4057e80a82b458aafc1e9dd90f0ba"><ce:given-name>E.</ce:given-name><ce:surname>Garcia-Solis</ce:surname><ce:cross-ref refid="aff0100" id="crf3220"><ce:sup>10</ce:sup></ce:cross-ref></ce:author><ce:author id="au2930" author-id="S0370269323000643-8f3671ab888761876ad72f8897499a10"><ce:given-name>K.</ce:given-name><ce:surname>Garg</ce:surname><ce:cross-ref refid="aff1150" id="crf3230"><ce:sup>115</ce:sup></ce:cross-ref></ce:author><ce:author id="au2940" author-id="S0370269323000643-987fe2eab608df9880768b4abb040eae"><ce:given-name>C.</ce:given-name><ce:surname>Gargiulo</ce:surname><ce:cross-ref refid="aff0340" id="crf3240"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au2950" author-id="S0370269323000643-0a421258e4c2e68267cbf36b6e5c0d5b"><ce:given-name>A.</ce:given-name><ce:surname>Garibli</ce:surname><ce:cross-ref refid="aff0880" 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author-id="S0370269323000643-ab877e6ad3aa564fc07ff9279bdb0114"><ce:given-name>R.</ce:given-name><ce:surname>Hannigan</ce:surname><ce:cross-ref refid="aff1190" id="crf3750"><ce:sup>119</ce:sup></ce:cross-ref></ce:author><ce:author id="au3430" author-id="S0370269323000643-6cb4376ff64678829e7af4b6b3584d70"><ce:given-name>M.R.</ce:given-name><ce:surname>Haque</ce:surname><ce:cross-ref refid="aff1430" id="crf3760"><ce:sup>143</ce:sup></ce:cross-ref></ce:author><ce:author id="au3440" author-id="S0370269323000643-1ad6c786673c0d88fb3ec98199ef2627"><ce:given-name>A.</ce:given-name><ce:surname>Harlenderova</ce:surname><ce:cross-ref refid="aff1080" id="crf3770"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au3450" author-id="S0370269323000643-093df752b6ef0875125996d2e249b17f"><ce:given-name>J.W.</ce:given-name><ce:surname>Harris</ce:surname><ce:cross-ref refid="aff1470" id="crf3780"><ce:sup>147</ce:sup></ce:cross-ref></ce:author><ce:author id="au3460" author-id="S0370269323000643-1cc09eaf1767efb447484c3576c326a3"><ce:given-name>A.</ce:given-name><ce:surname>Harton</ce:surname><ce:cross-ref refid="aff0100" id="crf3790"><ce:sup>10</ce:sup></ce:cross-ref></ce:author><ce:author id="au3470" author-id="S0370269323000643-8f1d14fcc366c98d7ffc616fd4020b3a"><ce:given-name>J.A.</ce:given-name><ce:surname>Hasenbichler</ce:surname><ce:cross-ref refid="aff0340" id="crf3800"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au3480" author-id="S0370269323000643-3c2864accc71830551ec0dc4bc72466c"><ce:given-name>H.</ce:given-name><ce:surname>Hassan</ce:surname><ce:cross-ref refid="aff0970" id="crf3810"><ce:sup>97</ce:sup></ce:cross-ref></ce:author><ce:author id="au3490" author-id="S0370269323000643-ec5233119b47dffa5f645e1bcad8c3ab"><ce:given-name>D.</ce:given-name><ce:surname>Hatzifotiadou</ce:surname><ce:cross-ref refid="aff0540" id="crf3820"><ce:sup>54</ce:sup></ce:cross-ref></ce:author><ce:author id="au3500" author-id="S0370269323000643-c494e289485a5b77e2698762fd7f4b6d"><ce:given-name>P.</ce:given-name><ce:surname>Hauer</ce:surname><ce:cross-ref refid="aff0430" id="crf3830"><ce:sup>43</ce:sup></ce:cross-ref></ce:author><ce:author id="au3510" author-id="S0370269323000643-d07756ab653a033a44665f2378719cc2"><ce:given-name>L.B.</ce:given-name><ce:surname>Havener</ce:surname><ce:cross-ref refid="aff1470" id="crf3840"><ce:sup>147</ce:sup></ce:cross-ref></ce:author><ce:author id="au3520" author-id="S0370269323000643-8e268fbd99e6fae4655760d64978b0ff"><ce:given-name>S.T.</ce:given-name><ce:surname>Heckel</ce:surname><ce:cross-ref refid="aff1060" id="crf3850"><ce:sup>106</ce:sup></ce:cross-ref></ce:author><ce:author id="au3530" author-id="S0370269323000643-7ad645538c2eb3346cfb0dd07e68d266"><ce:given-name>E.</ce:given-name><ce:surname>Hellbär</ce:surname><ce:cross-ref refid="aff1080" id="crf3860"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au3540" author-id="S0370269323000643-2ec6e519610c64506f6a6c4958b0ac54"><ce:given-name>H.</ce:given-name><ce:surname>Helstrup</ce:surname><ce:cross-ref refid="aff0360" id="crf3870"><ce:sup>36</ce:sup></ce:cross-ref></ce:author><ce:author id="au3550" author-id="S0370269323000643-51fb6f5f9415eda76fa12ad79ea6ef67"><ce:given-name>T.</ce:given-name><ce:surname>Herman</ce:surname><ce:cross-ref refid="aff0370" id="crf3880"><ce:sup>37</ce:sup></ce:cross-ref></ce:author><ce:author id="au3560" author-id="S0370269323000643-395e012e61feeafeb06dc1b1ee34d8d4"><ce:given-name>G.</ce:given-name><ce:surname>Herrera Corral</ce:surname><ce:cross-ref refid="aff0090" id="crf3890"><ce:sup>9</ce:sup></ce:cross-ref></ce:author><ce:author id="au3570" author-id="S0370269323000643-7a7a63e4af499b5ed75851524b48baa5"><ce:given-name>F.</ce:given-name><ce:surname>Herrmann</ce:surname><ce:cross-ref refid="aff1450" id="crf3900"><ce:sup>145</ce:sup></ce:cross-ref></ce:author><ce:author id="au3580" author-id="S0370269323000643-8fe0fd23ac8c4d7a4221a002b047a705"><ce:given-name>K.F.</ce:given-name><ce:surname>Hetland</ce:surname><ce:cross-ref refid="aff0360" id="crf3910"><ce:sup>36</ce:sup></ce:cross-ref></ce:author><ce:author id="au3590" author-id="S0370269323000643-5ab0a906131a8cf631664a48b8bb9672"><ce:given-name>B.</ce:given-name><ce:surname>Heybeck</ce:surname><ce:cross-ref refid="aff0680" id="crf3920"><ce:sup>68</ce:sup></ce:cross-ref></ce:author><ce:author id="au3600" author-id="S0370269323000643-d83a69a11b8778333bf72d127a9faaa4"><ce:given-name>H.</ce:given-name><ce:surname>Hillemanns</ce:surname><ce:cross-ref refid="aff0340" id="crf3930"><ce:sup>34</ce:sup></ce:cross-ref></ce:author><ce:author id="au3610" author-id="S0370269323000643-26a04d5f7344e810ea2275dad6767a9e"><ce:given-name>C.</ce:given-name><ce:surname>Hills</ce:surname><ce:cross-ref refid="aff1280" id="crf3940"><ce:sup>128</ce:sup></ce:cross-ref></ce:author><ce:author id="au3620" author-id="S0370269323000643-91692271254645e12e8687c69cd93550"><ce:given-name>B.</ce:given-name><ce:surname>Hippolyte</ce:surname><ce:cross-ref refid="aff1380" id="crf3950"><ce:sup>138</ce:sup></ce:cross-ref></ce:author><ce:author id="au3630" author-id="S0370269323000643-b2b41e3fa6c79c0bc760ec204badbf8e"><ce:given-name>B.</ce:given-name><ce:surname>Hofman</ce:surname><ce:cross-ref refid="aff0620" id="crf3960"><ce:sup>62</ce:sup></ce:cross-ref></ce:author><ce:author id="au3640" author-id="S0370269323000643-d68d4563c957d636f8ea0b438d75af8e"><ce:given-name>B.</ce:given-name><ce:surname>Hohlweger</ce:surname><ce:cross-ref refid="aff0910" id="crf3970"><ce:sup>91</ce:sup></ce:cross-ref></ce:author><ce:author id="au3650" author-id="S0370269323000643-6bcd750d66da852b78a422563f846dad"><ce:given-name>J.</ce:given-name><ce:surname>Honermann</ce:surname><ce:cross-ref refid="aff1450" id="crf3980"><ce:sup>145</ce:sup></ce:cross-ref></ce:author><ce:author id="au3660" author-id="S0370269323000643-fb87ef9112c084814dd4cd2584045cf3"><ce:given-name>G.H.</ce:given-name><ce:surname>Hong</ce:surname><ce:cross-ref refid="aff1480" id="crf3990"><ce:sup>148</ce:sup></ce:cross-ref></ce:author><ce:author id="au3670" author-id="S0370269323000643-bb0a4fa2cb3019f784b4aac68042d91d"><ce:given-name>D.</ce:given-name><ce:surname>Horak</ce:surname><ce:cross-ref refid="aff0370" id="crf4000"><ce:sup>37</ce:sup></ce:cross-ref></ce:author><ce:author id="au3680" author-id="S0370269323000643-58e97219d1f610ad3da8d74cd8d4df1d"><ce:given-name>S.</ce:given-name><ce:surname>Hornung</ce:surname><ce:cross-ref refid="aff1080" id="crf4010"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au3690" author-id="S0370269323000643-9a7d8925dcb9bade29afd917e864e548"><ce:given-name>A.</ce:given-name><ce:surname>Horzyk</ce:surname><ce:cross-ref refid="aff0020" id="crf4020"><ce:sup>2</ce:sup></ce:cross-ref></ce:author><ce:author id="au3700" 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author-id="S0370269323000643-c1701ed0ea0ca8b2c7644f838deb6a52"><ce:given-name>P.</ce:given-name><ce:surname>Huhn</ce:surname><ce:cross-ref refid="aff0680" id="crf4070"><ce:sup>68</ce:sup></ce:cross-ref></ce:author><ce:author id="au3750" author-id="S0370269323000643-1d44428a50028ba0167ad2e836cdccdd"><ce:given-name>L.M.</ce:given-name><ce:surname>Huhta</ce:surname><ce:cross-ref refid="aff1260" id="crf4080"><ce:sup>126</ce:sup></ce:cross-ref></ce:author><ce:author id="au3760" author-id="S0370269323000643-b77b5ca70871829f4a6caed289764804"><ce:given-name>C.V.</ce:given-name><ce:surname>Hulse</ce:surname><ce:cross-ref refid="aff0780" id="crf4090"><ce:sup>78</ce:sup></ce:cross-ref></ce:author><ce:author id="au3770" author-id="S0370269323000643-c0ec982d5ca2280aac9dadb166a34f51"><ce:given-name>T.J.</ce:given-name><ce:surname>Humanic</ce:surname><ce:cross-ref refid="aff0980" id="crf4100"><ce:sup>98</ce:sup></ce:cross-ref></ce:author><ce:author id="au3780" 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author-id="S0370269323000643-68c856e3e31ba9ae46e1a3cf0fd7b39f"><ce:given-name>A.</ce:given-name><ce:surname>Ohlson</ce:surname><ce:cross-ref refid="aff0810" id="crf7040"><ce:sup>81</ce:sup></ce:cross-ref></ce:author><ce:author id="au6560" author-id="S0370269323000643-b6dd7e1db4469e2b861a26c63c002f91"><ce:given-name>V.A.</ce:given-name><ce:surname>Okorokov</ce:surname><ce:cross-ref refid="aff0940" id="crf7050"><ce:sup>94</ce:sup></ce:cross-ref></ce:author><ce:author id="au6570" author-id="S0370269323000643-4b54637a0b3dbb9f03da377fb857f191"><ce:given-name>J.</ce:given-name><ce:surname>Oleniacz</ce:surname><ce:cross-ref refid="aff1430" id="crf7060"><ce:sup>143</ce:sup></ce:cross-ref></ce:author><ce:author id="au6580" author-id="S0370269323000643-e919d0a3ebb4bb927bf3a87e6fc67c88"><ce:given-name>A.C.</ce:given-name><ce:surname>Oliveira Da Silva</ce:surname><ce:cross-ref refid="aff1310" id="crf7070"><ce:sup>131</ce:sup></ce:cross-ref></ce:author><ce:author id="au6590" author-id="S0370269323000643-ee3f97c8e319fe4cdf98e0660785c85a"><ce:given-name>M.H.</ce:given-name><ce:surname>Oliver</ce:surname><ce:cross-ref refid="aff1470" id="crf7080"><ce:sup>147</ce:sup></ce:cross-ref></ce:author><ce:author id="au6600" author-id="S0370269323000643-e6f50f7f16b66af365a7af642930fd18"><ce:given-name>A.</ce:given-name><ce:surname>Onnerstad</ce:surname><ce:cross-ref refid="aff1260" id="crf7090"><ce:sup>126</ce:sup></ce:cross-ref></ce:author><ce:author id="au6610" author-id="S0370269323000643-dda435886a590cbfc7ea1d77af3e51f5"><ce:given-name>C.</ce:given-name><ce:surname>Oppedisano</ce:surname><ce:cross-ref refid="aff0590" id="crf7100"><ce:sup>59</ce:sup></ce:cross-ref></ce:author><ce:author id="au6620" author-id="S0370269323000643-c6a2d1996925547f52c523a908287916"><ce:given-name>A.</ce:given-name><ce:surname>Ortiz Velasquez</ce:surname><ce:cross-ref refid="aff0690" id="crf7110"><ce:sup>69</ce:sup></ce:cross-ref></ce:author><ce:author id="au6630" author-id="S0370269323000643-866acaca91118f974307720fd456828b"><ce:given-name>T.</ce:given-name><ce:surname>Osako</ce:surname><ce:cross-ref refid="aff0460" id="crf7120"><ce:sup>46</ce:sup></ce:cross-ref></ce:author><ce:author id="au6640" author-id="S0370269323000643-d7f1b48c18151d2adcfd0901daea63ca"><ce:given-name>A.</ce:given-name><ce:surname>Oskarsson</ce:surname><ce:cross-ref refid="aff0810" id="crf7130"><ce:sup>81</ce:sup></ce:cross-ref></ce:author><ce:author id="au6650" author-id="S0370269323000643-453c2920b4d8be43344a508683c61206"><ce:given-name>J.</ce:given-name><ce:surname>Otwinowski</ce:surname><ce:cross-ref refid="aff1180" id="crf7140"><ce:sup>118</ce:sup></ce:cross-ref></ce:author><ce:author id="au6660" author-id="S0370269323000643-3397848215c1376cf0f1ed7b9a934e8b"><ce:given-name>M.</ce:given-name><ce:surname>Oya</ce:surname><ce:cross-ref refid="aff0460" id="crf7150"><ce:sup>46</ce:sup></ce:cross-ref></ce:author><ce:author id="au6670" author-id="S0370269323000643-ef1f97862dcdec187e1b2f91615f60fb"><ce:given-name>K.</ce:given-name><ce:surname>Oyama</ce:surname><ce:cross-ref refid="aff0830" id="crf7160"><ce:sup>83</ce:sup></ce:cross-ref></ce:author><ce:author id="au6680" author-id="S0370269323000643-02951a4df40a044a3fdcb144a019433a"><ce:given-name>Y.</ce:given-name><ce:surname>Pachmayer</ce:surname><ce:cross-ref refid="aff1050" id="crf7170"><ce:sup>105</ce:sup></ce:cross-ref></ce:author><ce:author id="au6690" author-id="S0370269323000643-0b6285cc2d39707ea9f13a51bd6d5b54"><ce:given-name>S.</ce:given-name><ce:surname>Padhan</ce:surname><ce:cross-ref refid="aff0490" id="crf7180"><ce:sup>49</ce:sup></ce:cross-ref></ce:author><ce:author id="au6700" author-id="S0370269323000643-9069b38983788ec66d195a57d1524b7f"><ce:given-name>D.</ce:given-name><ce:surname>Pagano</ce:surname><ce:cross-ref refid="aff1410" id="crf7190"><ce:sup>141</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0580" 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author-id="S0370269323000643-0a168bc99240d66a8925ff06e271bdf5"><ce:given-name>J.</ce:given-name><ce:surname>Saetre</ce:surname><ce:cross-ref refid="aff0210" id="crf8360"><ce:sup>21</ce:sup></ce:cross-ref></ce:author><ce:author id="au7800" author-id="S0370269323000643-21a9d7d32dbc31b6d9fe58ee57047f0f"><ce:given-name>K.</ce:given-name><ce:surname>Šafařík</ce:surname><ce:cross-ref refid="aff0370" id="crf8370"><ce:sup>37</ce:sup></ce:cross-ref></ce:author><ce:author id="au7810" author-id="S0370269323000643-e124e41bd8c9a15152d347654ca43d3a"><ce:given-name>S.K.</ce:given-name><ce:surname>Saha</ce:surname><ce:cross-ref refid="aff1420" id="crf8380"><ce:sup>142</ce:sup></ce:cross-ref></ce:author><ce:author id="au7820" author-id="S0370269323000643-97780e442ae33eb1a0ab97a98d717272"><ce:given-name>S.</ce:given-name><ce:surname>Saha</ce:surname><ce:cross-ref refid="aff0870" id="crf8390"><ce:sup>87</ce:sup></ce:cross-ref></ce:author><ce:author id="au7830" author-id="S0370269323000643-2d0dbe8d0df0bf01a83ddf19e043addc"><ce:given-name>B.</ce:given-name><ce:surname>Sahoo</ce:surname><ce:cross-ref refid="aff0490" id="crf8400"><ce:sup>49</ce:sup></ce:cross-ref></ce:author><ce:author id="au7840" author-id="S0370269323000643-c91ceefdaf62924dbe41969b9214a864"><ce:given-name>P.</ce:given-name><ce:surname>Sahoo</ce:surname><ce:cross-ref refid="aff0490" id="crf8410"><ce:sup>49</ce:sup></ce:cross-ref></ce:author><ce:author id="au7850" author-id="S0370269323000643-bf022b9756aebb065155d194bfc5e116"><ce:given-name>R.</ce:given-name><ce:surname>Sahoo</ce:surname><ce:cross-ref refid="aff0500" id="crf8420"><ce:sup>50</ce:sup></ce:cross-ref></ce:author><ce:author id="au7860" author-id="S0370269323000643-d12dd0293e56fb8f196c824e945688ef"><ce:given-name>S.</ce:given-name><ce:surname>Sahoo</ce:surname><ce:cross-ref refid="aff0650" id="crf8430"><ce:sup>65</ce:sup></ce:cross-ref></ce:author><ce:author id="au7870" 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author-id="S0370269323000643-8a3806e0320b7d360b40762c9a667520"><ce:given-name>M.P.</ce:given-name><ce:surname>Salvan</ce:surname><ce:cross-ref refid="aff1080" id="crf8480"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au7920" author-id="S0370269323000643-1bc9f8186efa42ba63b483c287a4106e"><ce:given-name>S.</ce:given-name><ce:surname>Sambyal</ce:surname><ce:cross-ref refid="aff1020" id="crf8490"><ce:sup>102</ce:sup></ce:cross-ref></ce:author><ce:author id="au7930" author-id="S0370269323000643-cd56f5a9e647bb69890a3a180b5ed4f6"><ce:given-name>T.B.</ce:given-name><ce:surname>Saramela</ce:surname><ce:cross-ref refid="aff1210" id="crf8500"><ce:sup>121</ce:sup></ce:cross-ref></ce:author><ce:author id="au7940" author-id="S0370269323000643-4e7b512d816400994223e9f92ecca2ec"><ce:given-name>D.</ce:given-name><ce:surname>Sarkar</ce:surname><ce:cross-ref refid="aff1440" id="crf8510"><ce:sup>144</ce:sup></ce:cross-ref></ce:author><ce:author id="au7950" author-id="S0370269323000643-2142bcc0044d7022f40dbf34b4e197ca"><ce:given-name>N.</ce:given-name><ce:surname>Sarkar</ce:surname><ce:cross-ref refid="aff1420" id="crf8520"><ce:sup>142</ce:sup></ce:cross-ref></ce:author><ce:author id="au7960" author-id="S0370269323000643-a3851d6513ae8b68029b7958a5abed81"><ce:given-name>P.</ce:given-name><ce:surname>Sarma</ce:surname><ce:cross-ref refid="aff0420" id="crf8530"><ce:sup>42</ce:sup></ce:cross-ref></ce:author><ce:author id="au7970" author-id="S0370269323000643-cc5ffcd83ec77f5fd9325b0b7118b774"><ce:given-name>V.M.</ce:given-name><ce:surname>Sarti</ce:surname><ce:cross-ref refid="aff1060" id="crf8540"><ce:sup>106</ce:sup></ce:cross-ref></ce:author><ce:author id="au7980" author-id="S0370269323000643-4f6b354f623cbb6890bb23228e05b5fe"><ce:given-name>M.H.P.</ce:given-name><ce:surname>Sas</ce:surname><ce:cross-ref refid="aff1470" id="crf8550"><ce:sup>147</ce:sup></ce:cross-ref></ce:author><ce:author id="au7990" 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author-id="S0370269323000643-0217b26765b58bf272de6570444ed5dc"><ce:given-name>A.</ce:given-name><ce:surname>Schmah</ce:surname><ce:cross-ref refid="aff1050" id="crf8600"><ce:sup>105</ce:sup></ce:cross-ref></ce:author><ce:author id="au8040" author-id="S0370269323000643-f94f964488b585f9462e6ab4015791c9"><ce:given-name>C.</ce:given-name><ce:surname>Schmidt</ce:surname><ce:cross-ref refid="aff1080" id="crf8610"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au8050" author-id="S0370269323000643-fd43f769f1f7612b2725ab8467cb3988"><ce:given-name>H.R.</ce:given-name><ce:surname>Schmidt</ce:surname><ce:cross-ref refid="aff1040" id="crf8620"><ce:sup>104</ce:sup></ce:cross-ref></ce:author><ce:author id="au8060" author-id="S0370269323000643-000dbff16850e64d09345b19cc7e2cae"><ce:given-name>M.O.</ce:given-name><ce:surname>Schmidt</ce:surname><ce:cross-ref refid="aff0340" id="crf8630"><ce:sup>34</ce:sup></ce:cross-ref><ce:cross-ref refid="aff1050" 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author-id="S0370269323000643-6ed7035a79dd824bd6a0b2572108463b"><ce:given-name>C.</ce:given-name><ce:surname>Wang</ce:surname><ce:cross-ref refid="aff0400" id="crf10370"><ce:sup>40</ce:sup></ce:cross-ref></ce:author><ce:author id="au9720" author-id="S0370269323000643-acba1a294e63a54789e94f0c48170a7c"><ce:given-name>D.</ce:given-name><ce:surname>Wang</ce:surname><ce:cross-ref refid="aff0400" id="crf10380"><ce:sup>40</ce:sup></ce:cross-ref></ce:author><ce:author id="au9730" author-id="S0370269323000643-86448d3a1cf868fca209fc2d36f58f5e"><ce:given-name>M.</ce:given-name><ce:surname>Weber</ce:surname><ce:cross-ref refid="aff1140" id="crf10390"><ce:sup>114</ce:sup></ce:cross-ref></ce:author><ce:author id="au9740" author-id="S0370269323000643-b19740465a27da9581dafc73eb6a141a"><ce:given-name>R.J.G.V.</ce:given-name><ce:surname>Weelden</ce:surname><ce:cross-ref refid="aff0910" id="crf10400"><ce:sup>91</ce:sup></ce:cross-ref></ce:author><ce:author id="au9750" 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author-id="S0370269323000643-d35206c8771dcf8151d8531c8b5a4ad9"><ce:given-name>J.</ce:given-name><ce:surname>Wiechula</ce:surname><ce:cross-ref refid="aff0680" id="crf10450"><ce:sup>68</ce:sup></ce:cross-ref></ce:author><ce:author id="au9800" author-id="S0370269323000643-924aa82bb7057758dfd6b448c22248a6"><ce:given-name>J.</ce:given-name><ce:surname>Wikne</ce:surname><ce:cross-ref refid="aff0200" id="crf10460"><ce:sup>20</ce:sup></ce:cross-ref></ce:author><ce:author id="au9810" author-id="S0370269323000643-c6cf2d96fecc56427c920ff58c5f8f6c"><ce:given-name>G.</ce:given-name><ce:surname>Wilk</ce:surname><ce:cross-ref refid="aff0860" id="crf10470"><ce:sup>86</ce:sup></ce:cross-ref></ce:author><ce:author id="au9820" author-id="S0370269323000643-defbde1b1ee45aba41243923803c24ee"><ce:given-name>J.</ce:given-name><ce:surname>Wilkinson</ce:surname><ce:cross-ref refid="aff1080" id="crf10480"><ce:sup>108</ce:sup></ce:cross-ref></ce:author><ce:author id="au9830" 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id="crf10810"><ce:sup>129</ce:sup></ce:cross-ref></ce:author><ce:author id="au10150" author-id="S0370269323000643-9feeecb04ac9bd35d770466fb86792be"><ce:given-name>V.</ce:given-name><ce:surname>Zherebchevskii</ce:surname><ce:cross-ref refid="aff1130" id="crf10820"><ce:sup>113</ce:sup></ce:cross-ref></ce:author><ce:author id="au10160" author-id="S0370269323000643-617c42c28cb1b1d3dab680558c293767"><ce:given-name>Y.</ce:given-name><ce:surname>Zhi</ce:surname><ce:cross-ref refid="aff0110" id="crf10830"><ce:sup>11</ce:sup></ce:cross-ref></ce:author><ce:author id="au10170" author-id="S0370269323000643-3038f7cd79250ffa05fa0aff7644d2b3"><ce:given-name>N.</ce:given-name><ce:surname>Zhigareva</ce:surname><ce:cross-ref refid="aff0930" id="crf10840"><ce:sup>93</ce:sup></ce:cross-ref></ce:author><ce:author id="au10180" author-id="S0370269323000643-e9bc0465b352cd706eac4fc8b795a4c2"><ce:given-name>D.</ce:given-name><ce:surname>Zhou</ce:surname><ce:cross-ref refid="aff0070" id="crf10850"><ce:sup>7</ce:sup></ce:cross-ref></ce:author><ce:author id="au10190" author-id="S0370269323000643-4a0cdba9c352af9310759dd83e906db9"><ce:given-name>Y.</ce:given-name><ce:surname>Zhou</ce:surname><ce:cross-ref refid="aff0900" id="crf10860"><ce:sup>90</ce:sup></ce:cross-ref></ce:author><ce:author id="au10200" author-id="S0370269323000643-b564bd6149c66f06bbea2780670b8f50"><ce:given-name>J.</ce:given-name><ce:surname>Zhu</ce:surname><ce:cross-ref refid="aff1080" id="crf10870"><ce:sup>108</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0070" id="crf10880"><ce:sup>7</ce:sup></ce:cross-ref></ce:author><ce:author id="au10210" author-id="S0370269323000643-1fb0ab925cfcdaacb160193d82c1a978"><ce:given-name>Y.</ce:given-name><ce:surname>Zhu</ce:surname><ce:cross-ref refid="aff0070" id="crf10890"><ce:sup>7</ce:sup></ce:cross-ref></ce:author><ce:author id="au10220" author-id="S0370269323000643-2ad055472e2fac995111c51c08396d0c"><ce:given-name>G.</ce:given-name><ce:surname>Zinovjev</ce:surname><ce:cross-ref refid="aff0030" id="crf10900"><ce:sup>3</ce:sup></ce:cross-ref><ce:cross-ref refid="fn0010" id="crf10910"><ce:sup>I</ce:sup></ce:cross-ref></ce:author><ce:author id="au10230" author-id="S0370269323000643-92b5ecf5612e2e849fc1bba72c6cff99"><ce:given-name>N.</ce:given-name><ce:surname>Zurlo</ce:surname><ce:cross-ref refid="aff1410" id="crf10920"><ce:sup>141</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0580" id="crf10930"><ce:sup>58</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="aff0010" affiliation-id="S0370269323000643-79d30baa35325e84d46378ba6ce12c18"><ce:label>1</ce:label><ce:textfn>A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia</ce:textfn><sa:affiliation><sa:organization>A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation</sa:organization><sa:city>Yerevan</sa:city><sa:country>Armenia</sa:country></sa:affiliation><ce:source-text id="srct0005">A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia</ce:source-text></ce:affiliation><ce:affiliation id="aff0020" affiliation-id="S0370269323000643-65754d218cf7f84bf1e02306b80caca0"><ce:label>2</ce:label><ce:textfn>AGH University of Science and Technology, Cracow, Poland</ce:textfn><sa:affiliation><sa:organization>AGH University of Science and Technology</sa:organization><sa:city>Cracow</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0010">AGH University of Science and Technology, Cracow, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff0030" affiliation-id="S0370269323000643-e916a2f48a17bc32220b61ae0e9b8e05"><ce:label>3</ce:label><ce:textfn>Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine</ce:textfn><sa:affiliation><sa:organization>Bogolyubov Institute for Theoretical Physics</sa:organization><sa:organization>National Academy of Sciences of Ukraine</sa:organization><sa:city>Kiev</sa:city><sa:country>Ukraine</sa:country></sa:affiliation><ce:source-text id="srct0015">Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine</ce:source-text></ce:affiliation><ce:affiliation id="aff0040" affiliation-id="S0370269323000643-9869d95133b2275e34836b8ad2f235c3"><ce:label>4</ce:label><ce:textfn>Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Bose Institute</sa:organization><sa:organization>Department of Physics</sa:organization><sa:organization>Centre for Astroparticle Physics and Space Science (CAPSS)</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0020">Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0050" affiliation-id="S0370269323000643-187ae13619e9f75c78fa64b9926e2952"><ce:label>5</ce:label><ce:textfn>Budker Institute for Nuclear Physics, Novosibirsk, Russia</ce:textfn><sa:affiliation><sa:organization>Budker Institute for Nuclear Physics</sa:organization><sa:city>Novosibirsk</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0025">Budker Institute for Nuclear Physics, Novosibirsk, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0060" affiliation-id="S0370269323000643-b7f796ef6c934d79a497288cdec192f6"><ce:label>6</ce:label><ce:textfn>California Polytechnic State University, San Luis Obispo, CA, United States</ce:textfn><sa:affiliation><sa:organization>California Polytechnic State University</sa:organization><sa:city>San Luis Obispo</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0030">California Polytechnic State University, San Luis Obispo, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0070" affiliation-id="S0370269323000643-3dcd6ffc2e8f27d6ed2f6237a209a384"><ce:label>7</ce:label><ce:textfn>Central China Normal University, Wuhan, China</ce:textfn><sa:affiliation><sa:organization>Central China Normal University</sa:organization><sa:city>Wuhan</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0035">Central China Normal University, Wuhan, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0080" affiliation-id="S0370269323000643-110460c7f2fbb319ec6d52e7cc3fc1d5"><ce:label>8</ce:label><ce:textfn>Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba</ce:textfn><sa:affiliation><sa:organization>Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN)</sa:organization><sa:city>Havana</sa:city><sa:country>Cuba</sa:country></sa:affiliation><ce:source-text id="srct0040">Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba</ce:source-text></ce:affiliation><ce:affiliation id="aff0090" affiliation-id="S0370269323000643-641aa526558990d110c090840e6f3d0c"><ce:label>9</ce:label><ce:textfn>Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico</ce:textfn><sa:affiliation><sa:organization>Centro de Investigación y de Estudios Avanzados (CINVESTAV)</sa:organization><sa:city>Mexico City and Mérida</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0045">Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0100" affiliation-id="S0370269323000643-9c73ece39ab638447cd10f268e329b2e"><ce:label>10</ce:label><ce:textfn>Chicago State University, Chicago, IL, United States</ce:textfn><sa:affiliation><sa:organization>Chicago State University</sa:organization><sa:city>Chicago</sa:city><sa:state>IL</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0050">Chicago State University, Chicago, Illinois, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0110" affiliation-id="S0370269323000643-d7da93bf3f02be0a46bee74f439b5930"><ce:label>11</ce:label><ce:textfn>China Institute of Atomic Energy, Beijing, China</ce:textfn><sa:affiliation><sa:organization>China Institute of Atomic Energy</sa:organization><sa:city>Beijing</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0055">China Institute of Atomic Energy, Beijing, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0120" affiliation-id="S0370269323000643-d477851cd020cd39da135784676a96ff"><ce:label>12</ce:label><ce:textfn>Chungbuk National University, Cheongju, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Chungbuk National University</sa:organization><sa:city>Cheongju</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0060">Chungbuk National University, Cheongju, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0130" affiliation-id="S0370269323000643-ffca702bc50b4f17be4153d6998acb8d"><ce:label>13</ce:label><ce:textfn>Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovakia</ce:textfn><sa:affiliation><sa:organization>Comenius University Bratislava</sa:organization><sa:organization>Faculty of Mathematics, Physics and Informatics</sa:organization><sa:city>Bratislava</sa:city><sa:country>Slovakia</sa:country></sa:affiliation><ce:source-text id="srct0065">Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovakia</ce:source-text></ce:affiliation><ce:affiliation id="aff0140" affiliation-id="S0370269323000643-78cc8333b14d598fc5e3c0230d1de22e"><ce:label>14</ce:label><ce:textfn>COMSATS University Islamabad, Islamabad, Pakistan</ce:textfn><sa:affiliation><sa:organization>COMSATS University Islamabad</sa:organization><sa:city>Islamabad</sa:city><sa:country>Pakistan</sa:country></sa:affiliation><ce:source-text id="srct0070">COMSATS University Islamabad, Islamabad, Pakistan</ce:source-text></ce:affiliation><ce:affiliation id="aff0150" affiliation-id="S0370269323000643-42161ea7466da89325c5b37601107c55"><ce:label>15</ce:label><ce:textfn>Creighton University, Omaha, NE, United States</ce:textfn><sa:affiliation><sa:organization>Creighton University</sa:organization><sa:city>Omaha</sa:city><sa:state>NE</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0075">Creighton University, Omaha, Nebraska, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0160" affiliation-id="S0370269323000643-bd5e6b818668501c1deea76e96cac833"><ce:label>16</ce:label><ce:textfn>Department of Physics, Aligarh Muslim University, Aligarh, India</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Aligarh Muslim University</sa:organization><sa:city>Aligarh</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0080">Department of Physics, Aligarh Muslim University, Aligarh, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0170" affiliation-id="S0370269323000643-ebe7875fb705d3c179953d196aa8b94b"><ce:label>17</ce:label><ce:textfn>Department of Physics, Pusan National University, Pusan, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Pusan National University</sa:organization><sa:city>Pusan</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0085">Department of Physics, Pusan National University, Pusan, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0180" affiliation-id="S0370269323000643-24ddc81b37e640977b88412ba28336a4"><ce:label>18</ce:label><ce:textfn>Department of Physics, Sejong University, Seoul, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Sejong University</sa:organization><sa:city>Seoul</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0090">Department of Physics, Sejong University, Seoul, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0190" affiliation-id="S0370269323000643-66245837acb0d7fb7ada51de0fd95043"><ce:label>19</ce:label><ce:textfn>Department of Physics, University of California, Berkeley, CA, United States</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>University of California</sa:organization><sa:city>Berkeley</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0095">Department of Physics, University of California, Berkeley, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0200" affiliation-id="S0370269323000643-63776eabafa9e32856b35562692a4488"><ce:label>20</ce:label><ce:textfn>Department of Physics, University of Oslo, Oslo, Norway</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>University of Oslo</sa:organization><sa:city>Oslo</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0100">Department of Physics, University of Oslo, Oslo, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0210" affiliation-id="S0370269323000643-020451a59d5e257505166bdb7847cdd7"><ce:label>21</ce:label><ce:textfn>Department of Physics and Technology, University of Bergen, Bergen, Norway</ce:textfn><sa:affiliation><sa:organization>Department of Physics and Technology</sa:organization><sa:organization>University of Bergen</sa:organization><sa:city>Bergen</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0105">Department of Physics and Technology, University of Bergen, Bergen, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0220" affiliation-id="S0370269323000643-81d87d2cd8c9f6aa2a1c7f4b13115ef3"><ce:label>22</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Cagliari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0110">Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0230" affiliation-id="S0370269323000643-849d0ba7888fcb2a09aad7ac1095ec15"><ce:label>23</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Trieste</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0115">Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0240" affiliation-id="S0370269323000643-68c63fd1cb9e8623af62118bbef39c9e"><ce:label>24</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0120">Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0250" affiliation-id="S0370269323000643-7a40dab487121429c63e59c3de15ccfa"><ce:label>25</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bologna</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0125">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0260" affiliation-id="S0370269323000643-72e3fe624de7d3d3223574366a22ef30"><ce:label>26</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Catania</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0130">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0270" affiliation-id="S0370269323000643-6145c67e3009fb117e82f5429b2af282"><ce:label>27</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Padova</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0135">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0280" affiliation-id="S0370269323000643-ed02ab18bfc8872ec646d178987b6fb8"><ce:label>28</ce:label><ce:textfn>Dipartimento di Fisica e Nucleare e Teorica, Università di Pavia, Pavia, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Nucleare e Teorica</sa:organization><sa:organization>Università di Pavia</sa:organization><sa:city>Pavia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0140">Dipartimento di Fisica e Nucleare e Teorica, Università di Pavia, Pavia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0290" affiliation-id="S0370269323000643-a001bc692b1f1f84377401c9e2632e51"><ce:label>29</ce:label><ce:textfn>Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università and Gruppo Collegato INFN, Salerno, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università</sa:organization><sa:organization>Gruppo Collegato INFN</sa:organization><sa:city>Salerno</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0145">Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università and Gruppo Collegato INFN, Salerno, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0300" affiliation-id="S0370269323000643-362983c57dbe79a62add2550bfe565dc"><ce:label>30</ce:label><ce:textfn>Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento DISAT del Politecnico</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0150">Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0310" affiliation-id="S0370269323000643-be2c5e0552d256581ca35805d61bfd27"><ce:label>31</ce:label><ce:textfn>Dipartimento di Scienze e Innovazione Tecnologica dell'Università del Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Scienze e Innovazione Tecnologica dell'Università del Piemonte Orientale</sa:organization><sa:organization>INFN Sezione di Torino</sa:organization><sa:city>Alessandria</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0155">Dipartimento di Scienze e Innovazione Tecnologica dell'Università del Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0320" affiliation-id="S0370269323000643-7a82e32411929fc5768d11b72609a4d8"><ce:label>32</ce:label><ce:textfn>Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Scienze MIFT</sa:organization><sa:organization>Università di Messina</sa:organization><sa:city>Messina</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0160">Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0330" affiliation-id="S0370269323000643-0bfba0b176e2b6e1b67e6564c87308b5"><ce:label>33</ce:label><ce:textfn>Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento Interateneo di Fisica ‘M. Merlin’</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0165">Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0340" affiliation-id="S0370269323000643-44f92095d23d6e3d2fb8e8fc998c51f2"><ce:label>34</ce:label><ce:textfn>European Organization for Nuclear Research (CERN), Geneva, Switzerland</ce:textfn><sa:affiliation><sa:organization>European Organization for Nuclear Research (CERN)</sa:organization><sa:city>Geneva</sa:city><sa:country>Switzerland</sa:country></sa:affiliation><ce:source-text id="srct0170">European Organization for Nuclear Research (CERN), Geneva, Switzerland</ce:source-text></ce:affiliation><ce:affiliation id="aff0350" affiliation-id="S0370269323000643-f220870dd6ed2747e8ec11b2ba624bf5"><ce:label>35</ce:label><ce:textfn>Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia</ce:textfn><sa:affiliation><sa:organization>Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture</sa:organization><sa:organization>University of Split</sa:organization><sa:city>Split</sa:city><sa:country>Croatia</sa:country></sa:affiliation><ce:source-text id="srct0175">Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia</ce:source-text></ce:affiliation><ce:affiliation id="aff0360" affiliation-id="S0370269323000643-5bf4b9d297f0544e6037addfb7689840"><ce:label>36</ce:label><ce:textfn>Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway</ce:textfn><sa:affiliation><sa:organization>Faculty of Engineering and Science</sa:organization><sa:organization>Western Norway University of Applied Sciences</sa:organization><sa:city>Bergen</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0180">Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0370" affiliation-id="S0370269323000643-56d412e1c3fe114d9a18225fa76274c9"><ce:label>37</ce:label><ce:textfn>Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Faculty of Nuclear Sciences and Physical Engineering</sa:organization><sa:organization>Czech Technical University in Prague</sa:organization><sa:city>Prague</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0185">Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0380" affiliation-id="S0370269323000643-fb857933012a43dca7eed2503db4f3dd"><ce:label>38</ce:label><ce:textfn>Faculty of Science, P.J. Šafárik University, Košice, Slovakia</ce:textfn><sa:affiliation><sa:organization>Faculty of Science</sa:organization><sa:organization>P.J. Šafárik University</sa:organization><sa:city>Košice</sa:city><sa:country>Slovakia</sa:country></sa:affiliation><ce:source-text id="srct0190">Faculty of Science, P.J. Šafárik University, Košice, Slovakia</ce:source-text></ce:affiliation><ce:affiliation id="aff0390" affiliation-id="S0370269323000643-a18716885f5bc83f5d1aee8d0d80c7af"><ce:label>39</ce:label><ce:textfn>Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Frankfurt Institute for Advanced Studies</sa:organization><sa:organization>Johann Wolfgang Goethe-Universität Frankfurt</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0195">Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0400" affiliation-id="S0370269323000643-f0b0a2b18fef5547dcd39253b5714404"><ce:label>40</ce:label><ce:textfn>Fudan University, Shanghai, China</ce:textfn><sa:affiliation><sa:organization>Fudan University</sa:organization><sa:city>Shanghai</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0200">Fudan University, Shanghai, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0410" affiliation-id="S0370269323000643-a2398937a38c48ebf6ac3ff5e5d60b95"><ce:label>41</ce:label><ce:textfn>Gangneung-Wonju National University, Gangneung, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Gangneung-Wonju National University</sa:organization><sa:city>Gangneung</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0205">Gangneung-Wonju National University, Gangneung, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0420" affiliation-id="S0370269323000643-73bc80872cd116960a09bc477e035838"><ce:label>42</ce:label><ce:textfn>Gauhati University, Department of Physics, Guwahati, India</ce:textfn><sa:affiliation><sa:organization>Gauhati University</sa:organization><sa:organization>Department of Physics</sa:organization><sa:city>Guwahati</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0210">Gauhati University, Department of Physics, Guwahati, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0430" affiliation-id="S0370269323000643-9fd27eebf76465366fe59cb8d9620ae8"><ce:label>43</ce:label><ce:textfn>Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany</ce:textfn><sa:affiliation><sa:organization>Helmholtz-Institut für Strahlen- und Kernphysik</sa:organization><sa:organization>Rheinische Friedrich-Wilhelms-Universität Bonn</sa:organization><sa:city>Bonn</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0215">Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0440" affiliation-id="S0370269323000643-66b9f8889421b091fa908925e638d91e"><ce:label>44</ce:label><ce:textfn>Helsinki Institute of Physics (HIP), Helsinki, Finland</ce:textfn><sa:affiliation><sa:organization>Helsinki Institute of Physics (HIP)</sa:organization><sa:city>Helsinki</sa:city><sa:country>Finland</sa:country></sa:affiliation><ce:source-text id="srct0220">Helsinki Institute of Physics (HIP), Helsinki, Finland</ce:source-text></ce:affiliation><ce:affiliation id="aff0450" affiliation-id="S0370269323000643-6968018d6fe1bd12a4413918be70ab85"><ce:label>45</ce:label><ce:textfn>High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico</ce:textfn><sa:affiliation><sa:organization>High Energy Physics Group</sa:organization><sa:organization>Universidad Autónoma de Puebla</sa:organization><sa:city>Puebla</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0225">High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0460" affiliation-id="S0370269323000643-8454dbc80caf49377b12cbde1879d7a0"><ce:label>46</ce:label><ce:textfn>Hiroshima University, Hiroshima, Japan</ce:textfn><sa:affiliation><sa:organization>Hiroshima University</sa:organization><sa:city>Hiroshima</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0230">Hiroshima University, Hiroshima, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0470" affiliation-id="S0370269323000643-9c00b97a376e62a36465178b70ef1f89"><ce:label>47</ce:label><ce:textfn>Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation (ZTT), Worms, Germany</ce:textfn><sa:affiliation><sa:organization>Hochschule Worms</sa:organization><sa:organization>Zentrum für Technologietransfer und Telekommunikation (ZTT)</sa:organization><sa:city>Worms</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0235">Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation (ZTT), Worms, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0480" affiliation-id="S0370269323000643-16bdf6e3577a480da99159f5d54452b1"><ce:label>48</ce:label><ce:textfn>Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>Horia Hulubei National Institute of Physics and Nuclear Engineering</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0240">Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff0490" affiliation-id="S0370269323000643-78df0ad3e050545612fd8d71a4776a19"><ce:label>49</ce:label><ce:textfn>Indian Institute of Technology Bombay (IIT), Mumbai, India</ce:textfn><sa:affiliation><sa:organization>Indian Institute of Technology Bombay (IIT)</sa:organization><sa:city>Mumbai</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0245">Indian Institute of Technology Bombay (IIT), Mumbai, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0500" affiliation-id="S0370269323000643-b8ad6c9375b7a89b768adb13f27427b4"><ce:label>50</ce:label><ce:textfn>Indian Institute of Technology Indore, Indore, India</ce:textfn><sa:affiliation><sa:organization>Indian Institute of Technology Indore</sa:organization><sa:city>Indore</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0250">Indian Institute of Technology Indore, Indore, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0510" affiliation-id="S0370269323000643-98c636440432201908fa6753b352596d"><ce:label>51</ce:label><ce:textfn>Indonesian Institute of Sciences, Jakarta, Indonesia</ce:textfn><sa:affiliation><sa:organization>Indonesian Institute of Sciences</sa:organization><sa:city>Jakarta</sa:city><sa:country>Indonesia</sa:country></sa:affiliation><ce:source-text id="srct0255">Indonesian Institute of Sciences, Jakarta, Indonesia</ce:source-text></ce:affiliation><ce:affiliation id="aff0520" affiliation-id="S0370269323000643-25658fba725d22058ae2a8649ceeb084"><ce:label>52</ce:label><ce:textfn>INFN, Laboratori Nazionali di Frascati, Frascati, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Laboratori Nazionali di Frascati</sa:organization><sa:city>Frascati</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0260">INFN, Laboratori Nazionali di Frascati, Frascati, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0530" affiliation-id="S0370269323000643-5d9dee68bdf16e34f2f3f01d930f367d"><ce:label>53</ce:label><ce:textfn>INFN, Sezione di Bari, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Bari</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0265">INFN, Sezione di Bari, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0540" affiliation-id="S0370269323000643-340d750afed4ff705cf1dd72bf688ca4"><ce:label>54</ce:label><ce:textfn>INFN, Sezione di Bologna, Bologna, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Bologna</sa:organization><sa:city>Bologna</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0270">INFN, Sezione di Bologna, Bologna, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0550" affiliation-id="S0370269323000643-436e8c989d6c10cae74944a81714a4e2"><ce:label>55</ce:label><ce:textfn>INFN, Sezione di Cagliari, Cagliari, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Cagliari</sa:organization><sa:city>Cagliari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0275">INFN, 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Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Trieste</sa:organization><sa:city>Trieste</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0300">INFN, Sezione di Trieste, Trieste, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0610" affiliation-id="S0370269323000643-22e3a6d8462b39bb6c155bce0c9b21cd"><ce:label>61</ce:label><ce:textfn>Inha University, Incheon, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Inha University</sa:organization><sa:city>Incheon</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0305">Inha University, Incheon, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0620" affiliation-id="S0370269323000643-d4941ebd67ab9bbf96b1e6a906f4397c"><ce:label>62</ce:label><ce:textfn>Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands</ce:textfn><sa:affiliation><sa:organization>Institute for Gravitational and Subatomic Physics (GRASP)</sa:organization><sa:organization>Utrecht University/Nikhef</sa:organization><sa:city>Utrecht</sa:city><sa:country>Netherlands</sa:country></sa:affiliation><ce:source-text id="srct0310">Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands</ce:source-text></ce:affiliation><ce:affiliation id="aff0630" affiliation-id="S0370269323000643-ffd3d07e3cd462f194d82b1a01fbabeb"><ce:label>63</ce:label><ce:textfn>Institute for Nuclear Research, Academy of Sciences, Moscow, Russia</ce:textfn><sa:affiliation><sa:organization>Institute for Nuclear Research</sa:organization><sa:organization>Academy of Sciences</sa:organization><sa:city>Moscow</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0315">Institute for Nuclear Research, Academy of Sciences, Moscow, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0640" affiliation-id="S0370269323000643-c0ed70d0f039726935bd0309f74fcd8a"><ce:label>64</ce:label><ce:textfn>Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia</ce:textfn><sa:affiliation><sa:organization>Institute of Experimental Physics</sa:organization><sa:organization>Slovak Academy of Sciences</sa:organization><sa:city>Košice</sa:city><sa:country>Slovakia</sa:country></sa:affiliation><ce:source-text id="srct0320">Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia</ce:source-text></ce:affiliation><ce:affiliation id="aff0650" affiliation-id="S0370269323000643-6fa96ff1fcc4aaff09633c1217f06ff0"><ce:label>65</ce:label><ce:textfn>Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India</ce:textfn><sa:affiliation><sa:organization>Institute of Physics</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Bhubaneswar</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0325">Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0660" affiliation-id="S0370269323000643-9f77e70dcbde10c6ed3d37b3b0071107"><ce:label>66</ce:label><ce:textfn>Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Institute of Physics of the Czech Academy of Sciences</sa:organization><sa:city>Prague</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0330">Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0670" affiliation-id="S0370269323000643-3309ea65bfe3ae2f96b232545cb67043"><ce:label>67</ce:label><ce:textfn>Institute of Space Science (ISS), Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>Institute of Space Science (ISS)</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0335">Institute of Space Science (ISS), Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff0680" affiliation-id="S0370269323000643-2759820a41f8b0ece8c42aa319465ed5"><ce:label>68</ce:label><ce:textfn>Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Institut für Kernphysik</sa:organization><sa:organization>Johann Wolfgang Goethe-Universität Frankfurt</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0340">Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0690" affiliation-id="S0370269323000643-b18018b82a469dc4e4eb94f595cf4812"><ce:label>69</ce:label><ce:textfn>Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:textfn><sa:affiliation><sa:organization>Instituto de Ciencias Nucleares</sa:organization><sa:organization>Universidad Nacional Autónoma de México</sa:organization><sa:city>Mexico City</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0345">Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0700" affiliation-id="S0370269323000643-2aa194eda46d62198ea9b929240200b8"><ce:label>70</ce:label><ce:textfn>Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil</ce:textfn><sa:affiliation><sa:organization>Instituto de Física</sa:organization><sa:organization>Universidade Federal do Rio Grande do Sul (UFRGS)</sa:organization><sa:city>Porto Alegre</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0350">Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff0710" affiliation-id="S0370269323000643-9e9b95fa2c082308cb0efc3488541c67"><ce:label>71</ce:label><ce:textfn>Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:textfn><sa:affiliation><sa:organization>Instituto de Física</sa:organization><sa:organization>Universidad Nacional Autónoma de México</sa:organization><sa:city>Mexico City</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0355">Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0720" affiliation-id="S0370269323000643-4ef9aeae6b2f74e366edd12aafbea4cf"><ce:label>72</ce:label><ce:textfn>iThemba LABS, National Research Foundation, Somerset West, South Africa</ce:textfn><sa:affiliation><sa:organization>iThemba LABS</sa:organization><sa:organization>National Research Foundation</sa:organization><sa:city>Somerset West</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0360">iThemba LABS, National Research Foundation, Somerset West, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff0730" affiliation-id="S0370269323000643-2f4c4db0447d27fd2e43117b90bb74d4"><ce:label>73</ce:label><ce:textfn>Jeonbuk National University, Jeonju, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Jeonbuk National University</sa:organization><sa:city>Jeonju</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0365">Jeonbuk National University, Jeonju, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0740" affiliation-id="S0370269323000643-81ec7d4c8c69486a57ef1ca6a567076c"><ce:label>74</ce:label><ce:textfn>Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik</sa:organization><sa:organization>Fachbereich Informatik und Mathematik</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0370">Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0750" affiliation-id="S0370269323000643-53c65b0bfaf5dee3f840aa6c9e183978"><ce:label>75</ce:label><ce:textfn>Joint Institute for Nuclear Research (JINR), Dubna, Russia</ce:textfn><sa:affiliation><sa:organization>Joint Institute for Nuclear Research (JINR)</sa:organization><sa:city>Dubna</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0375">Joint Institute for Nuclear Research (JINR), Dubna, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0760" affiliation-id="S0370269323000643-c1d849875c0de0db6476f9cc96b07dd2"><ce:label>76</ce:label><ce:textfn>Korea Institute of Science and Technology Information, Daejeon, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Korea Institute of Science and Technology Information</sa:organization><sa:city>Daejeon</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0380">Korea Institute of Science and Technology Information, Daejeon, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0770" affiliation-id="S0370269323000643-c5cd420fa3d6b7c79dc3ab9e7c2dfb06"><ce:label>77</ce:label><ce:textfn>KTO Karatay University, Konya, Turkey</ce:textfn><sa:affiliation><sa:organization>KTO Karatay University</sa:organization><sa:city>Konya</sa:city><sa:country>Turkey</sa:country></sa:affiliation><ce:source-text id="srct0385">KTO Karatay University, Konya, Turkey</ce:source-text></ce:affiliation><ce:affiliation id="aff0780" affiliation-id="S0370269323000643-0bdd954451acb47f491c8c4996fa87da"><ce:label>78</ce:label><ce:textfn>Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France</ce:textfn><sa:affiliation><sa:organization>Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie</sa:organization><sa:city>Orsay</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0390">Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France</ce:source-text></ce:affiliation><ce:affiliation id="aff0790" affiliation-id="S0370269323000643-1487532e5bfe30325bc10c0583f7c38e"><ce:label>79</ce:label><ce:textfn>Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France</ce:textfn><sa:affiliation><sa:organization>Laboratoire de Physique Subatomique et de Cosmologie</sa:organization><sa:organization>Université Grenoble-Alpes</sa:organization><sa:organization>CNRS-IN2P3</sa:organization><sa:city>Grenoble</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0395">Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France</ce:source-text></ce:affiliation><ce:affiliation id="aff0800" affiliation-id="S0370269323000643-dd5512fbf2faf90b56635e0b411d44a2"><ce:label>80</ce:label><ce:textfn>Lawrence Berkeley National Laboratory, Berkeley, CA, United States</ce:textfn><sa:affiliation><sa:organization>Lawrence Berkeley National Laboratory</sa:organization><sa:city>Berkeley</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0400">Lawrence Berkeley National Laboratory, Berkeley, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0810" affiliation-id="S0370269323000643-ed03eefd58007822249c697d882deffc"><ce:label>81</ce:label><ce:textfn>Lund University Department of Physics, Division of Particle Physics, Lund, Sweden</ce:textfn><sa:affiliation><sa:organization>Lund University Department of Physics</sa:organization><sa:organization>Division of Particle Physics</sa:organization><sa:city>Lund</sa:city><sa:country>Sweden</sa:country></sa:affiliation><ce:source-text id="srct0405">Lund University Department of Physics, Division of Particle Physics, Lund, Sweden</ce:source-text></ce:affiliation><ce:affiliation id="aff0820" affiliation-id="S0370269323000643-ecd2d28e753f4396ad11c81a0c3e50ba"><ce:label>82</ce:label><ce:textfn>Moscow Institute for Physics and Technology, Moscow, Russia</ce:textfn><sa:affiliation><sa:organization>Moscow Institute for Physics and Technology</sa:organization><sa:city>Moscow</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0410">Moscow Institute for Physics and Technology, Moscow, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0830" affiliation-id="S0370269323000643-5b47ca06644b7f88e2267e9accbe867b"><ce:label>83</ce:label><ce:textfn>Nagasaki Institute of Applied Science, Nagasaki, Japan</ce:textfn><sa:affiliation><sa:organization>Nagasaki Institute of Applied Science</sa:organization><sa:city>Nagasaki</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0415">Nagasaki Institute of Applied Science, Nagasaki, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0840" affiliation-id="S0370269323000643-61cd55f0b96e9831d6063318d9967d43"><ce:label>84</ce:label><ce:textfn>Nara Women's University (NWU), Nara, Japan</ce:textfn><sa:affiliation><sa:organization>Nara Women's University (NWU)</sa:organization><sa:city>Nara</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0420">Nara Women's University (NWU), Nara, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0850" affiliation-id="S0370269323000643-7f50d99eb340e408fbc323cbce0c8905"><ce:label>85</ce:label><ce:textfn>National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece</ce:textfn><sa:affiliation><sa:organization>National and Kapodistrian University of Athens</sa:organization><sa:organization>School of Science</sa:organization><sa:organization>Department of Physics</sa:organization><sa:city>Athens</sa:city><sa:country>Greece</sa:country></sa:affiliation><ce:source-text id="srct0425">National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece</ce:source-text></ce:affiliation><ce:affiliation id="aff0860" affiliation-id="S0370269323000643-58192de9f95a93cb8bf485b9a5647bf5"><ce:label>86</ce:label><ce:textfn>National Centre for Nuclear Research, Warsaw, Poland</ce:textfn><sa:affiliation><sa:organization>National Centre for Nuclear Research</sa:organization><sa:city>Warsaw</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0430">National Centre for Nuclear Research, Warsaw, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff0870" affiliation-id="S0370269323000643-5c73c93f2c9aba1b7fcfa30741ce17ca"><ce:label>87</ce:label><ce:textfn>National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India</ce:textfn><sa:affiliation><sa:organization>National Institute of Science Education and Research</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Jatni</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0435">National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0880" affiliation-id="S0370269323000643-1e229a9758219f877a7f903c2dde9c2b"><ce:label>88</ce:label><ce:textfn>National Nuclear Research Center, Baku, Azerbaijan</ce:textfn><sa:affiliation><sa:organization>National Nuclear Research Center</sa:organization><sa:city>Baku</sa:city><sa:country>Azerbaijan</sa:country></sa:affiliation><ce:source-text id="srct0440">National Nuclear Research Center, Baku, Azerbaijan</ce:source-text></ce:affiliation><ce:affiliation id="aff0890" affiliation-id="S0370269323000643-ef899e70d7f524a8b4dc85511af17a46"><ce:label>89</ce:label><ce:textfn>National Research Centre Kurchatov Institute, Moscow, Russia</ce:textfn><sa:affiliation><sa:organization>National Research Centre Kurchatov Institute</sa:organization><sa:city>Moscow</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0445">National Research Centre Kurchatov Institute, Moscow, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0900" affiliation-id="S0370269323000643-3e7cb9222cae4e75df0d93d2ea96329f"><ce:label>90</ce:label><ce:textfn>Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark</ce:textfn><sa:affiliation><sa:organization>Niels Bohr Institute</sa:organization><sa:organization>University of Copenhagen</sa:organization><sa:city>Copenhagen</sa:city><sa:country>Denmark</sa:country></sa:affiliation><ce:source-text id="srct0450">Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark</ce:source-text></ce:affiliation><ce:affiliation id="aff0910" affiliation-id="S0370269323000643-11b144edfb6d992e108d17991cc7681b"><ce:label>91</ce:label><ce:textfn>Nikhef, National Institute for Subatomic Physics, Amsterdam, Netherlands</ce:textfn><sa:affiliation><sa:organization>Nikhef</sa:organization><sa:organization>National Institute for Subatomic Physics</sa:organization><sa:city>Amsterdam</sa:city><sa:country>Netherlands</sa:country></sa:affiliation><ce:source-text id="srct0455">Nikhef, National institute for subatomic physics, Amsterdam, Netherlands</ce:source-text></ce:affiliation><ce:affiliation id="aff0920" affiliation-id="S0370269323000643-3c430648d0afb815a8e84258861895f0"><ce:label>92</ce:label><ce:textfn>NRC Kurchatov Institute IHEP, Protvino, Russia</ce:textfn><sa:affiliation><sa:organization>NRC Kurchatov Institute</sa:organization><sa:organization>IHEP</sa:organization><sa:city>Protvino</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0460">NRC Kurchatov Institute IHEP, Protvino, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0930" affiliation-id="S0370269323000643-3928c351c9b4797dafcf46dd65eee0cd"><ce:label>93</ce:label><ce:textfn>NRC «Kurchatov»Institute – ITEP, Moscow, Russia</ce:textfn><sa:affiliation><sa:organization>NRC «Kurchatov»Institute – ITEP</sa:organization><sa:city>Moscow</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0465">NRC «Kurchatov»Institute – ITEP, Moscow, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0940" affiliation-id="S0370269323000643-b4ad181607b9ab8fbc66e2278cdf943a"><ce:label>94</ce:label><ce:textfn>NRNU Moscow Engineering Physics Institute, Moscow, Russia</ce:textfn><sa:affiliation><sa:organization>NRNU Moscow Engineering Physics Institute</sa:organization><sa:city>Moscow</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0470">NRNU Moscow Engineering Physics Institute, Moscow, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff0950" affiliation-id="S0370269323000643-7ffc71d9a245ae3927bb4d373ff478ed"><ce:label>95</ce:label><ce:textfn>Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Group</sa:organization><sa:organization>STFC Daresbury Laboratory</sa:organization><sa:city>Daresbury</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0475">Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff0960" affiliation-id="S0370269323000643-35f0d35e6b51b7c24d9b3bc939a45b30"><ce:label>96</ce:label><ce:textfn>Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Institute of the Czech Academy of Sciences</sa:organization><sa:city>Řež u Prahy</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0480">Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0970" affiliation-id="S0370269323000643-43c0b745ad096562d858417552718575"><ce:label>97</ce:label><ce:textfn>Oak Ridge National Laboratory, Oak Ridge, TN, United States</ce:textfn><sa:affiliation><sa:organization>Oak Ridge National Laboratory</sa:organization><sa:city>Oak Ridge</sa:city><sa:state>TN</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0485">Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0980" affiliation-id="S0370269323000643-e4e32a64c7e436b0f2a688687dedfe5e"><ce:label>98</ce:label><ce:textfn>Ohio State University, Columbus, OH, United States</ce:textfn><sa:affiliation><sa:organization>Ohio State University</sa:organization><sa:city>Columbus</sa:city><sa:state>OH</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0490">Ohio State University, Columbus, Ohio, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0990" affiliation-id="S0370269323000643-90511898a8b3f881cbde7907eee7df47"><ce:label>99</ce:label><ce:textfn>Petersburg Nuclear Physics Institute, Gatchina, Russia</ce:textfn><sa:affiliation><sa:organization>Petersburg Nuclear Physics Institute</sa:organization><sa:city>Gatchina</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0495">Petersburg Nuclear Physics Institute, Gatchina, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff1000" affiliation-id="S0370269323000643-3c4eb171ede7112c64f1bc2c164f4732"><ce:label>100</ce:label><ce:textfn>Physics Department, Faculty of Science, University of Zagreb, Zagreb, Croatia</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>Faculty of Science</sa:organization><sa:organization>University of Zagreb</sa:organization><sa:city>Zagreb</sa:city><sa:country>Croatia</sa:country></sa:affiliation><ce:source-text id="srct0500">Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia</ce:source-text></ce:affiliation><ce:affiliation id="aff1010" affiliation-id="S0370269323000643-30d6dbb96747c914840798f88bf250cb"><ce:label>101</ce:label><ce:textfn>Physics Department, Panjab University, Chandigarh, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>Panjab University</sa:organization><sa:city>Chandigarh</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0505">Physics Department, Panjab University, Chandigarh, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1020" affiliation-id="S0370269323000643-712bc80495e97050498604da603cb5cc"><ce:label>102</ce:label><ce:textfn>Physics Department, University of Jammu, Jammu, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>University of Jammu</sa:organization><sa:city>Jammu</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0510">Physics Department, University of Jammu, Jammu, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1030" affiliation-id="S0370269323000643-500771749243cadcbaeeac3f726e6a62"><ce:label>103</ce:label><ce:textfn>Physics Department, University of Rajasthan, Jaipur, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>University of Rajasthan</sa:organization><sa:city>Jaipur</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0515">Physics Department, University of Rajasthan, Jaipur, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1040" affiliation-id="S0370269323000643-0886f70da565231fd0e43398021fd604"><ce:label>104</ce:label><ce:textfn>Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany</ce:textfn><sa:affiliation><sa:organization>Physikalisches Institut</sa:organization><sa:organization>Eberhard-Karls-Universität Tübingen</sa:organization><sa:city>Tübingen</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0520">Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1050" affiliation-id="S0370269323000643-1c67678e124982de924355cdd6d6b91b"><ce:label>105</ce:label><ce:textfn>Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany</ce:textfn><sa:affiliation><sa:organization>Physikalisches Institut</sa:organization><sa:organization>Ruprecht-Karls-Universität Heidelberg</sa:organization><sa:city>Heidelberg</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0525">Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1060" affiliation-id="S0370269323000643-7a5765cc3928149c8b77128cd52462ae"><ce:label>106</ce:label><ce:textfn>Physik Department, Technische Universität München, Munich, Germany</ce:textfn><sa:affiliation><sa:organization>Physik Department</sa:organization><sa:organization>Technische Universität München</sa:organization><sa:city>Munich</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0530">Physik Department, Technische Universität München, Munich, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1070" affiliation-id="S0370269323000643-0a06f4f5fb60a25c8f1936ed71c96cfe"><ce:label>107</ce:label><ce:textfn>Politecnico di Bari and Sezione INFN, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>Politecnico di Bari</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0535">Politecnico di Bari and Sezione INFN, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1080" affiliation-id="S0370269323000643-3451d358e90b759225813c9b9a6f098c"><ce:label>108</ce:label><ce:textfn>Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany</ce:textfn><sa:affiliation><sa:organization>Research Division</sa:organization><sa:organization>ExtreMe Matter Institute EMMI</sa:organization><sa:organization>GSI Helmholtzzentrum für Schwerionenforschung GmbH</sa:organization><sa:city>Darmstadt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0540">Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1090" affiliation-id="S0370269323000643-bdc21ecb323f7b76ee73d71fa687e6c3"><ce:label>109</ce:label><ce:textfn>Russian Federal Nuclear Center (VNIIEF), Sarov, Russia</ce:textfn><sa:affiliation><sa:organization>Russian Federal Nuclear Center (VNIIEF)</sa:organization><sa:city>Sarov</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0545">Russian Federal Nuclear Center (VNIIEF), Sarov, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff1100" affiliation-id="S0370269323000643-2771a2ae295804be11a4a937b894a546"><ce:label>110</ce:label><ce:textfn>Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Saha Institute of Nuclear Physics</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0550">Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1110" affiliation-id="S0370269323000643-b602cde70213041ca40ec5720e4e3e75"><ce:label>111</ce:label><ce:textfn>School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom</ce:textfn><sa:affiliation><sa:organization>School of Physics and Astronomy</sa:organization><sa:organization>University of Birmingham</sa:organization><sa:city>Birmingham</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0555">School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff1120" affiliation-id="S0370269323000643-1a1f0f3ae33ba0323ad4da08a216f4c3"><ce:label>112</ce:label><ce:textfn>Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru</ce:textfn><sa:affiliation><sa:organization>Sección Física</sa:organization><sa:organization>Departamento de Ciencias</sa:organization><sa:organization>Pontificia Universidad Católica del Perú</sa:organization><sa:city>Lima</sa:city><sa:country>Peru</sa:country></sa:affiliation><ce:source-text id="srct0560">Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru</ce:source-text></ce:affiliation><ce:affiliation id="aff1130" affiliation-id="S0370269323000643-e0932a7de5094071a8671adb41fe6812"><ce:label>113</ce:label><ce:textfn>St. Petersburg State University, St. Petersburg, Russia</ce:textfn><sa:affiliation><sa:organization>St. Petersburg State University</sa:organization><sa:city>St. Petersburg</sa:city><sa:country>Russia</sa:country></sa:affiliation><ce:source-text id="srct0565">St. Petersburg State University, St. Petersburg, Russia</ce:source-text></ce:affiliation><ce:affiliation id="aff1140" affiliation-id="S0370269323000643-24347400090c9a1efb87e7537af7461b"><ce:label>114</ce:label><ce:textfn>Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria</ce:textfn><sa:affiliation><sa:organization>Stefan Meyer Institut für Subatomare Physik (SMI)</sa:organization><sa:city>Vienna</sa:city><sa:country>Austria</sa:country></sa:affiliation><ce:source-text id="srct0570">Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria</ce:source-text></ce:affiliation><ce:affiliation id="aff1150" affiliation-id="S0370269323000643-920c67cc008319f3917103203f6dd703"><ce:label>115</ce:label><ce:textfn>SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes, France</ce:textfn><sa:affiliation><sa:organization>SUBATECH</sa:organization><sa:organization>IMT Atlantique</sa:organization><sa:organization>Université de Nantes</sa:organization><sa:organization>CNRS-IN2P3</sa:organization><sa:city>Nantes</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0575">SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1160" affiliation-id="S0370269323000643-c3972f6af24eef12cc2ae3a53d6a7623"><ce:label>116</ce:label><ce:textfn>Suranaree University of Technology, Nakhon Ratchasima, Thailand</ce:textfn><sa:affiliation><sa:organization>Suranaree University of Technology</sa:organization><sa:city>Nakhon Ratchasima</sa:city><sa:country>Thailand</sa:country></sa:affiliation><ce:source-text id="srct0580">Suranaree University of Technology, Nakhon Ratchasima, Thailand</ce:source-text></ce:affiliation><ce:affiliation id="aff1170" affiliation-id="S0370269323000643-c78ab7c321a491b1927107824e5f2c30"><ce:label>117</ce:label><ce:textfn>Technical University of Košice, Košice, Slovakia</ce:textfn><sa:affiliation><sa:organization>Technical University of Košice</sa:organization><sa:city>Košice</sa:city><sa:country>Slovakia</sa:country></sa:affiliation><ce:source-text id="srct0585">Technical University of Košice, Košice, Slovakia</ce:source-text></ce:affiliation><ce:affiliation id="aff1180" affiliation-id="S0370269323000643-425836ada61fe0d3ee4ebf5b87598919"><ce:label>118</ce:label><ce:textfn>The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland</ce:textfn><sa:affiliation><sa:organization>The Henryk Niewodniczanski Institute of Nuclear Physics</sa:organization><sa:organization>Polish Academy of Sciences</sa:organization><sa:city>Cracow</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0590">The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff1190" affiliation-id="S0370269323000643-302a66584ead3636c3e74b86b51cf474"><ce:label>119</ce:label><ce:textfn>The University of Texas at Austin, Austin, TX, United States</ce:textfn><sa:affiliation><sa:organization>The University of Texas at Austin</sa:organization><sa:city>Austin</sa:city><sa:state>TX</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0595">The University of Texas at Austin, Austin, Texas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1200" affiliation-id="S0370269323000643-48ea3700cc067946e6c3032832f0ce0a"><ce:label>120</ce:label><ce:textfn>Universidad Autónoma de Sinaloa, Culiacán, Mexico</ce:textfn><sa:affiliation><sa:organization>Universidad Autónoma de Sinaloa</sa:organization><sa:city>Culiacán</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0600">Universidad Autónoma de Sinaloa, Culiacán, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff1210" affiliation-id="S0370269323000643-310945a13cafb9845fe651fc7aa661b6"><ce:label>121</ce:label><ce:textfn>Universidade de São Paulo (USP), São Paulo, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade de São Paulo (USP)</sa:organization><sa:city>São Paulo</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0605">Universidade de São Paulo (USP), São Paulo, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1220" affiliation-id="S0370269323000643-226f6a475b7a1634e1530e2077d7590e"><ce:label>122</ce:label><ce:textfn>Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade Estadual de Campinas (UNICAMP)</sa:organization><sa:city>Campinas</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0610">Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1230" affiliation-id="S0370269323000643-7582533209ee7a368c4d249782e93c03"><ce:label>123</ce:label><ce:textfn>Universidade Federal do ABC, Santo Andre, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade Federal do ABC</sa:organization><sa:city>Santo Andre</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0615">Universidade Federal do ABC, Santo Andre, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1240" affiliation-id="S0370269323000643-f37f0b36132124cce5539b1f9c518079"><ce:label>124</ce:label><ce:textfn>University of Cape Town, Cape Town, South Africa</ce:textfn><sa:affiliation><sa:organization>University of Cape Town</sa:organization><sa:city>Cape Town</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0620">University of Cape Town, Cape Town, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff1250" affiliation-id="S0370269323000643-58fe35a5cc9d91eea34fd6e19293599b"><ce:label>125</ce:label><ce:textfn>University of Houston, Houston, TX, United States</ce:textfn><sa:affiliation><sa:organization>University of Houston</sa:organization><sa:city>Houston</sa:city><sa:state>TX</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0625">University of Houston, Houston, Texas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1260" affiliation-id="S0370269323000643-e9be2b1885c8285b7cb7b6534ca93d98"><ce:label>126</ce:label><ce:textfn>University of Jyväskylä, Jyväskylä, Finland</ce:textfn><sa:affiliation><sa:organization>University of Jyväskylä</sa:organization><sa:city>Jyväskylä</sa:city><sa:country>Finland</sa:country></sa:affiliation><ce:source-text id="srct0630">University of Jyväskylä, Jyväskylä, Finland</ce:source-text></ce:affiliation><ce:affiliation id="aff1270" affiliation-id="S0370269323000643-6e2074f6fd0b88987501b22e5d74f9c9"><ce:label>127</ce:label><ce:textfn>University of Kansas, Lawrence, KS, United States</ce:textfn><sa:affiliation><sa:organization>University of Kansas</sa:organization><sa:city>Lawrence</sa:city><sa:state>KS</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0635">University of Kansas, Lawrence, Kansas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1280" affiliation-id="S0370269323000643-f538fc5a59ec1d1c8f86d2a83cb4ced6"><ce:label>128</ce:label><ce:textfn>University of Liverpool, Liverpool, United Kingdom</ce:textfn><sa:affiliation><sa:organization>University of Liverpool</sa:organization><sa:city>Liverpool</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0640">University of Liverpool, Liverpool, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff1290" affiliation-id="S0370269323000643-5507e3344bf7b46ba698c2c045a8aba6"><ce:label>129</ce:label><ce:textfn>University of Science and Technology of China, Hefei, China</ce:textfn><sa:affiliation><sa:organization>University of Science and Technology of China</sa:organization><sa:city>Hefei</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0645">University of Science and Technology of China, Hefei, China</ce:source-text></ce:affiliation><ce:affiliation id="aff1300" affiliation-id="S0370269323000643-b080b99a0650ae4321512ce14fe0060e"><ce:label>130</ce:label><ce:textfn>University of South-Eastern Norway, Tonsberg, Norway</ce:textfn><sa:affiliation><sa:organization>University of South-Eastern Norway</sa:organization><sa:city>Tonsberg</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0650">University of South-Eastern Norway, Tonsberg, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff1310" affiliation-id="S0370269323000643-d6620449e5365c80224ff22fe1ca4e4a"><ce:label>131</ce:label><ce:textfn>University of Tennessee, Knoxville, TN, United States</ce:textfn><sa:affiliation><sa:organization>University of Tennessee</sa:organization><sa:city>Knoxville</sa:city><sa:state>TN</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0655">University of Tennessee, Knoxville, Tennessee, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1320" affiliation-id="S0370269323000643-6d43022152ccaa62119e4287bf3b5270"><ce:label>132</ce:label><ce:textfn>University of the Witwatersrand, Johannesburg, South Africa</ce:textfn><sa:affiliation><sa:organization>University of the Witwatersrand</sa:organization><sa:city>Johannesburg</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0660">University of the Witwatersrand, Johannesburg, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff1330" affiliation-id="S0370269323000643-9b9071fc23073da0e4317b3863c91b9e"><ce:label>133</ce:label><ce:textfn>University of Tokyo, Tokyo, Japan</ce:textfn><sa:affiliation><sa:organization>University of Tokyo</sa:organization><sa:city>Tokyo</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0665">University of Tokyo, Tokyo, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff1340" affiliation-id="S0370269323000643-da3fb88e8779358429638ee8f8dc4cb5"><ce:label>134</ce:label><ce:textfn>University of Tsukuba, Tsukuba, Japan</ce:textfn><sa:affiliation><sa:organization>University of Tsukuba</sa:organization><sa:city>Tsukuba</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0670">University of Tsukuba, Tsukuba, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff1350" affiliation-id="S0370269323000643-9fc1715eadb7b2ba79b1a87cef8015cd"><ce:label>135</ce:label><ce:textfn>University Politehnica of Bucharest, Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>University Politehnica of Bucharest</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0675">University Politehnica of Bucharest, Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff1360" affiliation-id="S0370269323000643-43e2ac82e11c4ca2fabbc7e457c74b37"><ce:label>136</ce:label><ce:textfn>Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France</ce:textfn><sa:affiliation><sa:organization>Université Clermont Auvergne</sa:organization><sa:organization>CNRS/IN2P3</sa:organization><sa:organization>LPC</sa:organization><sa:city>Clermont-Ferrand</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0680">Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1370" affiliation-id="S0370269323000643-08f13150d05b32d440951efec6f80d29"><ce:label>137</ce:label><ce:textfn>Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France</ce:textfn><sa:affiliation><sa:organization>Université de Lyon</sa:organization><sa:organization>CNRS/IN2P3</sa:organization><sa:organization>Institut de Physique des 2 Infinis de Lyon</sa:organization><sa:city>Lyon</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0685">Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1380" affiliation-id="S0370269323000643-766d0075165899c16752f5f7ff1b8771"><ce:label>138</ce:label><ce:textfn>Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France</ce:textfn><sa:affiliation><sa:organization>Université de Strasbourg</sa:organization><sa:organization>CNRS</sa:organization><sa:organization>IPHC UMR 7178</sa:organization><sa:city>Strasbourg</sa:city><sa:postal-code>F-67000</sa:postal-code><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0690">Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1390" affiliation-id="S0370269323000643-36117066550bb6faaf9708b1b0ea670e"><ce:label>139</ce:label><ce:textfn>Université Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France</ce:textfn><sa:affiliation><sa:organization>Université Paris-Saclay</sa:organization><sa:organization>Centre d'Etudes de Saclay (CEA)</sa:organization><sa:organization>IRFU</sa:organization><sa:organization>Départment de Physique Nucléaire (DPhN)</sa:organization><sa:city>Saclay</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0695">Université Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1400" affiliation-id="S0370269323000643-47ef552cdaf90d05d791574b14a8dcf9"><ce:label>140</ce:label><ce:textfn>Università degli Studi di Foggia, Foggia, Italy</ce:textfn><sa:affiliation><sa:organization>Università degli Studi di Foggia</sa:organization><sa:city>Foggia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0700">Università degli Studi di Foggia, Foggia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1410" affiliation-id="S0370269323000643-83cd0c0929f483f88e3d2cb59f064287"><ce:label>141</ce:label><ce:textfn>Università di Brescia, Brescia, Italy</ce:textfn><sa:affiliation><sa:organization>Università di Brescia</sa:organization><sa:city>Brescia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0705">Università di Brescia, Brescia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1420" affiliation-id="S0370269323000643-f1ae52f852d4d7d99988b3e872f887e4"><ce:label>142</ce:label><ce:textfn>Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Variable Energy Cyclotron Centre</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0710">Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1430" affiliation-id="S0370269323000643-bfd4fe0d3b8675ea55d96ce64033d817"><ce:label>143</ce:label><ce:textfn>Warsaw University of Technology, Warsaw, Poland</ce:textfn><sa:affiliation><sa:organization>Warsaw University of Technology</sa:organization><sa:city>Warsaw</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0715">Warsaw University of Technology, Warsaw, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff1440" affiliation-id="S0370269323000643-4e11d38f810a3540206ffb5151a35d3b"><ce:label>144</ce:label><ce:textfn>Wayne State University, Detroit, MI, United States</ce:textfn><sa:affiliation><sa:organization>Wayne State University</sa:organization><sa:city>Detroit</sa:city><sa:state>MI</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0720">Wayne State University, Detroit, Michigan, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1450" affiliation-id="S0370269323000643-0b76517a6de7ff0579ec14780f79d81b"><ce:label>145</ce:label><ce:textfn>Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany</ce:textfn><sa:affiliation><sa:organization>Westfälische Wilhelms-Universität Münster</sa:organization><sa:organization>Institut für Kernphysik</sa:organization><sa:city>Münster</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0725">Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1460" affiliation-id="S0370269323000643-6b1ae45228f1f040c55767dc107b589d"><ce:label>146</ce:label><ce:textfn>Wigner Research Centre for Physics, Budapest, Hungary</ce:textfn><sa:affiliation><sa:organization>Wigner Research Centre for Physics</sa:organization><sa:city>Budapest</sa:city><sa:country>Hungary</sa:country></sa:affiliation><ce:source-text id="srct0730">Wigner Research Centre for Physics, Budapest, Hungary</ce:source-text></ce:affiliation><ce:affiliation id="aff1470" affiliation-id="S0370269323000643-96e79e7381eab77664732c3553e247e8"><ce:label>147</ce:label><ce:textfn>Yale University, New Haven, CT, United States</ce:textfn><sa:affiliation><sa:organization>Yale University</sa:organization><sa:city>New Haven</sa:city><sa:state>CT</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0735">Yale University, New Haven, Connecticut, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1480" affiliation-id="S0370269323000643-f3db17ceaf6ea4190e2176a6a129c96b"><ce:label>148</ce:label><ce:textfn>Yonsei University, Seoul, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Yonsei University</sa:organization><sa:city>Seoul</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0740">Yonsei University, Seoul, Republic of Korea</ce:source-text></ce:affiliation><ce:footnote id="fn0010"><ce:label>I</ce:label><ce:note-para id="np0010">Deceased.</ce:note-para></ce:footnote><ce:footnote id="fn0020"><ce:label>II</ce:label><ce:note-para id="np0020">Also at: Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA), Bologna, Italy.</ce:note-para></ce:footnote><ce:footnote id="fn0030"><ce:label>III</ce:label><ce:note-para id="np0030">Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy.</ce:note-para></ce:footnote><ce:footnote id="fn0040"><ce:label>IV</ce:label><ce:note-para id="np0040">Also at: M.V. Lomonosov Moscow State University, D.V. Skobeltsyn Institute of Nuclear, Physics, Moscow, Russia.</ce:note-para></ce:footnote><ce:footnote id="fn0050"><ce:label>V</ce:label><ce:note-para id="np0050">Also at: Department of Applied Physics, Aligarh Muslim University, Aligarh, India.</ce:note-para></ce:footnote><ce:footnote id="fn0060"><ce:label>VI</ce:label><ce:note-para id="np0060">Also at: Institute of Theoretical Physics, University of Wroclaw, Poland.</ce:note-para></ce:footnote><ce:footnote id="fn0070"><ce:label>VII</ce:label><ce:note-para id="np0070">Also at: University of Kansas, Lawrence, Kansas, United States.</ce:note-para></ce:footnote></ce:author-group></ce:collaboration><ce:footnote id="fn0080"><ce:label>⋆</ce:label><ce:note-para id="np0080"><ce:italic>E-mail address:</ce:italic> <ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/text/html" xlink:href="mailto:alice-publications@cern.ch" id="inf0020">alice-publications@cern.ch</ce:inter-ref>.</ce:note-para></ce:footnote></ce:author-group><ce:date-received day="20" month="5" year="2022"/><ce:date-revised day="17" month="1" year="2023"/><ce:date-accepted day="17" month="1" year="2023"/><ce:miscellaneous id="ms0010">Editor: M. Doser</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0070">We present the first systematic comparison of the charged-particle pseudorapidity densities for three widely different collision systems, pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb, at the top energy of the Large Hadron Collider (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>) measured over a wide pseudorapidity range (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.5</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5</mml:mn></mml:math>), the widest possible among the four experiments at that facility. The systematic uncertainties are minimised since the measurements are recorded by the same experimental apparatus (ALICE). The distributions for p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb collisions are determined as a function of the centrality of the collisions, while results from pp collisions are reported for inelastic events with at least one charged particle at midrapidity. The charged-particle pseudorapidity densities are, under simple and robust assumptions, transformed to charged-particle rapidity densities. This allows for the calculation and the presentation of the evolution of the width of the rapidity distributions and of a lower bound on the Bjorken energy density, as a function of the number of participants in all three collision systems. We find a decreasing width of the particle production, and roughly a smooth ten fold increase in the energy density, as the system size grows, which is consistent with a gradually higher dense phase of matter.</ce:simple-para></ce:abstract-sec></ce:abstract></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">The number of charged particles produced in energetic nuclear collisions is an important indicator for the strong interaction processes that determine the particle production at the sub-nucleonic level. In particular, the production of charged particles is expected to reflect the number of quark and gluon collisions occurring during the initial stages of the reaction. The total number of particles produced also provides information on the energy transfer available from the initial colliding beams to particle production, as a consequence of nuclear stopping <ce:cross-ref refid="br0010" id="crf10940">[1]</ce:cross-ref>. In order to help unravel this complex scenario it is important to compare the particle production amongst collision systems of different sizes over a wide kinematic range.</ce:para><ce:para id="pr0020">We present the measured charged-particle pseudorapidity density, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math>, for pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb (previously published <ce:cross-ref refid="br0020" id="crf10950">[2]</ce:cross-ref>) collisions at the same collision energy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> in the nucleon–nucleon centre-of-mass reference frame. This is, at present, the maximum available energy at CERN's Large Hadron Collider (LHC) for Pb<ce:glyph name="sbnd"/>Pb collisions. The measurements were carried out using ALICE at LHC (for earlier <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math> results see for example Refs. <ce:cross-refs refid="br0030 br0040 br0050" id="crs0010">[3–5]</ce:cross-refs>). The three studied reactions have different characteristics probing widely different particle production yields and mechanisms. In Pb<ce:glyph name="sbnd"/>Pb collisions, the total particle yield for central collisions is of the order 10<ce:sup>4</ce:sup> <ce:cross-ref refid="br0020" id="crf10960">[2]</ce:cross-ref>, and a strongly coupled plasma of quarks and gluons (sQGP) is formed <ce:cross-refs refid="br0060 br0070 br0080 br0090" id="crs0020">[6–9]</ce:cross-refs>, whose collective and transport properties are currently under intense study. On the other hand, pp collisions represent the simplest possible nuclear collision system, where the average total particle production is much smaller (≈80, by integrating the measured distributions), and is to first approximation much less subject to collective effects <ce:cross-ref refid="br0100" id="crf10970">[10]</ce:cross-ref>. The p<ce:glyph name="sbnd"/>Pb system is intermediate to the other reactions, corresponding to the situation where a single nucleon probes the nucleons in a narrow cylinder of the target nucleus. The extent to which p<ce:glyph name="sbnd"/>Pb is governed by the initial state cold nuclear matter of the lead ion or whether collective phenomena in the hot and dense medium play an important role is, at present, a matter under scrutiny by the community <ce:cross-refs refid="br0100 br0110" id="crs0030">[10,11]</ce:cross-refs>.</ce:para><ce:para id="pr0030">In this letter, we compare the three reactions and present the ratios of the charged-particle pseudorapidity density distributions (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math>) of the more complex reactions to the pp distribution. Owing to ALICE's unique large acceptance in pseudorapidity, and using simple and robust assumptions, we transform the measured charged-particle pseudorapidity density distributions into charged-particle rapidity density distributions (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math>). This allows us to calculate the width of the rapidity distributions as a function of the number of participating nucleons. The parameters of the transformation also allow us to estimate a lower bound on the energy density using the well-known formula from Bjorken <ce:cross-ref refid="br0120" id="crf10980">[12]</ce:cross-ref>. An energy density exceeding the critical energy density of roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> <ce:cross-ref refid="br0130" id="crf10990">[13]</ce:cross-ref> is a necessary condition for the formation of deconfined matter of quarks and gluons, and thus it is of the utmost interest to understand the development of these energy densities across different collision systems.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">Experimental set-up, data sample, analysis method, systematic uncertainties</ce:section-title><ce:para id="pr0040">A detailed description of the ALICE detector and its performance can be found elsewhere <ce:cross-refs refid="br0140 br0150" id="crs0040">[14,15]</ce:cross-refs>. The present analysis uses the Silicon Pixel Detector (SPD) to determine the pseudorapidity densities in the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math> and the Forward Multiplicity Detector (FMD) in the ranges <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.5</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1.8</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mn>1.8</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5</mml:mn></mml:math>. The V0, comprised of two plastic scintillator discs covering <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.7</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1.7</mml:mn></mml:math> (V0C) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mn>2.8</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5.1</mml:mn></mml:math> (V0A), and the ZDC, two zero-degree calorimeters located <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>112.5</mml:mn><mml:mspace width="0.2em"/><mml:mtext>m</mml:mtext></mml:math> from the interaction point, measurements determine the collision centrality and are used for offline event selection <ce:cross-ref refid="br0020" id="crf11000">[2]</ce:cross-ref>.</ce:para><ce:para id="pr0050">The results presented are based on data from collisions at a centre-of-mass energy per nucleon pair of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> as collected by ALICE during LHC Run 1 (2013) for p<ce:glyph name="sbnd"/>Pb, and during Run 2 (2015) for pp and Pb<ce:glyph name="sbnd"/>Pb. The FMD suffered high levels of background noise during the 2016 p<ce:glyph name="sbnd"/>Pb campaign, due to the high collision rate, and this data is therefore not used for the present analysis. About 10<ce:sup>5</ce:sup> events with a minimum bias trigger requirement <ce:cross-ref refid="br0020" id="crf11010">[2]</ce:cross-ref> were analysed in the centrality range from 0% to 90% and 0% to 100% of the visible cross section for Pb<ce:glyph name="sbnd"/>Pb and p<ce:glyph name="sbnd"/>Pb collisions, respectively. The minimum bias trigger for p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb collisions in ALICE was defined as a coincidence between the V0A and V0C sides of the V0 detector.</ce:para><ce:para id="pr0060">The data from the p<ce:glyph name="sbnd"/>Pb collisions were taken in two beam configurations: one where the lead ion travelled toward positive pseudorapidity and one where it travelled toward negative pseudorapidity. The results from the latter collisions are mirrored around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>. The centre-of-mass frame in p<ce:glyph name="sbnd"/>Pb collisions does not coincide with the laboratory frame, due to the single magnetic field in the LHC, and thus the rapidity of the centre-of-mass is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">CM</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>±</mml:mo><mml:mn>0.465</mml:mn></mml:math> for the two directions, respectively, in the laboratory frame. For this reason, pseudorapidity, calculated with respect to the laboratory frame, is denoted <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub></mml:math> whenever p<ce:glyph name="sbnd"/>Pb results are presented.</ce:para><ce:para id="pr0070">Likewise, for the pp collisions, about 10<ce:sup>5</ce:sup> events with coincidence between V0A and V0C and at least one charged particle in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1</mml:mn></mml:math> were analysed. By requiring at least one charged particle at midrapidity, the so-called INEL>0 event class, the systematic uncertainty, related to the absolute normalisation to the full inelastic cross section, is reduced, while still sampling a large fraction (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>75</mml:mn><mml:mtext>%</mml:mtext></mml:math>) of the hadronic cross section <ce:cross-refs refid="br0160 br0170" id="crs0050">[16,17]</ce:cross-refs>.</ce:para><ce:para id="pr0080">The standard ALICE event selection <ce:cross-ref refid="br0180" id="crf11020">[18]</ce:cross-ref> and centrality estimator based on the V0 amplitude <ce:cross-refs refid="br0190 br0200" id="crs0060">[19,20]</ce:cross-refs> are used in this analysis. The event selection consists of: a) exclusion of background events using the timing information from the ZDC (for Pb<ce:glyph name="sbnd"/>Pb and p<ce:glyph name="sbnd"/>Pb, e.g., beam–gas interactions) and V0 detectors, b) verification of the trigger conditions, and c) a reconstructed position of the collision (primary vertex). In Pb<ce:glyph name="sbnd"/>Pb collisions, centrality is obtained from the sum amplitude in both V0 detector arrays (V0M). For p<ce:glyph name="sbnd"/>Pb only the amplitude in the array on the lead-going side (V0A or V0C) is used. In Pb<ce:glyph name="sbnd"/>Pb collisions, the 10% most peripheral collisions have substantial contributions from electromagnetic processes and are therefore not included in the results presented here <ce:cross-ref refid="br0190" id="crf11030">[19]</ce:cross-ref>.</ce:para><ce:para id="pr0090">A primary charged particle is defined as a charged particle with a mean proper lifetime <ce:italic>τ</ce:italic> larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mtext>cm</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mtext>c</mml:mtext></mml:math>, which is either a) produced directly in the interaction, or b) from decays of particles with <ce:italic>τ</ce:italic> smaller than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mtext>cm</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mtext>c</mml:mtext></mml:math> <ce:cross-ref refid="br0210" id="crf11040">[21]</ce:cross-ref>. All quantities reported here are for primary, charged particles, though “primary” is omitted in the following for brevity.</ce:para><ce:para id="pr0100">The analysis method is identical to that of previous publications <ce:cross-ref refid="br0020" id="crf11050">[2]</ce:cross-ref>: the measurement of the charged-particle pseudorapidity density at midrapidity is obtained from counting particle trajectories determined using the two layers of the SPD. The SPD has a lower transverse momentum acceptance of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:mn>50</mml:mn><mml:mspace width="0.2em"/><mml:mtext>MeV</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="italic">c</mml:mi></mml:math>, and the yield is extrapolated down to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mtext>MeV</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="italic">c</mml:mi></mml:math> via simulations. In the forward regions, the measurement is provided by the analysis of the deposited energy signal in the FMD and a statistical method is employed to calculate the inclusive number of charged particles. A data-driven correction <ce:cross-ref refid="br0220" id="crf11060">[22]</ce:cross-ref>, based on separate measurements exploiting displaced collision vertices, is applied to remove the background from secondary particles.</ce:para><ce:para id="pr0110">Systematic uncertainty estimations for the midrapidity measurements are detailed elsewhere <ce:cross-refs refid="br0020 br0160 br0200" id="crs0070">[2,16,20]</ce:cross-refs>, and are from background suppression, transverse momentum extrapolation, weak decays, and simulations. The estimates are obtained through variation of thresholds and simulation studies. For pp (p<ce:glyph name="sbnd"/>Pb), the total systematic uncertainty amounts to 1.5% (2.7%) over the whole pseudorapidity range; while for Pb<ce:glyph name="sbnd"/>Pb the total systematic uncertainty is 2.6% at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> and 2.9% at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2</mml:mn></mml:math>. The systematic uncertainty is mostly correlated over pseudorapidity for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math>, and largely independent of centrality. The uncertainty in the forward region, estimated via variations of thresholds and simulation studies, is the same for all collision systems and is uncorrelated across <ce:italic>η</ce:italic>, amounting to 6.9% for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>3.5</mml:mn></mml:math> and 6.4% elsewhere within the forward regions <ce:cross-ref refid="br0220" id="crf11070">[22]</ce:cross-ref>. In the figures of this letter, uncorrelated, local in pseudorapidity, systematic uncertainties are indicated by open boxes on the data points, while correlated systematic uncertainties, those that affect the overall scale and typically from event classification and selection, are indicated by filled boxes to the right of the data. The systematic uncertainty on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math>, due to the centrality class definition in Pb<ce:glyph name="sbnd"/>Pb, is estimated to vary from 0.6% for the most central to 9.5% for the most peripheral class <ce:cross-ref refid="br0230" id="crf11080">[23]</ce:cross-ref>. The 80% to 90% centrality class has residual contamination from electromagnetic processes as detailed elsewhere <ce:cross-ref refid="br0190" id="crf11090">[19]</ce:cross-ref>, which gives rise to an additional 4% systematic uncertainty in the measurements. No overall systematic uncertainty has been estimated for p<ce:glyph name="sbnd"/>Pb collisions, as the centrality selection in that collision system is inherently difficult to map to the underlying dynamics of the collisions <ce:cross-ref refid="br0200" id="crf11100">[20]</ce:cross-ref>.</ce:para></ce:section><ce:section id="se0030" role="results"><ce:label>3</ce:label><ce:section-title id="st0040">Results</ce:section-title><ce:para id="pr0120"><ce:cross-ref refid="fg0010" id="crf11420">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/> shows the measured pseudorapidity densities in pp, and in central p<ce:glyph name="sbnd"/>Pb, and the previously published results for Pb<ce:glyph name="sbnd"/>Pb <ce:cross-ref refid="br0020" id="crf11120">[2]</ce:cross-ref> collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> for primary particles.</ce:para><ce:para id="pr0130">For the 5% most central Pb<ce:glyph name="sbnd"/>Pb collisions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:mo>≈</mml:mo><mml:mn>2000</mml:mn></mml:math> at midrapidity (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>) <ce:cross-ref refid="br0020" id="crf11130">[2]</ce:cross-ref>, while for p<ce:glyph name="sbnd"/>Pb collisions the distribution peaks at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mn>60</mml:mn></mml:math> around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>3</mml:mn></mml:math> in the lead-going direction (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>). For pp collisions with the INEL>0 trigger condition discussed above, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.7</mml:mn><mml:mo>±</mml:mo><mml:mn>0.2</mml:mn></mml:math> at midrapidity, consistent with previous results derived from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra <ce:cross-ref refid="br0240" id="crf11140">[24]</ce:cross-ref>.</ce:para><ce:para id="pr0140"><ce:cross-ref refid="fg0020" id="crf11430">Fig. 2</ce:cross-ref><ce:float-anchor refid="fg0020"/> shows, as a function of centrality, the measured charged-particle pseudorapidity densities for p<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>. The strategy of centrality selection for proton on nucleus reactions is explained elsewhere <ce:cross-ref refid="br0200" id="crf11160">[20]</ce:cross-ref>. The ALICE Collaboration has previously presented <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math> for Pb<ce:glyph name="sbnd"/>Pb collisions at this energy <ce:cross-ref refid="br0020" id="crf11170">[2]</ce:cross-ref>.</ce:para><ce:para id="pr0150">In <ce:cross-ref refid="fg0030" id="crf11180">Fig. 3</ce:cross-ref><ce:float-anchor refid="fg0030"/>, the charged-particle pseudorapidity densities in p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb reactions are divided by the pp distributions corresponding to the INEL>0 trigger class. The ratio is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pp</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>, where <ce:italic>X</ce:italic> labels p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb collisions, in centrality classes, as a function of pseudorapidity. In the ratios, systematic uncertainties, of common origin, are partially cancelled, and, as an estimate, the magnitude of the resulting systematic uncertainties are given only by the uncertainties in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:math> measurements, since the uncertainties are independent of the collision system. In p<ce:glyph name="sbnd"/>Pb collisions the rapidity of the centre-of-mass is non-zero, which is not taken into account in the ratios. Such a correction would require prior determination of the full Jacobian of the transformation from pseudorapidity to rapidity, which is not possible to perform reliably with the ALICE apparatus.</ce:para><ce:para id="pr0160">The ratio of the p<ce:glyph name="sbnd"/>Pb relative to the pp distributions increases with pseudorapidity from the p-going to the Pb-going direction for central collisions, which Brodsky et al. and Adil et al. <ce:cross-refs refid="br0250 br0260" id="crs0080">[25,26]</ce:cross-refs> suggest is a sign of scaling of the pp distribution with the increasing number of participants as the lead nucleus is probed by the incident proton, and thus independent proton–nucleon scatterings on the lead-ion side. A similar scaling, however, does not hold for the Pb<ce:glyph name="sbnd"/>Pb reaction. The ratios cannot be obtained by simple scaling of the elementary pp distributions. Instead, the ratio of the Pb<ce:glyph name="sbnd"/>Pb relative to the pp distributions exhibits an enhancement of particle production around midrapidity for the more central collisions which is indicative of the formation of the sQGP <ce:cross-ref refid="br0070" id="crf11190">[7]</ce:cross-ref>. Likewise, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="normal">pPb</mml:mtext></mml:mrow></mml:msub></mml:math> increases for all but the two most peripheral centrality classes as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:mn>3</mml:mn></mml:math>. In Pb<ce:glyph name="sbnd"/>Pb collisions it is seen that the various mechanisms behind the pseudorapidity distributions are more transversely directed than in pp collisions by the increase of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="normal">PbPb</mml:mtext></mml:mrow></mml:msub></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mn>0</mml:mn></mml:math></ce:para></ce:section><ce:section id="se0040"><ce:label>4</ce:label><ce:section-title id="st0050">Rapidity and energy-density dependence on system size and discussion</ce:section-title><ce:para id="pr0170">It has been shown that the charged-particle <ce:italic>rapidity</ce:italic> density (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math>) in Pb<ce:glyph name="sbnd"/>Pb collisions, to a good accuracy, follows a normal distribution over the considered rapidity interval (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo>≲</mml:mo><mml:mn>5</mml:mn></mml:math>) <ce:cross-refs refid="br0020 br0270" id="crs0090">[2,27]</ce:cross-refs>. Those results relied on calculating the average Jacobian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>J</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math> using the full <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra, at midrapidity, of charged pions and kaons as well as protons and antiprotons. Here, we use the approximation<ce:display><ce:formula id="fm0010"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mrow><mml:mi>y</mml:mi><mml:mo>≈</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">cos</mml:mi><mml:mo>⁡</mml:mo><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>ϑ</ce:italic> is the polar angle of emission, and identify <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mi>a</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>m</mml:mi></mml:math> with an effective ratio of transverse momentum over mass. With this, the effective Jacobian can be written as<ce:display><ce:formula id="fm0020"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">cosh</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>⁡</mml:mo><mml:mi>η</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0180">We further make the ansatz that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> is normal distributed for symmetric collision systems (pp and Pb<ce:glyph name="sbnd"/>Pb), so that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math> can be parameterised as<ce:display><ce:formula id="fm0030"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mi>A</mml:mi><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:msqrt><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">{</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>A</ce:italic> and <ce:italic>σ</ce:italic> are the total integral and width of the distribution, respectively, and <ce:italic>y</ce:italic> the rapidity in the centre-of-mass frame. Motivated by the observed approximate linearity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pPb</mml:mi></mml:mrow></mml:msub></mml:math> (see lower panel of <ce:cross-ref refid="fg0030" id="crf11200">Fig. 3</ce:cross-ref>), we replace <ce:italic>A</ce:italic> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mi>y</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> for the asymmetric system (p<ce:glyph name="sbnd"/>Pb) and parameterise <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub></mml:math> as<ce:display><ce:formula id="fm0040"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo id="mmlbr0001" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>α</mml:mi><mml:mi>y</mml:mi><mml:mo stretchy="false">{</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="true" linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="after">)</mml:mo></mml:mrow><mml:mspace width="1em"/><mml:mo>×</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:msqrt><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">{</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">CM</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0190">The functions <ce:italic>f</ce:italic> and <ce:italic>g</ce:italic> defined in Eq. <ce:cross-ref refid="fm0030" id="crf11210">(1)</ce:cross-ref> and Eq. <ce:cross-ref refid="fm0040" id="crf11220">(2)</ce:cross-ref>, respectively, describe the measurements within the measured region with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> per degrees of freedom (<ce:italic>ν</ce:italic>) in the range of 0.1 to 0.5. The small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mi>ν</mml:mi></mml:math> values are a consequence of the relatively large uncorrelated systematic uncertainties on the measurements. That is, the charged-particle distributions for pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> follow a normal distribution in rapidity, with free parameters <ce:italic>A</ce:italic>, <ce:italic>a</ce:italic>, <ce:italic>σ</ce:italic>, and <ce:italic>α</ce:italic> in the asymmetric case.</ce:para><ce:para id="pr0200">The top panel of <ce:cross-ref refid="fg0040" id="crf11230">Fig. 4</ce:cross-ref><ce:float-anchor refid="fg0040"/> shows the best-fit parameter values of the normal width (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math>) for all three collision systems as a function of the average number of participating nucleons (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo></mml:math>) calculated using a Glauber model <ce:cross-ref refid="br0280" id="crf11240">[28]</ce:cross-ref>. The best-fit parameters are found taking statistical and uncorrelated systematic uncertainties into account. The result using the above procedure, for the most central Pb<ce:glyph name="sbnd"/>Pb collisions, is found to be compatible with previous results extracted by unfolding with the mean Jacobian estimated from transverse momentum spectra <ce:cross-ref refid="br0020" id="crf11250">[2]</ce:cross-ref>. The open points (crosses) and dashed lines on the figure are from evaluations of Eq. <ce:cross-ref refid="fm0030" id="crf11260">(1)</ce:cross-ref> and Eq. <ce:cross-ref refid="fm0040" id="crf11270">(2)</ce:cross-ref>, and direct calculations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math>, respectively, using model calculations with EPOS-LHC <ce:cross-ref refid="br0290" id="crf11280">[29]</ce:cross-ref>. EPOS-LHC was chosen as it provides predictions for all three collision systems. The parameterisation, in terms of the two functions, of this model calculation generally reproduces the widths of the charged-particle rapidity densities, except in the asymmetric case where a direct evaluation of the standard deviation is less motivated.</ce:para><ce:para id="pr0210">The general trend is that the widths decrease as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo></mml:math> increases, consistent with the behaviour of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="normal">PbPb</mml:mtext></mml:mrow></mml:msub></mml:math> ratios. Notably, the width of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> distributions in p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb, for low number of participant nucleons in the collisions, approaches the width of the pp distribution, which, presumably, is dominated by kinematic and phase space constraints.</ce:para><ce:para id="pr0220">The lower panel of <ce:cross-ref refid="fg0040" id="crf11290">Fig. 4</ce:cross-ref> shows the dependence of <ce:italic>a</ce:italic> on the average number of participants. The right-hand ordinate is the same, but multiplied by the average mass <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>0.215</mml:mn><mml:mo>±</mml:mo><mml:mn>0.001</mml:mn><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mtext>GeV</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="italic">c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> estimated from measurements of identified particles in Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> <ce:cross-ref refid="br0300" id="crf11300">[30]</ce:cross-ref>. To better understand the parameter <ce:italic>a</ce:italic>, this parameter extracted from the EPOS-LHC calculations, using the above procedure, is also shown in the figure. The dotted lines show the average <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>m</mml:mi></mml:math> predicted by EPOS-LHC <ce:cross-ref refid="br0290" id="crf11310">[29]</ce:cross-ref>. The EPOS-LHC calculations indicate that the extracted effective transverse momentum to mass ratio <ce:italic>a</ce:italic> is consistently smaller than the ratio of the average transverse momentum to the average mass. Thus <ce:italic>a</ce:italic> gives a lower bound on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math>.</ce:para><ce:para id="pr0230">We can estimate the energy density that is reached in the collisions as a function of the number of participants for the three systems. A conventional approach is to use the model originally proposed by Bjorken <ce:cross-ref refid="br0120" id="crf11320">[12]</ce:cross-ref> in which the energy density (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub></mml:math>) depends on the rapidity density of particles and the volume of a longitudinal cylinder with cross sectional area determined by the overlap between the colliding partners and length determined by a characteristic particle formation time<ce:display><ce:formula id="fm0050"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>τ</mml:mi></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">〈</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">〉</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Here, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mi>π</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>≈</mml:mo><mml:mi>π</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msubsup></mml:math> is the transverse area spanned by the participating nucleons, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> is the transverse-energy rapidity density, and <ce:italic>τ</ce:italic> is the formation time. While a formation time of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:mi>τ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">c</mml:mi></mml:math> is often assumed, it is left as a free parameter here. With <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt></mml:math>, the transverse-energy rapidity density can be approximated by<ce:display><ce:formula id="fm0060"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:mrow><mml:mrow><mml:mo stretchy="true">〈</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">〉</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0.55</mml:mn><mml:mo>±</mml:mo><mml:mn>0.01</mml:mn></mml:math>, the ratio of charged particles to all particles <ce:cross-ref refid="br0310" id="crf11330">[31]</ce:cross-ref>, accounts for neutral particles not measured in the experiment, and is assumed the same for all collision systems. Substituting the derived <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> and the effective <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mi>a</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>m</mml:mi><mml:mo>≲</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math> results in a lower bound estimate for the Bjorken energy density (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">LB</mml:mi></mml:mrow></mml:msub></mml:math>)<ce:display><ce:formula id="fm0070"><ce:label>(3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi><mml:mo>≥</mml:mo><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">LB</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">cosh</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>⁡</mml:mo><mml:mi>η</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>a</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math> are as in the top panel of <ce:cross-ref refid="fg0040" id="crf11340">Fig. 4</ce:cross-ref>.</ce:para><ce:para id="pr0240">The transverse area <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> is estimated in a numerical Glauber model <ce:cross-refs refid="br0320 br0330" id="crs0100">[32,33]</ce:cross-refs> as shown in <ce:cross-ref refid="fg0050" id="crf11350">Fig. 5</ce:cross-ref><ce:float-anchor refid="fg0050"/>. We consider two extremes for the transverse area spanned by the participating nucleons: a) the <ce:italic>exclusive</ce:italic> (or direct) overlap between participating nucleons, ∩ and open markers in <ce:cross-ref refid="fg0050" id="crf11360">Fig. 5</ce:cross-ref>, and b) the <ce:italic>inclusive</ce:italic> (or full) area of all participating nucleons, ∪ and full markers in <ce:cross-ref refid="fg0050" id="crf11370">Fig. 5</ce:cross-ref>.</ce:para><ce:para id="pr0250"><ce:cross-ref refid="fg0060" id="crf11440">Fig. 6</ce:cross-ref><ce:float-anchor refid="fg0060"/> shows the lower-bound energy density estimate, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">LB</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi></mml:math>, as a function of the number of participants, which reaches values between 10 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:mn>20</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> in the most central Pb<ce:glyph name="sbnd"/>Pb collisions. The uncertainties are from standard error propagation of Eq. <ce:cross-ref refid="fm0070" id="crf11390">(3)</ce:cross-ref> of uncertainties on the best-fit parameter values, the number of participants, mean mass, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:math>. A rise from roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si79.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> to over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:mn>10</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is observed if the transverse area is assumed to be the inclusive area of participating nucleons. This trend is illustrated by a power-law (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si81.svg"><mml:mi>C</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msubsup></mml:math>) fit to the data in the figure, with the parameter values <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si82.svg"><mml:mi>C</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>0.8</mml:mn><mml:mo>±</mml:mo><mml:mn>0.3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi>p</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0.44</mml:mn><mml:mo>±</mml:mo><mml:mn>0.08</mml:mn></mml:math>. On the other hand, if the transverse area is assumed to be the smaller exclusive overlap area, we observe a substantially larger lower bound on the energy density, but a less dramatic increase with increasing number of participating nucleons. Also shown in the figure are estimates of the Bjorken energy density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi></mml:math> for Pb<ce:glyph name="sbnd"/>Pb reactions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> <ce:cross-ref refid="br0310" id="crf11400">[31]</ce:cross-ref>. These results where obtained from measurements of the transverse energy in the collisions and using the inclusive estimate of the transverse area <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>. The trend of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> results is similar to these earlier results. Bearing in mind that for the largest LHC collision energy we show a lower bound estimate of the energy density in <ce:cross-ref refid="fg0060" id="crf11410">Fig. 6</ce:cross-ref>, we find a likely overall increase in the energy density from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>.</ce:para></ce:section><ce:section id="se0050"><ce:label>5</ce:label><ce:section-title id="st0060">Summary and conclusions</ce:section-title><ce:para id="pr0260">We have measured the charged particle pseudorapidity density in pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> over the widest possible pseudorapidity range available at the LHC. The distributions where determined using the same experimental apparatus and methods, and systematic uncertainties have been minimised to within the capabilities of the set-up. While the particle production in central Pb<ce:glyph name="sbnd"/>Pb collisions clearly exhibits an enhancement as compared to pp collisions, particle production in p<ce:glyph name="sbnd"/>Pb collisions is consistent with dominantly incoherent nucleon–nucleon collisions. By transforming the measured pseudorapidity distributions to rapidity distributions we have obtained systematic trends for the width of the rapidity distributions and a lower bound on the energy density, which shows a clear scaling behaviour as a function of the average number of participant nucleons. The decreasing width of the deduced rapidity distributions with increasing participant number suggests that the kinematic spread of particles, including longitudinal degrees of freedom, is reduced due to interactions in the early stages of the collisions. This is also reflected in the accompanying growth of the energy density. Both observations are consistent with the gradual establishment of a high-density phase of matter with increasing size of the collision domain.</ce:para></ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0080">Declaration of Competing Interest</ce:section-title><ce:para id="pr0270">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0090">Acknowledgements</ce:section-title><ce:para id="pr0280">The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: <ce:grant-sponsor id="gsp0010">A. I. 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<ce:grant-sponsor id="gsp0690" sponsor-id="https://doi.org/10.13039/100020381">Turkish Energy, Nuclear and Mineral Research Agency</ce:grant-sponsor> (TENMAK), Turkey; <ce:grant-sponsor id="gsp0700" sponsor-id="https://doi.org/10.13039/501100004742">National Academy of Sciences of Ukraine</ce:grant-sponsor>, Ukraine; <ce:grant-sponsor id="gsp0710" sponsor-id="https://doi.org/10.13039/501100000271">Science and Technology Facilities Council</ce:grant-sponsor> (STFC), United Kingdom; National Science Foundation of the United States of America (<ce:grant-sponsor id="gsp0720" sponsor-id="https://doi.org/10.13039/100000001">NSF</ce:grant-sponsor>) and United States Department of Energy, Office of Nuclear Physics (<ce:grant-sponsor id="gsp0730" sponsor-id="https://doi.org/10.13039/100006209">DOE NP</ce:grant-sponsor>), United States of America.</ce:para></ce:acknowledgment></body><tail><ce:bibliography id="bl0010"><ce:section-title id="st0070">References</ce:section-title><ce:bibliography-sec id="bs0010"><ce:bib-reference id="br0010"><ce:label>[1]</ce:label><sb:reference id="bib866BD36D5E8AFE73B567F49DDE14CD65s1"><sb:contribution><sb:authors><sb:collaboration>BRAHMS Collaboration</sb:collaboration><sb:author><ce:given-name>I.C.</ce:given-name><ce:surname>Arsene</ce:surname></sb:author><sb:et-al/></sb:authors><sb:title><sb:maintitle>Nuclear stopping and rapidity loss in Au+Au collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>62.4</mml:mn><mml:mspace width="0.2em"/><mml:mtext>GeV</mml:mtext></mml:math></sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Phys. 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C94 (2016) 024914, arXiv:1603.07375 [nucl-ex].</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> diff --git a/tests/units/elsevier/data/CERNQ000000010011/S0370269323000850/main.xml b/tests/units/elsevier/data/CERNQ000000010011/S0370269323000850/main.xml index ab6ba370..2b0a788d 100644 --- a/tests/units/elsevier/data/CERNQ000000010011/S0370269323000850/main.xml +++ b/tests/units/elsevier/data/CERNQ000000010011/S0370269323000850/main.xml @@ -1 +1 @@ -<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd"><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="sco" xml:lang="en"><item-info><jid>PLB</jid><aid>137751</aid><ce:article-number>137751</ce:article-number><ce:pii>S0370-2693(23)00085-0</ce:pii><ce:doi>10.1016/j.physletb.2023.137751</ce:doi><ce:copyright year="2023" type="other">The Author(s)</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>Astrophysics and Cosmology</ce:text></ce:doctopic></ce:doctopics></item-info><head><ce:title id="ti0010">Cosmological stability in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity</ce:title><ce:author-group id="ag0010"><ce:author id="au0010" author-id="S0370269323000850-ade115eed9bce9e185e44b0aa9b50010"><ce:given-name>Shinji</ce:given-name><ce:surname>Tsujikawa</ce:surname><ce:e-address type="email" xlink:href="mailto:tsujikawa@waseda.jp" id="ea0010">tsujikawa@waseda.jp</ce:e-address></ce:author><ce:affiliation id="aff0010" affiliation-id="S0370269323000850-2391d88b911951b1d0cff5f6ba23fdf8"><ce:textfn>Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Waseda University</sa:organization><sa:address-line>3-4-1 Okubo</sa:address-line><sa:city>Tokyo</sa:city><sa:state>Shinjuku</sa:state><sa:postal-code>169-8555</sa:postal-code><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0005">Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan</ce:source-text></ce:affiliation></ce:author-group><ce:date-received day="20" month="12" year="2022"/><ce:date-revised day="2" month="2" year="2023"/><ce:date-accepted day="2" month="2" year="2023"/><ce:miscellaneous id="ms0010">Editor: M. Trodden</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0010">In gravitational theories where a canonical scalar field <ce:italic>ϕ</ce:italic> with a potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is coupled to a Gauss-Bonnet (GB) term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> with the Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, we study the cosmological stability of tensor and scalar perturbations in the presence of a perfect fluid. We show that, in decelerating cosmological epochs with a positive tensor propagation speed squared, the existence of nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> in <ce:italic>f</ce:italic> always induces Laplacian instability of a dynamical scalar perturbation associated with the GB term. This is also the case for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity, where the presence of nonlinear GB functions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is not allowed during the radiation- and matter-dominated epochs. A linearly coupled GB term with <ce:italic>ϕ</ce:italic> of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> can be consistent with all the stability conditions, provided that the scalar-GB coupling is subdominant to the background cosmological dynamics.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0140">Data availability</ce:section-title><ce:para id="pr0440">No data was used for the research described in the article.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">General Relativity (GR) is a fundamental theory of gravity whose validity has been probed in Solar System experiments <ce:cross-ref refid="br0010" id="crf0010">[1]</ce:cross-ref> and submillimeter laboratory tests <ce:cross-refs refid="br0020 br0030" id="crs0010">[2,3]</ce:cross-refs>. Despite the success of GR describing gravitational interactions in the Solar System, there have been long-standing cosmological problems such as the origins of inflation, dark energy, and dark matter. To address these problems, one typically introduces additional degrees of freedom (DOFs) beyond those appearing in GR <ce:cross-refs refid="br0040 br0050 br0060 br0070 br0080 br0090 br0100" id="crs0020">[4–10]</ce:cross-refs>. One of such new DOFs is a canonical scalar field <ce:italic>ϕ</ce:italic> with a potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> <ce:cross-refs refid="br0110 br0120 br0130 br0140 br0150 br0160 br0170 br0180 br0190 br0200 br0210 br0220" id="crs0030">[11–22]</ce:cross-refs>. If the scalar field evolves slowly along the potential, it is possible to realize cosmic acceleration responsible for inflation or dark energy. An oscillating scalar field around the potential minimum can be also the source for dark matter.</ce:para><ce:para id="pr0020">The other way of introducing a new dynamical DOF is to modify the gravitational sector from GR. The Lagrangian in GR is given by an Einstein-Hilbert term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow></mml:msub></mml:math> is the reduced Planck mass and <ce:italic>R</ce:italic> is the Ricci scalar. If we consider theories containing nonlinear functions of <ce:italic>R</ce:italic> of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, there is one scalar DOF arising from the modification of gravity <ce:cross-refs refid="br0230 br0240" id="crs0040">[23,24]</ce:cross-refs>. One well known example is the Starobinsky's model, in which the presence of a quadratic curvature term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> drives cosmic acceleration <ce:cross-ref refid="br0250" id="crf0020">[25]</ce:cross-ref>. It is also possible to construct <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> models of late-time cosmic acceleration <ce:cross-refs refid="br0260 br0270 br0280 br0290 br0300 br0310 br0320" id="crs0050">[26–32]</ce:cross-refs>, while being consistent with local gravity constraints.</ce:para><ce:para id="pr0030">The Einstein tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> obtained by varying the Einstein-Hilbert action satisfies the conserved relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math> is a covariant derivative operator), with the property of second-order field equations of motion in metrics. If we demand such conserved and second-order properties for 2-rank symmetric tensors, GR is the unique theory of gravity in 4 dimensions <ce:cross-ref refid="br0330" id="crf0030">[33]</ce:cross-ref>. In spacetime dimensions higher than 4, there is a particular combination known as a Gauss-Bonnet (GB) term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> consistent with those demands <ce:cross-ref refid="br0340" id="crf0040">[34]</ce:cross-ref>. In 4 dimensions, the GB term is a topological surface term and hence it does not contribute to the field equations of motion. In the presence of a coupling between a scalar field <ce:italic>ϕ</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math>, the spacetime dynamics is modified by the time or spatial variation of <ce:italic>ϕ</ce:italic>. Indeed, this type of scalar-GB coupling appears in the context of low energy effective string theory <ce:cross-refs refid="br0350 br0360 br0370" id="crs0060">[35–37]</ce:cross-refs>. The cosmological application of the coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> has been extensively performed in the literature <ce:cross-refs refid="br0380 br0390 br0400 br0410 br0420 br0430 br0440 br0450 br0460 br0470 br0480 br0490 br0500 br0510 br0520 br0530 br0540 br0550 br0560 br0570 br0580 br0590 br0600 br0610 br0620 br0630" id="crs0070">[38–63]</ce:cross-refs>. Moreover, it is known that the same coupling gives rise to spherically symmetric solutions of hairy black holes and neutron stars <ce:cross-refs refid="br0640 br0650 br0660 br0670 br0680 br0690 br0700 br0710 br0720 br0730 br0740 br0750 br0760 br0770 br0780 br0790 br0800" id="crs0080">[64–80]</ce:cross-refs>. The Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> containing nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> also generates nontrivial contributions to the spacetime dynamics <ce:cross-refs refid="br0810 br0820 br0830 br0840 br0850 br0860 br0870 br0880 br0890 br0900" id="crs0090">[81–90]</ce:cross-refs>.</ce:para><ce:para id="pr0040">In Ref. <ce:cross-ref refid="br0910" id="crf0050">[91]</ce:cross-ref>, De Felice and Suyama studied the stability of scalar perturbations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity on a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) background. In theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:msubsup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mi>R</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>f</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo>∂</mml:mo><mml:mi>R</mml:mi></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mi>R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>f</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>f</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="script">G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, there is an unusual scale-dependent sound speed which propagates superluminally in the short-wavelength limit, unless the vacuum is in a de Sitter state (see also Ref. <ce:cross-ref refid="br0920" id="crf0060">[92]</ce:cross-ref> for the analysis in an anisotropic cosmological background). We note that this problem does not arise for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity. In Ref. <ce:cross-ref refid="br0930" id="crf0070">[93]</ce:cross-ref>, the same authors extended the analysis to a more general Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> with a canonical scalar field <ce:italic>ϕ</ce:italic> and showed that the property of the scale-dependent sound speed is not modified by the presence of <ce:italic>ϕ</ce:italic>. Taking a perfect fluid (radiation or nonrelativistic matter) into account in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity, the cosmological stability and evolution of matter perturbations were studied in Refs. <ce:cross-refs refid="br0940 br0950 br0960" id="crs0100">[94–96]</ce:cross-refs>.</ce:para><ce:para id="pr0050">In Einstein-scalar-GB gravity given by the Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, where <ce:italic>ϕ</ce:italic> is a canonical scalar field, the problem of scale-dependent sound speeds mentioned above is not present. In this theory, the propagation of scalar perturbations on the flat FLRW background was studied in Ref. <ce:cross-ref refid="br0930" id="crf0080">[93]</ce:cross-ref> without taking into account matter. While the sound speed associated with the field <ce:italic>ϕ</ce:italic> is luminal for theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, the propagation speed squared <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> arising from a nonlinear GB term deviates from that of light and it can be even negative. In Ref. <ce:cross-ref refid="br0930" id="crf0090">[93]</ce:cross-ref>, the authors discussed the possibility for satisfying the Laplacian stability condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>. In the presence of matter, however, the stability conditions are subject to modifications from those in the vacuum. To understand what happens for the dynamics of cosmological perturbations during radiation- and matter-dominated epochs, we need to study their stabilities by incorporating radiation or nonrelativistic matter.</ce:para><ce:para id="pr0060">In this letter, we will derive general conditions for the absence of ghosts and Laplacian instabilities in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity, where <ce:italic>ϕ</ce:italic> is a canonical scalar field with a potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. In theories where the scalar field <ce:italic>ϕ</ce:italic> is coupled to the linear GB term, i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math>, there is only one dynamical scalar DOF <ce:italic>ϕ</ce:italic>. In theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, the Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> can be expressed in terms of two scalar fields <ce:italic>ϕ</ce:italic> and <ce:italic>χ</ce:italic> coupled to the linear GB term, where <ce:italic>χ</ce:italic> arises from the nonlinearity in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math>. Hence the latter theory has two dynamical scalar DOFs. To study the cosmological stability of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, we take a perfect fluid into account as a form of the Schutz-Sorkin action <ce:cross-refs refid="br0970 br0980 br0990" id="crs0110">[97–99]</ce:cross-refs>. We will show that the squared sound speed arising from nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> is negative during decelerating cosmological epochs including radiation and matter eras. To reach this conclusion, we exploit the fact that the propagation speed squared <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> of tensor perturbations must be positive to avoid Laplacian instability of gravitational waves.</ce:para><ce:para id="pr0070">The same Laplacian instability of scalar perturbations is also present in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with any nonlinear function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> in <ce:italic>f</ce:italic>. We note that, in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> models of late-time cosmic acceleration, violent instabilities of matter density perturbations during the radiation and matter eras were reported in Ref. <ce:cross-ref refid="br1000" id="crf0100">[100]</ce:cross-ref>. This can be regarded as the consequence of a negative sound speed squared of the scalar perturbation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mi>δ</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:math> arising from the nonlinearity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> in <ce:italic>f</ce:italic>. Since <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mi>δ</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:math> is coupled to the matter perturbation <ce:italic>δρ</ce:italic>, the background cosmological evolution during the radiation and matter eras is spoiled by the rapid growth of <ce:italic>δρ</ce:italic>. Our analysis in this letter shows that similar catastrophic instabilities persist for more general scalar-GB couplings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>.</ce:para><ce:para id="pr0080">This letter is organized as follows. In Sec. <ce:cross-ref refid="se0020" id="crf0110">2</ce:cross-ref>, we revisit cosmological stability conditions in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity with a canonical scalar field <ce:italic>ϕ</ce:italic>, which can be accommodated in a subclass of Horndeski theories with a single scalar DOF <ce:cross-refs refid="br1010 br1020 br1030 br1040" id="crs0120">[101–104]</ce:cross-refs>. This is an exceptional case satisfying the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, under which the Laplacian instability of scalar perturbations can be avoided. In Sec. <ce:cross-ref refid="se0030" id="crf0120">3</ce:cross-ref>, we derive the background equations and stability conditions of tensor perturbations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math> by incorporating a perfect fluid. In Sec. <ce:cross-ref refid="se0060" id="crf0130">4</ce:cross-ref>, we proceed to the derivation of a second-order action of scalar perturbations and obtain conditions for the absence of ghosts and Laplacian instabilities in the scalar sector. In particular, we show that an effective cosmological equation of state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub></mml:math> needs to be in the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mn>6</mml:mn></mml:math> to ensure Laplacian stabilities of the perturbation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mi>δ</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:math>. Sec. <ce:cross-ref refid="se0090" id="crf0140">5</ce:cross-ref> is devoted to conclusions.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity</ce:section-title><ce:para id="pr0090">We first briefly revisit the cosmological stability in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity given by the action<ce:display><ce:formula id="fm0010"><ce:label>(2.1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mi mathvariant="script">S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>R</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>η</mml:mi><mml:mi>X</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>g</ce:italic> is a determinant of the metric tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>, <ce:italic>η</ce:italic> is a constant, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:mi>X</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi></mml:math> is a kinetic term of the scalar field <ce:italic>ϕ</ce:italic>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> are functions of <ce:italic>ϕ</ce:italic>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> is a GB term defined by<ce:display><ce:formula id="fm0020"><ce:label>(2.2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mi mathvariant="script">G</mml:mi><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub></mml:math> being the Ricci and Riemann tensors, respectively. For the matter action <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math>, we consider a perfect fluid minimally coupled to gravity.</ce:para><ce:para id="pr0100">The action <ce:cross-ref refid="fm0010" id="crf0150">(2.1)</ce:cross-ref> contains one scalar DOF <ce:italic>ϕ</ce:italic> besides the matter field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math>. If we consider Horndeski theories <ce:cross-ref refid="br1010" id="crf0160">[101]</ce:cross-ref> given by the action<ce:display><ce:formula id="fm0030"><ce:label>(2.3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mi mathvariant="script">S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo id="mmlbr0001">∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mspace width="0.2em"/><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>□</mml:mo><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>R</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>□</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">}</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="before">+</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>□</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mo>□</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">}</mml:mo></mml:mrow><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="before">+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> then the theory <ce:cross-ref refid="fm0010" id="crf0170">(2.1)</ce:cross-ref> can be accommodated by choosing the coupling functions <ce:cross-ref refid="br1030" id="crf0180">[103]</ce:cross-ref><ce:display><ce:formula id="fm0040"><ce:label>(2.4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub id="mmlbr0002"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>η</mml:mi><mml:mi>X</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">ln</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>7</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mi mathvariant="normal">ln</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="after">,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">ln</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="normal">ln</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where we use the notations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>∂</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:mi>X</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>∂</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:mi>ϕ</mml:mi></mml:math> for any arbitrary function <ce:italic>F</ce:italic>.</ce:para><ce:para id="pr0110">Let us consider a spatially flat FLRW background given by the line element <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:mi>a</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is a time-dependent scale factor. The perfect fluid has a density <ce:italic>ρ</ce:italic> and pressure <ce:italic>P</ce:italic>. The background equations as well as the perturbation equations in full Horndeski theories were derived in Refs. <ce:cross-refs refid="br1030 br1050 br1060 br1070" id="crs0130">[103,105–107]</ce:cross-refs>. On using the correspondence <ce:cross-ref refid="fm0040" id="crf0190">(2.4)</ce:cross-ref>, the background equations of motion in theories given by the action <ce:cross-ref refid="fm0010" id="crf0200">(2.1)</ce:cross-ref> are<ce:display><ce:formula id="fm0050"><ce:label>(2.5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mrow><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>ρ</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0060"><ce:label>(2.6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>P</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0070"><ce:label>(2.7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:mrow><mml:mi>η</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant="script">G</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0080"><ce:label>(2.8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:mi>H</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">/</mml:mo><mml:mi>a</mml:mi></mml:math> is the Hubble expansion rate, a dot represents the derivative with respect to <ce:italic>t</ce:italic>, and<ce:display><ce:formula id="fm0090"><ce:label>(2.9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0100"><ce:label>(2.10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0110"><ce:label>(2.11)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mrow><mml:mi mathvariant="script">G</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>24</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0120">In the presence of tensor perturbations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> with the perturbed line element <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:math>, the second-order action of traceless and divergence-free modes of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> was already derived in full Horndeski theories <ce:cross-refs refid="br1030 br1060 br1070" id="crs0140">[103,106,107]</ce:cross-refs>. In the current theory, the conditions for the absence of ghosts and Laplacian instabilities are<ce:display><ce:formula id="fm0120"><ce:label>(2.12)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0130"><ce:label>(2.13)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> are defined by Eqs. <ce:cross-ref refid="fm0090" id="crf0210">(2.9)</ce:cross-ref> and <ce:cross-ref refid="fm0100" id="crf0220">(2.10)</ce:cross-ref>, respectively. Note that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> determines the sign of a kinetic term of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math>, while <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> corresponds to the propagation speed squared of tensor perturbations.</ce:para><ce:para id="pr0130">For the scalar sector, we choose the perturbed line element <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:mi>α</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>B</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:math> in the flat gauge, where <ce:italic>α</ce:italic> and <ce:italic>B</ce:italic> are scalar metric perturbations. There is also a scalar-field perturbation <ce:italic>δϕ</ce:italic> besides the matter perturbation <ce:italic>δρ</ce:italic> and the fluid velocity potential <ce:italic>v</ce:italic>. After deriving the quadratic-order action of scalar perturbations, we can eliminate nondynamical variables <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, and <ce:italic>v</ce:italic> from the action. Then, we are left with the two dynamical perturbations <ce:italic>δϕ</ce:italic> and <ce:italic>δρ</ce:italic> in the second-order action. In the short-wavelength limit, there is neither ghost nor Laplacian instability for <ce:italic>δϕ</ce:italic> under the conditions <ce:cross-refs refid="br1030 br1060 br1070" id="crs0150">[103,106,107]</ce:cross-refs><ce:display><ce:formula id="fm0140"><ce:label>(2.14)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>96</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0150"><ce:label>(2.15)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>32</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>96</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math> corresponds to the propagation speed of <ce:italic>δϕ</ce:italic>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub></mml:math> is the cosmological effective equation of state defined by<ce:display><ce:formula id="fm0160"><ce:label>(2.16)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The stability conditions for <ce:italic>δρ</ce:italic> are given by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> is the matter sound speed squared.</ce:para><ce:para id="pr0140">Under the stability condition <ce:cross-ref refid="fm0120" id="crf0230">(2.12)</ce:cross-ref> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, the scalar no-ghost condition <ce:cross-ref refid="fm0140" id="crf0240">(2.14)</ce:cross-ref> is satisfied. Let us consider the case in which contributions of the scalar-GB coupling are suppressed, such that<ce:display><ce:formula id="fm0170"><ce:label>(2.17)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:mo stretchy="false">{</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo>,</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo>,</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">}</mml:mo><mml:mo>≪</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>≪</mml:mo><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Then, it follows that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>≃</mml:mo><mml:mn>1</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mn>2</mml:mn><mml:mi>η</mml:mi><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>≃</mml:mo><mml:mn>1</mml:mn></mml:math>. In such cases, provided that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, all the stability conditions are consistently satisfied. If the scalar-GB coupling contributes to the late-time cosmological dynamics, there is an observational bound on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> constrained from the GW170817 event together with the electromagnetic counterpart, i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si79.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>15</mml:mn></mml:mrow></mml:msup><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mn>7</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>16</mml:mn></mml:mrow></mml:msup></mml:math> <ce:cross-ref refid="br1080" id="crf0250">[108]</ce:cross-ref> for the redshift <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:mi>z</mml:mi><mml:mo>≤</mml:mo><mml:mn>0.009</mml:mn></mml:math>. This translates to the limit<ce:display><ce:formula id="fm0180"><ce:label>(2.18)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si81.svg"><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>≲</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>15</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> which gives a tight constraint on the amplitude of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. In this case, contributions of the scalar-GB coupling to the background Eqs. <ce:cross-ref refid="fm0050" id="crf0260">(2.5)</ce:cross-ref> and <ce:cross-ref refid="fm0060" id="crf0270">(2.6)</ce:cross-ref> are highly suppressed relative to the field density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si82.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and the matter density.</ce:para><ce:para id="pr0150">The bound <ce:cross-ref refid="fm0180" id="crf0280">(2.18)</ce:cross-ref> is not applied to early cosmological epochs including inflation, radiation, and matter eras. We note, however, that the dominance of the scalar-GB coupling to the background equations prevents the successful cosmic expansion history. This can also give rise to the violation of either of the stability conditions <ce:cross-ref refid="fm0120" id="crf0290">(2.12)</ce:cross-ref>-<ce:cross-ref refid="fm0150" id="crf0300">(2.15)</ce:cross-ref>. Provided the scalar-GB coupling is suppressed in such a way that inequalities <ce:cross-ref refid="fm0170" id="crf0310">(2.17)</ce:cross-ref> hold, the linear stabilities are ensured for both tensor and scalar perturbations.</ce:para></ce:section><ce:section id="se0030"><ce:label>3</ce:label><ce:section-title id="st0040"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity</ce:section-title><ce:para id="pr0160">We extend <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity to more general theories in which a canonical scalar field <ce:italic>ϕ</ce:italic> with a potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is coupled to the GB term of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. The action in such theories is given by<ce:display><ce:formula id="fm0190"><ce:label>(3.1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi mathvariant="script">S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>R</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>η</mml:mi><mml:mi>X</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where a matter field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math> is minimally coupled to gravity. It is more practical to introduce a scalar field <ce:italic>χ</ce:italic> and resort to the following action<ce:display><ce:formula id="fm0200"><ce:label>(3.2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:mi mathvariant="script">S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>R</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>η</mml:mi><mml:mi>X</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where<ce:display><ce:formula id="fm0210"><ce:label>(3.3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si85.svg"><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≡</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>χ</mml:mi><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> with the notation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si86.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>∂</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:mi>χ</mml:mi></mml:math>. Varying the action <ce:cross-ref refid="fm0200" id="crf0320">(3.2)</ce:cross-ref> with respect to <ce:italic>χ</ce:italic>, it follows that<ce:display><ce:formula id="fm0220"><ce:label>(3.4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>χ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> So long as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, we obtain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si89.svg"><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math>. In this case, the action <ce:cross-ref refid="fm0200" id="crf0330">(3.2)</ce:cross-ref> reduces to <ce:cross-ref refid="fm0190" id="crf0340">(3.1)</ce:cross-ref>. Thus, the equivalence of <ce:cross-ref refid="fm0200" id="crf0350">(3.2)</ce:cross-ref> with <ce:cross-ref refid="fm0190" id="crf0360">(3.1)</ce:cross-ref> holds for<ce:display><ce:formula id="fm0230"><ce:label>(3.5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si90.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> under which there is a new scalar DOF <ce:italic>χ</ce:italic> arising from the gravitational sector.</ce:para><ce:para id="pr0170">Theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> correspond to the coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:mi>f</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math>, in which case the cosmological stability conditions were already discussed in Sec. <ce:cross-ref refid="se0020" id="crf0370">2</ce:cross-ref>. In <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity, we do not have the additional scalar DOF <ce:italic>χ</ce:italic> arising from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math>, so the term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si92.svg"><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0220" id="crf0380">(3.4)</ce:cross-ref> does not have the meaning of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si93.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub></mml:math>. Thus, the action <ce:cross-ref refid="fm0200" id="crf0390">(3.2)</ce:cross-ref> with the new dynamical DOF <ce:italic>χ</ce:italic> does not reproduce the action <ce:cross-ref refid="fm0010" id="crf0400">(2.1)</ce:cross-ref> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity.</ce:para><ce:para id="pr0180">In the following, we will focus on theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, i.e., those containing the nonlinear dependence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> in <ce:italic>f</ce:italic>. For the matter field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math>, we incorporate a perfect fluid without a dynamical vector DOF. This matter sector is described by the Schutz-Sorkin action <ce:cross-refs refid="br0970 br0980 br0990" id="crs0160">[97–99]</ce:cross-refs><ce:display><ce:formula id="fm0240"><ce:label>(3.6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si94.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mspace width="0.2em"/><mml:mi>ρ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ℓ</mml:mi><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where the fluid density <ce:italic>ρ</ce:italic> is a function of its number density <ce:italic>n</ce:italic>. The vector field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math> is related to <ce:italic>n</ce:italic> according to the relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:mi>n</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si97.svg"><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mo stretchy="false">)</mml:mo></mml:math> is the fluid four velocity. A scalar quantity <ce:italic>ℓ</ce:italic> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math> is a Lagrange multiplier, with the notation of a partial derivative <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ℓ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>∂</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math>. Varying the matter action <ce:cross-ref refid="fm0240" id="crf0410">(3.6)</ce:cross-ref> with respect to <ce:italic>ℓ</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math>, respectively, we obtain<ce:display><ce:formula id="fm0250"><ce:label>(3.7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si99.svg"><mml:mrow><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0260"><ce:label>(3.8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si100.svg"><mml:mrow><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ℓ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>ρ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>n</mml:mi></mml:math>.</ce:para><ce:section id="se0040"><ce:label>3.1</ce:label><ce:section-title id="st0050">Background equations</ce:section-title><ce:para id="pr0190">We derive the background equations of motion on the spatially flat FLRW background given by the line element<ce:display><ce:formula id="fm0270"><ce:label>(3.9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si103.svg"><mml:mi>N</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is a lapse function. Since the fluid four velocity in its rest frame is given by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si104.svg"><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>, the vector field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math> has components <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.svg"><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>. From Eq. <ce:cross-ref refid="fm0250" id="crf0420">(3.7)</ce:cross-ref>, we obtain<ce:display><ce:formula id="fm0280"><ce:label>(3.10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si106.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mi>n</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mi mathvariant="normal">constant</mml:mi></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> which means that the total fluid number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si107.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is conserved. This translates to the differential equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si108.svg"><mml:mover accent="true"><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mi>n</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, which can be expressed as a form of the continuity equation<ce:display><ce:formula id="fm0290"><ce:label>(3.11)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si109.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>P</ce:italic> is a fluid pressure defined by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si110.svg"><mml:mi>P</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>n</mml:mi><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>ρ</mml:mi></mml:math>.</ce:para><ce:para id="pr0200">On the background <ce:cross-ref refid="fm0270" id="crf0430">(3.9)</ce:cross-ref>, the total action <ce:cross-ref refid="fm0200" id="crf0440">(3.2)</ce:cross-ref> is expressed in the form<ce:display><ce:formula id="fm0300"><ce:label>(3.12)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si111.svg"><mml:mi mathvariant="script">S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>a</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>N</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>8</mml:mn><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>N</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>n</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mover accent="true"><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> From Eq. <ce:cross-ref refid="fm0260" id="crf0450">(3.8)</ce:cross-ref>, we obtain the following relation<ce:display><ce:formula id="fm0310"><ce:label>(3.13)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si112.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>N</mml:mi><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Varying the action <ce:cross-ref refid="fm0300" id="crf0460">(3.12)</ce:cross-ref> with respect to <ce:italic>N</ce:italic>, <ce:italic>a</ce:italic>, <ce:italic>ϕ</ce:italic>, <ce:italic>χ</ce:italic> respectively and setting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si113.svg"><mml:mi>N</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:math> at the end, we obtain the background equations of motion<ce:display><ce:formula id="fm0320"><ce:label>(3.14)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si114.svg"><mml:mrow><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>ρ</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0330"><ce:label>(3.15)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si115.svg"><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>P</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0340"><ce:label>(3.16)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si116.svg"><mml:mrow><mml:mi>η</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0350"><ce:label>(3.17)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si117.svg"><mml:mrow><mml:mi>χ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>24</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where<ce:display><ce:formula id="fm0360"><ce:label>(3.18)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si118.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0370"><ce:label>(3.19)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si119.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0210">We recall that the perfect fluid obeys the continuity Eq. <ce:cross-ref refid="fm0290" id="crf0470">(3.11)</ce:cross-ref>. We notice that Eqs. <ce:cross-ref refid="fm0320" id="crf0480">(3.14)</ce:cross-ref>-<ce:cross-ref refid="fm0340" id="crf0490">(3.16)</ce:cross-ref> are of similar forms to Eqs. <ce:cross-ref refid="fm0050" id="crf0500">(2.5)</ce:cross-ref>-<ce:cross-ref refid="fm0070" id="crf0510">(2.7)</ce:cross-ref> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity, but the expressions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> are different from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, respectively, because of the appearance of time derivatives of <ce:italic>χ</ce:italic>. These <ce:italic>χ</ce:italic> derivatives do not vanish for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, i.e., for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>. As we will show in Sec. <ce:cross-ref refid="se0060" id="crf0520">4</ce:cross-ref>, nonlinearities of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> in <ce:italic>f</ce:italic> are responsible for the appearance of a new scalar propagating DOF <ce:italic>δχ</ce:italic>.</ce:para></ce:section><ce:section id="se0050"><ce:label>3.2</ce:label><ce:section-title id="st0060">Stabilities in the tensor sector</ce:section-title><ce:para id="pr0220">We proceed to the derivation of stability conditions for tensor perturbations in theories given by the action <ce:cross-ref refid="fm0200" id="crf0530">(3.2)</ce:cross-ref>. The perturbed line element including the tensor perturbation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> is<ce:display><ce:formula id="fm0380"><ce:label>(3.20)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where we impose the traceless and divergence-free gauge conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si122.svg"><mml:msub><mml:mrow><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si123.svg"><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>. For the gravitational wave propagating along the <ce:italic>z</ce:italic> direction, nonvanishing components of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> are expressed in the form<ce:display><ce:formula id="fm0390"><ce:label>(3.21)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si124.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where the two polarized modes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si126.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> are functions of <ce:italic>t</ce:italic> and <ce:italic>z</ce:italic>.</ce:para><ce:para id="pr0230">The second-order action arising from the matter action <ce:cross-ref refid="fm0240" id="crf0540">(3.6)</ce:cross-ref> can be expressed as<ce:display><ce:formula id="fm0400"><ce:label>(3.22)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si127.svg"><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>P</mml:mi><mml:msubsup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>P</ce:italic> can be eliminated by using the background Eq. <ce:cross-ref refid="fm0330" id="crf0550">(3.15)</ce:cross-ref>. Expanding the total action <ce:cross-ref refid="fm0200" id="crf0560">(3.2)</ce:cross-ref> up to quadratic order in tensor perturbations and integrating it by parts, the resulting second-order action reduces to<ce:display><ce:formula id="fm0410"><ce:label>(3.23)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si128.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si129.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. We recall that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> are given by Eqs. <ce:cross-ref refid="fm0360" id="crf0570">(3.18)</ce:cross-ref> and <ce:cross-ref refid="fm0370" id="crf0580">(3.19)</ce:cross-ref>, respectively.</ce:para><ce:para id="pr0240">To avoid the ghost and Laplacian instabilities in the tensor sector, we require the two conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, which translate to<ce:display><ce:formula id="fm0420"><ce:label>(3.24)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0430"><ce:label>(3.25)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si133.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> In <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity without the scalar field <ce:italic>ϕ</ce:italic>, tensor stability conditions can be obtained by setting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si134.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si135.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> in Eqs. <ce:cross-ref refid="fm0420" id="crf0590">(3.24)</ce:cross-ref> and <ce:cross-ref refid="fm0430" id="crf0600">(3.25)</ce:cross-ref>.</ce:para><ce:para id="pr0250">We vary the action <ce:cross-ref refid="fm0410" id="crf0610">(3.23)</ce:cross-ref> with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si136.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:math>) in Fourier space with a comoving wavenumber <ce:bold><ce:italic>k</ce:italic></ce:bold>. Then, we obtain the tensor perturbation equation of motion<ce:display><ce:formula id="fm0440"><ce:label>(3.26)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si138.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si139.svg"><mml:mi>k</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi mathvariant="bold-italic">k</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:math>. Since <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:mi>ξ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub></mml:math>, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> dependence in <ce:italic>f</ce:italic> leads to the modified evolution of gravitational waves in comparison to GR. If the energy densities of <ce:italic>ϕ</ce:italic> and <ce:italic>χ</ce:italic> are relevant to the late-time cosmological dynamics after the matter dominance, the observational constraint on the tensor propagation speed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> arising from the GW170817 event <ce:cross-ref refid="br1080" id="crf0620">[108]</ce:cross-ref> (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si142.svg"><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mo>≲</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>15</mml:mn></mml:mrow></mml:msup></mml:math>) gives a tight bound on the scalar-GB coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. Such a stringent limit is not applied to the cosmological dynamics in the early Universe, but the conditions <ce:cross-ref refid="fm0420" id="crf0630">(3.24)</ce:cross-ref> and <ce:cross-ref refid="fm0430" id="crf0640">(3.25)</ce:cross-ref> need to be still satisfied.</ce:para></ce:section></ce:section><ce:section id="se0060"><ce:label>4</ce:label><ce:section-title id="st0070">Stabilities of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity in the scalar sector</ce:section-title><ce:para id="pr0260">In this section, we will derive conditions for the absence of scalar ghosts and Laplacian instabilities in theories given by the action <ce:cross-ref refid="fm0200" id="crf0650">(3.2)</ce:cross-ref>. A perturbed line element containing scalar perturbations <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>ζ</ce:italic>, and <ce:italic>E</ce:italic> is of the form<ce:display><ce:formula id="fm0450"><ce:label>(4.1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si143.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:mi>α</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>B</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:mi>ζ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi>E</mml:mi><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> For the scalar fields <ce:italic>ϕ</ce:italic> and <ce:italic>χ</ce:italic>, we consider perturbations <ce:italic>δϕ</ce:italic> and <ce:italic>δχ</ce:italic> on the background values <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si144.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si145.svg"><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, respectively, such that<ce:display><ce:formula id="fm0460"><ce:label>(4.2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where we will omit a bar from the background quantities in the following.</ce:para><ce:para id="pr0270">In the matter sector, the temporal and spatial components of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math> are decomposed into the background and perturbed parts as<ce:display><ce:formula id="fm0470"><ce:label>(4.3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si147.svg"><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>δ</mml:mi><mml:mi>J</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>j</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>δJ</ce:italic> and <ce:italic>δj</ce:italic> are scalar perturbations. In terms of the velocity potential <ce:italic>v</ce:italic>, the spatial component of fluid four velocity is expressed as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si148.svg"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>v</mml:mi></mml:math>. From Eq. <ce:cross-ref refid="fm0260" id="crf0660">(3.8)</ce:cross-ref>, the scalar quantity <ce:italic>ℓ</ce:italic> has a background part obeying the relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si149.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> besides a perturbation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mi>v</mml:mi></mml:math>. Then, we have<ce:display><ce:formula id="fm0480"><ce:label>(4.4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si151.svg"><mml:mi>ℓ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover><mml:mo movablelimits="false">∫</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mi mathvariant="normal">d</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mi>v</mml:mi><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Defining the matter density perturbation<ce:display><ce:formula id="fm0490"><ce:label>(4.5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si152.svg"><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi>δ</mml:mi><mml:mi>J</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mi>ζ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>E</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> the fluid number density <ce:italic>n</ce:italic> has a perturbation <ce:cross-refs refid="br1070 br1090" id="crs0170">[107,109]</ce:cross-refs><ce:display><ce:formula id="fm0500"><ce:label>(4.6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si153.svg"><mml:mi>δ</mml:mi><mml:mi>n</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>∂</mml:mo><mml:mi>χ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>∂</mml:mo><mml:mi>δ</mml:mi><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mi>ζ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>E</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ζ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>E</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mi>ζ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>E</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> up to second order. The matter sound speed squared is given by<ce:display><ce:formula id="fm0510"><ce:label>(4.7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si154.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>n</mml:mi><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Expanding <ce:cross-ref refid="fm0240" id="crf0670">(3.6)</ce:cross-ref> up to quadratic order in perturbations, we obtain the second-order matter action same as that derived in Refs. <ce:cross-refs refid="br1070 br1090" id="crs0180">[107,109]</ce:cross-refs>. Varying this matter action with respect to <ce:italic>δj</ce:italic> leads to<ce:display><ce:formula id="fm0520"><ce:label>(4.8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si155.svg"><mml:mo>∂</mml:mo><mml:mi>δ</mml:mi><mml:mi>j</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>v</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>∂</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> whose relation will be used to eliminate <ce:italic>δj</ce:italic>.</ce:para><ce:para id="pr0280">In the following, we choose the gauge<ce:display><ce:formula id="fm0530"><ce:label>(4.9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si156.svg"><mml:mi>E</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> under which a scalar quantity <ce:italic>ξ</ce:italic> associated with the spatial gauge transformation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si157.svg"><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi>ξ</mml:mi></mml:math> is fixed. A scalar quantity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> associated with the temporal part of the gauge transformation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si159.svg"><mml:mi>t</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>t</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> can be fixed by choosing a flat gauge (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>ζ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>) or a unitary gauge (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si161.svg"><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>). We do not specify the temporal gauge condition in deriving the second-order action, but we will do so at the end.</ce:para><ce:para id="pr0290">Expanding the total action <ce:cross-ref refid="fm0200" id="crf0680">(3.2)</ce:cross-ref> up to quadratic order in scalar perturbations and integrating it by parts, the resulting second-order action is given by<ce:display><ce:formula id="fm0540"><ce:label>(4.10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si162.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">flat</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where<ce:display><ce:formula id="fm0550"><ce:label>(4.11)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si163.svg"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">flat</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup id="mmlbr0003"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi>δ</mml:mi><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">}</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mi>α</mml:mi><mml:mspace linebreak="newline"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>16</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mi>δ</mml:mi><mml:msup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">{</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">}</mml:mo><mml:mi>α</mml:mi><mml:mspace linebreak="newline"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi>v</mml:mi><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>v</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mi>v</mml:mi><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mspace linebreak="newline"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mi>δ</mml:mi><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>α</mml:mi><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0560"><ce:label>(4.12)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si164.svg"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup id="mmlbr0004"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">{</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mi>α</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>v</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">}</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mspace linebreak="newline"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>ζ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">{</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mi>α</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">}</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>ζ</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> are given by Eqs. <ce:cross-ref refid="fm0360" id="crf0690">(3.18)</ce:cross-ref> and <ce:cross-ref refid="fm0370" id="crf0700">(3.19)</ce:cross-ref>, respectively, and<ce:display><ce:formula id="fm0570"><ce:label>(4.13)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si165.svg"><mml:msub id="mmlbr0005"><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>24</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>η</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>24</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0005" linebreakstyle="after">,</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>8</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>8</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0005" linebreakstyle="after">,</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>8</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>η</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>8</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Now, we switch to the Fourier space with a comoving wavenumber <ce:bold><ce:italic>k</ce:italic></ce:bold>. Varying the total action <ce:cross-ref refid="fm0540" id="crf0710">(4.10)</ce:cross-ref> with respect to <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, and <ce:italic>v</ce:italic>, respectively, we obtain<ce:display><ce:formula id="fm0580"><ce:label>(4.14)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si166.svg"><mml:msub id="mmlbr0006"><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mi>α</mml:mi><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0006" linebreakstyle="before">+</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mi>ζ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi>B</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0590"><ce:label>(4.15)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si167.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi>α</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>v</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0600"><ce:label>(4.16)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si168.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0300">In the following, we choose the flat gauge given by<ce:display><ce:formula id="fm0610"><ce:label>(4.17)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si169.svg"><mml:mi>ζ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> to obtain stability conditions for scalar perturbations. We will discuss the two cases: (A) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity and (B) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity in turn.</ce:para><ce:section id="se0070"><ce:label>4.1</ce:label><ce:section-title id="st0080"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity</ce:section-title><ce:para id="pr0310">In <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, we can construct gauge-invariant scalar perturbations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si170.svg"><mml:mi>δ</mml:mi><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">f</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mi>ζ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>H</mml:mi></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si171.svg"><mml:mi>δ</mml:mi><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">f</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mi>ζ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>H</mml:mi></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.svg"><mml:mi>δ</mml:mi><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">f</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mi>ζ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>H</mml:mi></mml:math>. For the gauge choice <ce:cross-ref refid="fm0610" id="crf0720">(4.17)</ce:cross-ref>, they reduce, respectively, to <ce:italic>δϕ</ce:italic>, <ce:italic>δχ</ce:italic>, and <ce:italic>δρ</ce:italic>, which correspond to the dynamical scalar DOFs. Note that the perturbation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si173.svg"><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>δ</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:math> arises from nonlinearities in the GB term. We solve Eqs. <ce:cross-ref refid="fm0580" id="crf0730">(4.14)</ce:cross-ref>-<ce:cross-ref refid="fm0600" id="crf0740">(4.16)</ce:cross-ref> for <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>v</ce:italic> and substitute them into Eq. <ce:cross-ref refid="fm0540" id="crf0750">(4.10)</ce:cross-ref>. Then, the resulting quadratic-order action in Fourier space is expressed in the form<ce:display><ce:formula id="fm0620"><ce:label>(4.18)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si174.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="bold-italic">K</mml:mi><mml:mover accent="true"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="bold-italic">G</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="bold-italic">M</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="bold-italic">B</mml:mi><mml:mover accent="true"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:bold><ce:italic>K</ce:italic></ce:bold>, <ce:bold><ce:italic>G</ce:italic></ce:bold>, <ce:bold><ce:italic>M</ce:italic></ce:bold>, <ce:bold><ce:italic>B</ce:italic></ce:bold> are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si175.svg"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math> matrices, and<ce:display><ce:formula id="fm0630"><ce:label>(4.19)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si176.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The leading-order contributions to <ce:bold><ce:italic>M</ce:italic></ce:bold> and <ce:bold><ce:italic>B</ce:italic></ce:bold> are of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si177.svg"><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math>. Taking the small-scale limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si178.svg"><mml:mi>k</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mo>∞</mml:mo></mml:math>, nonvanishing components of the symmetric matrices <ce:bold><ce:italic>K</ce:italic></ce:bold> and <ce:bold><ce:italic>G</ce:italic></ce:bold> are<ce:display><ce:formula id="fm0640"><ce:label>(4.20)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si179.svg"><mml:msub id="mmlbr0007"><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0007" linebreakstyle="after">,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>η</mml:mi><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> and<ce:display><ce:formula id="fm0650"><ce:label>(4.21)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si180.svg"><mml:msub id="mmlbr0008"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">[</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="after">,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">[</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="after">,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:mi>η</mml:mi><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">{</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">}</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> To derive these coefficients, we have absorbed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si181.svg"><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>-dependent terms present in <ce:bold><ce:italic>B</ce:italic></ce:bold> into the components of <ce:bold><ce:italic>G</ce:italic></ce:bold> and used the relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si182.svg"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>η</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:math>, and<ce:display><ce:formula id="fm0660"><ce:label>(4.22)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si183.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>H</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>H</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>η</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>H</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>H</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>H</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>H</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The scalar ghosts are absent under the following three conditions<ce:display><ce:formula id="fm0670"><ce:label>(4.23)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si184.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0680"><ce:label>(4.24)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si185.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0690"><ce:label>(4.25)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si186.svg"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">det</mml:mi></mml:mrow><mml:mspace width="0.2em"/><mml:mi mathvariant="bold-italic">K</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Under the no-ghost condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> of tensor perturbations, inequalities <ce:cross-ref refid="fm0670" id="crf0760">(4.23)</ce:cross-ref>-<ce:cross-ref refid="fm0690" id="crf0770">(4.25)</ce:cross-ref> are satisfied for<ce:display><ce:formula id="fm0700"><ce:label>(4.26)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si187.svg"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0710"><ce:label>(4.27)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si188.svg"><mml:mrow><mml:mi>η</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0320">In the limit of large <ce:italic>k</ce:italic>, dominant contributions to the second-order action <ce:cross-ref refid="fm0620" id="crf0780">(4.18)</ce:cross-ref> arise from <ce:bold><ce:italic>K</ce:italic></ce:bold> and <ce:bold><ce:italic>G</ce:italic></ce:bold>. Then, the dispersion relation can be expressed in the form<ce:display><ce:formula id="fm0720"><ce:label>(4.28)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si189.svg"><mml:mrow><mml:mi mathvariant="normal">det</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi mathvariant="bold-italic">K</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="bold-italic">G</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si190.svg"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math> is the scalar propagation speed. Solving Eq. <ce:cross-ref refid="fm0720" id="crf0790">(4.28)</ce:cross-ref> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, we obtain the following three solutions<ce:display><ce:formula id="fm0730"><ce:label>(4.29)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si191.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0740"><ce:label>(4.30)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si192.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0750"><ce:label>(4.31)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si193.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> which correspond to the squared propagation speeds of <ce:italic>δϕ</ce:italic>, <ce:italic>δχ</ce:italic>, and <ce:italic>δρ</ce:italic>, respectively. The scalar perturbation <ce:italic>δϕ</ce:italic> has a luminal propagation speed, so it satisfies the Laplacian stability condition. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, the matter perturbation <ce:italic>δρ</ce:italic> is free from Laplacian instability. On using the background Eq. <ce:cross-ref refid="fm0330" id="crf0800">(3.15)</ce:cross-ref>, the sound speed squared <ce:cross-ref refid="fm0740" id="crf0810">(4.30)</ce:cross-ref> can be expressed as<ce:cross-ref refid="fn0010" id="crf0820"><ce:sup>1</ce:sup></ce:cross-ref><ce:footnote id="fn0010"><ce:label>1</ce:label><ce:note-para id="np0010">If we eliminate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> by using Eq. <ce:cross-ref refid="fm0330" id="crf0830">(3.15)</ce:cross-ref>, we can express Eq. <ce:cross-ref refid="fm0740" id="crf0840">(4.30)</ce:cross-ref> in the form<ce:display><ce:formula id="fm0760"><ce:label>(4.32)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si194.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> From this expression, it seems that the existence of the last term can lead to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si195.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> even in the decelerating Universe. In the absence of matter (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si196.svg"><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>P</mml:mi></mml:math>), this possibility was suggested in Ref. <ce:cross-ref refid="br0930" id="crf0850">[93]</ce:cross-ref>. Eliminating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> instead of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> from Eq. <ce:cross-ref refid="fm0740" id="crf0860">(4.30)</ce:cross-ref>, it is clear that this possibility is forbidden even in the presence of matter.</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0770"><ce:label>(4.33)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si197.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>4</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub></mml:math> is the effective equation of state defined by Eq. <ce:cross-ref refid="fm0160" id="crf0870">(2.16)</ce:cross-ref>. The Laplacian stability of <ce:italic>δχ</ce:italic> is ensured for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si195.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, i.e.,<ce:display><ce:formula id="fm0780"><ce:label>(4.34)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si198.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Since we need the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> for the absence of Laplacian instability in the tensor sector, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub></mml:math> must be in the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si199.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:math>. This translates to the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si200.svg"><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">/</mml:mo><mml:mi>a</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, so the Laplacian stability of <ce:italic>δχ</ce:italic> requires that the Universe is accelerating. In decelerating cosmological epochs, the condition <ce:cross-ref refid="fm0780" id="crf0880">(4.34)</ce:cross-ref> is always violated for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>. During the radiation-dominated (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si201.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:math>) and matter-dominated (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si202.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>) eras, we have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si203.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>4</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si204.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:math>, respectively, which are both negative for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>.</ce:para><ce:para id="pr0330">We thus showed that, for scalar-GB couplings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> containing nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math>, <ce:italic>δχ</ce:italic> is prone to the Laplacian instability during the radiation and matter eras. Hence nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> should not be present in decelerating cosmological epochs. Even if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si205.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> is positive in the inflationary epoch, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si205.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> changes its sign during the transition to a reheating epoch (in which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si206.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mn>0</mml:mn></mml:math> for a standard reheating scenario). During the epoch of late-time cosmic acceleration, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si205.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> can be positive, but it changes the sign as we go back to the matter era. Since <ce:italic>δχ</ce:italic> is coupled to <ce:italic>δϕ</ce:italic> and <ce:italic>δρ</ce:italic>, the instability of <ce:italic>δχ</ce:italic> leads to the growth of <ce:italic>δϕ</ce:italic> and <ce:italic>δρ</ce:italic> for perturbations deep inside the Hubble radius. This violates the successful background evolution during the decelerating cosmological epochs.</ce:para><ce:para id="pr0340">The squared propagation speeds <ce:cross-ref refid="fm0730" id="crf0890">(4.29)</ce:cross-ref>-<ce:cross-ref refid="fm0750" id="crf0900">(4.31)</ce:cross-ref> have been derived by choosing the flat gauge <ce:cross-ref refid="fm0610" id="crf0910">(4.17)</ce:cross-ref>, but they are independent of the gauge choices. Indeed, we will show in Appendix <ce:cross-ref refid="se0100" id="crf1290">A</ce:cross-ref> that the same values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si207.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si205.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si208.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> can be obtained by choosing the unitary gauge. We also note that the scalar propagation speed squared <ce:cross-ref refid="fm0150" id="crf0920">(2.15)</ce:cross-ref> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity is not equivalent to the value <ce:cross-ref refid="fm0730" id="crf0930">(4.29)</ce:cross-ref>. As we observe in Eq. <ce:cross-ref refid="fm0150" id="crf0940">(2.15)</ce:cross-ref>, the propagation of <ce:italic>ϕ</ce:italic> is affected by the coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> with the linear GB term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math>. In <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> theory with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, the new scalar field <ce:italic>χ</ce:italic> plays a role of the dynamical DOF arising from the nonlinear GB term. In this latter case, the propagation of the other field <ce:italic>ϕ</ce:italic> does not practically acquire the effect of a coupling with the GB term and hence <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si209.svg"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> reduces to the luminal value.</ce:para></ce:section><ce:section id="se0080"><ce:label>4.2</ce:label><ce:section-title id="st0090"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity</ce:section-title><ce:para id="pr0350">Finally, we also study the stability of scalar perturbations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity given by the action<ce:display><ce:formula id="fm0790"><ce:label>(4.35)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si210.svg"><mml:mi mathvariant="script">S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>R</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> In this case, there is no scalar field <ce:italic>ϕ</ce:italic> coupled to the GB term. The action <ce:cross-ref refid="fm0790" id="crf0950">(4.35)</ce:cross-ref> is equivalent to Eq. <ce:cross-ref refid="fm0200" id="crf0960">(3.2)</ce:cross-ref> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si211.svg"><mml:mi>ϕ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si212.svg"><mml:mi>X</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si213.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si214.svg"><mml:mi>U</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>χ</mml:mi><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si215.svg"><mml:mi>ξ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. As shown in Ref. <ce:cross-ref refid="br1030" id="crf0970">[103]</ce:cross-ref>, this theory belongs to a subclass of Horndeski theories with one scalar DOF <ce:italic>χ</ce:italic> besides a matter fluid.</ce:para><ce:para id="pr0360">In <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity, the second-order action of scalar perturbations is obtained by setting <ce:italic>ϕ</ce:italic>, <ce:italic>δϕ</ce:italic>, and their derivatives 0 in Eqs. <ce:cross-ref refid="fm0550" id="crf0980">(4.11)</ce:cross-ref> and <ce:cross-ref refid="fm0560" id="crf0990">(4.12)</ce:cross-ref>. We choose the flat gauge <ce:cross-ref refid="fm0610" id="crf1000">(4.17)</ce:cross-ref> and eliminate <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>v</ce:italic> from the action by using Eqs. <ce:cross-ref refid="fm0580" id="crf1010">(4.14)</ce:cross-ref>-<ce:cross-ref refid="fm0600" id="crf1020">(4.16)</ce:cross-ref>. Then, the second-order scalar action reduces to the form <ce:cross-ref refid="fm0620" id="crf1030">(4.18)</ce:cross-ref> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si216.svg"><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:mn>2</mml:mn></mml:math> matrices <ce:bold><ce:italic>K</ce:italic></ce:bold>, <ce:bold><ce:italic>G</ce:italic></ce:bold>, <ce:bold><ce:italic>M</ce:italic></ce:bold>, <ce:bold><ce:italic>B</ce:italic></ce:bold> and two dynamical perturbations<ce:display><ce:formula id="fm0800"><ce:label>(4.36)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si217.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> In the small-scale limit, nonvanishing components of <ce:bold><ce:italic>K</ce:italic></ce:bold> and <ce:bold><ce:italic>G</ce:italic></ce:bold> are given by<ce:display><ce:formula id="fm0810"><ce:label>(4.37)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si218.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0820"><ce:label>(4.38)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si219.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">[</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> The no-ghost conditions correspond to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si220.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si221.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, which are satisfied for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>. The propagation speed squared for <ce:italic>δχ</ce:italic> is<ce:display><ce:formula id="fm0830"><ce:label>(4.39)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si222.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where, in the last equality, we used the background Eq. <ce:cross-ref refid="fm0330" id="crf1040">(3.15)</ce:cross-ref> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si134.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>. The other matter propagation speed squared is given by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si223.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>. Since the last expression of Eq. <ce:cross-ref refid="fm0830" id="crf1050">(4.39)</ce:cross-ref> is of the same form as Eq. <ce:cross-ref refid="fm0770" id="crf1060">(4.33)</ce:cross-ref>, the Laplacian instability of <ce:italic>δχ</ce:italic> is present in decelerating cosmological epochs. In Ref. <ce:cross-ref refid="br1000" id="crf1070">[100]</ce:cross-ref>, violent growth of matter perturbations was found during the radiation and matter eras for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> models of late-time cosmic acceleration. This is attributed to the Laplacian instability of <ce:italic>δχ</ce:italic> coupled to <ce:italic>δρ</ce:italic>, which inevitably occurs for nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>.</ce:para></ce:section></ce:section><ce:section id="se0090" role="conclusion"><ce:label>5</ce:label><ce:section-title id="st0100">Conclusions</ce:section-title><ce:para id="pr0370">In this letter, we studied the stability of cosmological perturbations on the spatially flat FLRW background in scalar-GB theories given by the action <ce:cross-ref refid="fm0190" id="crf1080">(3.1)</ce:cross-ref>. Provided that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, the action <ce:cross-ref refid="fm0190" id="crf1090">(3.1)</ce:cross-ref> is equivalent to <ce:cross-ref refid="fm0200" id="crf1100">(3.2)</ce:cross-ref> with a new scalar DOF <ce:italic>χ</ce:italic> arising from nonlinear GB terms. Theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> correspond to a linear GB term coupled to a scalar field <ce:italic>ϕ</ce:italic> of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math>, which belongs to a subclass of Horndeski theories. To make a comparison with the scalar-GB coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> containing nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math>, we first revisited stabilities of cosmological perturbations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity in Sec. <ce:cross-ref refid="se0020" id="crf1110">2</ce:cross-ref>. In this latter theory, provided that the scalar-GB coupling is subdominant to the background equations of motion, the stability conditions of tensor and scalar perturbations can be consistently satisfied.</ce:para><ce:para id="pr0380">In Sec. <ce:cross-ref refid="se0030" id="crf1120">3</ce:cross-ref>, we derived the background equations and stability conditions of tensor perturbations for the scalar-GB coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>. Besides a canonical scalar field <ce:italic>ϕ</ce:italic> with the kinetic term <ce:italic>ηX</ce:italic> and the potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, we incorporate a perfect fluid given by the Schutz-Sorkin action <ce:cross-ref refid="fm0240" id="crf1130">(3.6)</ce:cross-ref>. The absence of ghosts and Laplacian instabilities requires that the quantities <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> defined by Eqs. <ce:cross-ref refid="fm0360" id="crf1140">(3.18)</ce:cross-ref> and <ce:cross-ref refid="fm0370" id="crf1150">(3.19)</ce:cross-ref> are both positive. In terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, the background equations of motion in the gravitational sector can be expressed in a simple manner as Eqs. <ce:cross-ref refid="fm0320" id="crf1160">(3.14)</ce:cross-ref> and <ce:cross-ref refid="fm0330" id="crf1170">(3.15)</ce:cross-ref>, where the latter is used to simplify a scalar sound speed later.</ce:para><ce:para id="pr0390">In Sec. <ce:cross-ref refid="se0060" id="crf1180">4</ce:cross-ref>, we expanded the action in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math> up to quadratic order in scalar perturbations. After eliminating nondynamical variables <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, and <ce:italic>v</ce:italic>, the second-order action is of the form <ce:cross-ref refid="fm0620" id="crf1190">(4.18)</ce:cross-ref> with three dynamical perturbations <ce:cross-ref refid="fm0630" id="crf1200">(4.19)</ce:cross-ref>. With the no-ghost condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> of tensor perturbations, the scalar ghosts are absent for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>. The sound speeds of perturbations <ce:italic>δϕ</ce:italic> and <ce:italic>δρ</ce:italic> have the standard values 1 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si224.svg"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math>, respectively. However, the squared propagation speed of <ce:italic>δχ</ce:italic>, which arises from nonlinear GB functions in <ce:italic>f</ce:italic>, has a nontrivial value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si225.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:math>. Since the positivity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si205.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> requires that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mn>6</mml:mn></mml:math>, we have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si199.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:math> under the absence of Laplacian instability in the tensor sector (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>). This means that the scalar perturbation associated with nonlinearities of the GB term is subject to Laplacian instability during decelerating cosmological epochs including radiation and matter eras. The same property also holds for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>.</ce:para><ce:para id="pr0400">We thus showed that a canonical scalar field <ce:italic>ϕ</ce:italic> coupled to a nonlinear GB term does not modify the property of negative values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si205.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> in the decelerating Universe. During inflation or the epoch of late-time cosmic acceleration, it is possible to avoid Laplacian instability of the perturbation <ce:italic>δχ</ce:italic> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>. However, in the subsequent reheating period after inflation or in the preceding matter era before dark energy dominance, the Laplacian instability inevitably emerges to violate the successful background cosmological evolution. We have shown this for a canonical scalar field <ce:italic>ϕ</ce:italic>, but it may be interesting to see whether the same property persists for the scalar field <ce:italic>ϕ</ce:italic> arising in Horndeski theories and its extensions like DHOST theories <ce:cross-refs refid="br1100 br1110" id="crs0190">[110,111]</ce:cross-refs>. While we focused on the analysis on the FLRW background, it will be also of interest to study whether some instabilities are present for perturbations on a static and spherically symmetric background in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>. The latter is important for the construction of stable hairy black hole or neutron star solutions in theories beyond the scalar-GB coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math>. These issues are left for future works.</ce:para> </ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0150">Declaration of Competing Interest</ce:section-title><ce:para id="pr0450">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0110">Acknowledgements</ce:section-title><ce:para id="pr0410">ST is supported by the Grant-in-Aid for Scientific Research Fund of the <ce:grant-sponsor id="gsp0010" sponsor-id="https://doi.org/10.13039/501100001691">JSPS</ce:grant-sponsor> Nos. <ce:grant-number refid="gsp0010">19K03854</ce:grant-number> and <ce:grant-number refid="gsp0010">22K03642</ce:grant-number>.</ce:para></ce:acknowledgment><ce:appendices><ce:section id="se0100"><ce:label>Appendix A</ce:label><ce:section-title id="st0120">Stability conditions in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity in unitary gauge</ce:section-title><ce:para id="pr0420">In this Appendix, we derive stability conditions of scalar perturbations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity by choosing the unitary gauge<ce:display><ce:formula id="fm0840"><ce:label>(A.1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si226.svg"><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Then, the gauge-invariant perturbations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si227.svg"><mml:mi mathvariant="script">R</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>ζ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si228.svg"><mml:mi>δ</mml:mi><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">u</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si229.svg"><mml:mi>δ</mml:mi><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">u</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:math> reduce, respectively, to <ce:italic>ζ</ce:italic>, <ce:italic>δχ</ce:italic>, and <ce:italic>δρ</ce:italic>. After the elimination of nondynamical variables <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>v</ce:italic> from Eq. <ce:cross-ref refid="fm0540" id="crf1210">(4.10)</ce:cross-ref>, the second-order action reduces to the form <ce:cross-ref refid="fm0620" id="crf1220">(4.18)</ce:cross-ref> with the dynamical perturbations<ce:display><ce:formula id="fm0850"><ce:label>(A.2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si230.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>ζ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where nonvanishing matrix components of <ce:bold><ce:italic>K</ce:italic></ce:bold> and <ce:bold><ce:italic>G</ce:italic></ce:bold> are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si231.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si232.svg"><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub></mml:math>. In the short-wavelength limit, the ghosts are absent for<ce:display><ce:formula id="fm0860"><ce:label>(A.3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si184.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0870"><ce:label>(A.4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si234.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0880"><ce:label>(A.5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si235.svg"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">det</mml:mi></mml:mrow><mml:mspace width="0.2em"/><mml:mi mathvariant="bold-italic">K</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Under the tensor no-ghost condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, inequalities <ce:cross-ref refid="fm0860" id="crf1230">(A.3)</ce:cross-ref>-<ce:cross-ref refid="fm0880" id="crf1240">(A.5)</ce:cross-ref> are satisfied for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>. These conditions are the same as those derived by choosing the flat gauge.</ce:para><ce:para id="pr0430">The scalar propagation speed squared <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> can be derived by solving the dispersion relation <ce:cross-ref refid="fm0720" id="crf1250">(4.28)</ce:cross-ref>. On using the background Eq. <ce:cross-ref refid="fm0330" id="crf1260">(3.15)</ce:cross-ref>, we obtain the three values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> exactly the same as Eqs. <ce:cross-ref refid="fm0730" id="crf1270">(4.29)</ce:cross-ref>-<ce:cross-ref refid="fm0750" id="crf1280">(4.31)</ce:cross-ref>. 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Tasinato, JCAP 04, 044 (2016), arXiv:1602.03119 [hep-th].</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> \ No newline at end of file +<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd"><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="sco" xml:lang="en"><item-info><jid>PLB</jid><aid>137751</aid><ce:article-number>137751</ce:article-number><ce:pii>S0370-2693(23)00085-0</ce:pii><ce:doi>10.1016/j.physletb.2023.137751</ce:doi><ce:copyright year="2023" type="other">The Author(s)</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>Astrophysics and Cosmology</ce:text></ce:doctopic></ce:doctopics></item-info><head><ce:title id="ti0010">Cosmological stability in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity</ce:title><ce:author-group id="ag0010"><ce:author id="au0010" author-id="S0370269323000850-ade115eed9bce9e185e44b0aa9b50010"><ce:given-name>Shinji</ce:given-name><ce:surname>Tsujikawa</ce:surname><ce:e-address type="email" xlink:href="mailto:tsujikawa@waseda.jp" id="ea0010">tsujikawa@waseda.jp</ce:e-address></ce:author><ce:affiliation id="aff0010" affiliation-id="S0370269323000850-2391d88b911951b1d0cff5f6ba23fdf8"><ce:textfn>Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Waseda University</sa:organization><sa:address-line>3-4-1 Okubo</sa:address-line><sa:city>Tokyo</sa:city><sa:state>Shinjuku</sa:state><sa:postal-code>169-8555</sa:postal-code><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0005">Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan</ce:source-text></ce:affiliation></ce:author-group><ce:date-received day="20" month="12" year="2022"/><ce:date-revised day="2" month="2" year="2023"/><ce:date-accepted day="2" month="2" year="2023"/><ce:miscellaneous id="ms0010">Editor: M. Trodden</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0010">In gravitational theories where a canonical scalar field <ce:italic>ϕ</ce:italic> with a potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is coupled to a Gauss-Bonnet (GB) term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> with the Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, we study the cosmological stability of tensor and scalar perturbations in the presence of a perfect fluid. We show that, in decelerating cosmological epochs with a positive tensor propagation speed squared, the existence of nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> in <ce:italic>f</ce:italic> always induces Laplacian instability of a dynamical scalar perturbation associated with the GB term. This is also the case for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity, where the presence of nonlinear GB functions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is not allowed during the radiation- and matter-dominated epochs. A linearly coupled GB term with <ce:italic>ϕ</ce:italic> of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> can be consistent with all the stability conditions, provided that the scalar-GB coupling is subdominant to the background cosmological dynamics.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0140">Data availability</ce:section-title><ce:para id="pr0440">No data was used for the research described in the article.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">General Relativity (GR) is a fundamental theory of gravity whose validity has been probed in Solar System experiments <ce:cross-ref refid="br0010" id="crf0010">[1]</ce:cross-ref> and submillimeter laboratory tests <ce:cross-refs refid="br0020 br0030" id="crs0010">[2,3]</ce:cross-refs>. Despite the success of GR describing gravitational interactions in the Solar System, there have been long-standing cosmological problems such as the origins of inflation, dark energy, and dark matter. To address these problems, one typically introduces additional degrees of freedom (DOFs) beyond those appearing in GR <ce:cross-refs refid="br0040 br0050 br0060 br0070 br0080 br0090 br0100" id="crs0020">[4–10]</ce:cross-refs>. One of such new DOFs is a canonical scalar field <ce:italic>ϕ</ce:italic> with a potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> <ce:cross-refs refid="br0110 br0120 br0130 br0140 br0150 br0160 br0170 br0180 br0190 br0200 br0210 br0220" id="crs0030">[11–22]</ce:cross-refs>. If the scalar field evolves slowly along the potential, it is possible to realize cosmic acceleration responsible for inflation or dark energy. An oscillating scalar field around the potential minimum can be also the source for dark matter.</ce:para><ce:para id="pr0020">The other way of introducing a new dynamical DOF is to modify the gravitational sector from GR. The Lagrangian in GR is given by an Einstein-Hilbert term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow></mml:msub></mml:math> is the reduced Planck mass and <ce:italic>R</ce:italic> is the Ricci scalar. If we consider theories containing nonlinear functions of <ce:italic>R</ce:italic> of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, there is one scalar DOF arising from the modification of gravity <ce:cross-refs refid="br0230 br0240" id="crs0040">[23,24]</ce:cross-refs>. One well known example is the Starobinsky's model, in which the presence of a quadratic curvature term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> drives cosmic acceleration <ce:cross-ref refid="br0250" id="crf0020">[25]</ce:cross-ref>. It is also possible to construct <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> models of late-time cosmic acceleration <ce:cross-refs refid="br0260 br0270 br0280 br0290 br0300 br0310 br0320" id="crs0050">[26–32]</ce:cross-refs>, while being consistent with local gravity constraints.</ce:para><ce:para id="pr0030">The Einstein tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> obtained by varying the Einstein-Hilbert action satisfies the conserved relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math> is a covariant derivative operator), with the property of second-order field equations of motion in metrics. If we demand such conserved and second-order properties for 2-rank symmetric tensors, GR is the unique theory of gravity in 4 dimensions <ce:cross-ref refid="br0330" id="crf0030">[33]</ce:cross-ref>. In spacetime dimensions higher than 4, there is a particular combination known as a Gauss-Bonnet (GB) term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> consistent with those demands <ce:cross-ref refid="br0340" id="crf0040">[34]</ce:cross-ref>. In 4 dimensions, the GB term is a topological surface term and hence it does not contribute to the field equations of motion. In the presence of a coupling between a scalar field <ce:italic>ϕ</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math>, the spacetime dynamics is modified by the time or spatial variation of <ce:italic>ϕ</ce:italic>. Indeed, this type of scalar-GB coupling appears in the context of low energy effective string theory <ce:cross-refs refid="br0350 br0360 br0370" id="crs0060">[35–37]</ce:cross-refs>. The cosmological application of the coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> has been extensively performed in the literature <ce:cross-refs refid="br0380 br0390 br0400 br0410 br0420 br0430 br0440 br0450 br0460 br0470 br0480 br0490 br0500 br0510 br0520 br0530 br0540 br0550 br0560 br0570 br0580 br0590 br0600 br0610 br0620 br0630" id="crs0070">[38–63]</ce:cross-refs>. Moreover, it is known that the same coupling gives rise to spherically symmetric solutions of hairy black holes and neutron stars <ce:cross-refs refid="br0640 br0650 br0660 br0670 br0680 br0690 br0700 br0710 br0720 br0730 br0740 br0750 br0760 br0770 br0780 br0790 br0800" id="crs0080">[64–80]</ce:cross-refs>. The Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> containing nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> also generates nontrivial contributions to the spacetime dynamics <ce:cross-refs refid="br0810 br0820 br0830 br0840 br0850 br0860 br0870 br0880 br0890 br0900" id="crs0090">[81–90]</ce:cross-refs>.</ce:para><ce:para id="pr0040">In Ref. <ce:cross-ref refid="br0910" id="crf0050">[91]</ce:cross-ref>, De Felice and Suyama studied the stability of scalar perturbations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity on a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) background. In theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:msubsup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mi>R</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>f</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo>∂</mml:mo><mml:mi>R</mml:mi></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mi>R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>f</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>f</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="script">G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, there is an unusual scale-dependent sound speed which propagates superluminally in the short-wavelength limit, unless the vacuum is in a de Sitter state (see also Ref. <ce:cross-ref refid="br0920" id="crf0060">[92]</ce:cross-ref> for the analysis in an anisotropic cosmological background). We note that this problem does not arise for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity. In Ref. <ce:cross-ref refid="br0930" id="crf0070">[93]</ce:cross-ref>, the same authors extended the analysis to a more general Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> with a canonical scalar field <ce:italic>ϕ</ce:italic> and showed that the property of the scale-dependent sound speed is not modified by the presence of <ce:italic>ϕ</ce:italic>. Taking a perfect fluid (radiation or nonrelativistic matter) into account in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity, the cosmological stability and evolution of matter perturbations were studied in Refs. <ce:cross-refs refid="br0940 br0950 br0960" id="crs0100">[94–96]</ce:cross-refs>.</ce:para><ce:para id="pr0050">In Einstein-scalar-GB gravity given by the Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, where <ce:italic>ϕ</ce:italic> is a canonical scalar field, the problem of scale-dependent sound speeds mentioned above is not present. In this theory, the propagation of scalar perturbations on the flat FLRW background was studied in Ref. <ce:cross-ref refid="br0930" id="crf0080">[93]</ce:cross-ref> without taking into account matter. While the sound speed associated with the field <ce:italic>ϕ</ce:italic> is luminal for theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, the propagation speed squared <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> arising from a nonlinear GB term deviates from that of light and it can be even negative. In Ref. <ce:cross-ref refid="br0930" id="crf0090">[93]</ce:cross-ref>, the authors discussed the possibility for satisfying the Laplacian stability condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>. In the presence of matter, however, the stability conditions are subject to modifications from those in the vacuum. To understand what happens for the dynamics of cosmological perturbations during radiation- and matter-dominated epochs, we need to study their stabilities by incorporating radiation or nonrelativistic matter.</ce:para><ce:para id="pr0060">In this letter, we will derive general conditions for the absence of ghosts and Laplacian instabilities in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity, where <ce:italic>ϕ</ce:italic> is a canonical scalar field with a potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. In theories where the scalar field <ce:italic>ϕ</ce:italic> is coupled to the linear GB term, i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math>, there is only one dynamical scalar DOF <ce:italic>ϕ</ce:italic>. In theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, the Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> can be expressed in terms of two scalar fields <ce:italic>ϕ</ce:italic> and <ce:italic>χ</ce:italic> coupled to the linear GB term, where <ce:italic>χ</ce:italic> arises from the nonlinearity in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math>. Hence the latter theory has two dynamical scalar DOFs. To study the cosmological stability of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, we take a perfect fluid into account as a form of the Schutz-Sorkin action <ce:cross-refs refid="br0970 br0980 br0990" id="crs0110">[97–99]</ce:cross-refs>. We will show that the squared sound speed arising from nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> is negative during decelerating cosmological epochs including radiation and matter eras. To reach this conclusion, we exploit the fact that the propagation speed squared <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> of tensor perturbations must be positive to avoid Laplacian instability of gravitational waves.</ce:para><ce:para id="pr0070">The same Laplacian instability of scalar perturbations is also present in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with any nonlinear function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> in <ce:italic>f</ce:italic>. We note that, in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> models of late-time cosmic acceleration, violent instabilities of matter density perturbations during the radiation and matter eras were reported in Ref. <ce:cross-ref refid="br1000" id="crf0100">[100]</ce:cross-ref>. This can be regarded as the consequence of a negative sound speed squared of the scalar perturbation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mi>δ</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:math> arising from the nonlinearity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> in <ce:italic>f</ce:italic>. Since <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mi>δ</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:math> is coupled to the matter perturbation <ce:italic>δρ</ce:italic>, the background cosmological evolution during the radiation and matter eras is spoiled by the rapid growth of <ce:italic>δρ</ce:italic>. Our analysis in this letter shows that similar catastrophic instabilities persist for more general scalar-GB couplings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>.</ce:para><ce:para id="pr0080">This letter is organized as follows. In Sec. <ce:cross-ref refid="se0020" id="crf0110">2</ce:cross-ref>, we revisit cosmological stability conditions in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity with a canonical scalar field <ce:italic>ϕ</ce:italic>, which can be accommodated in a subclass of Horndeski theories with a single scalar DOF <ce:cross-refs refid="br1010 br1020 br1030 br1040" id="crs0120">[101–104]</ce:cross-refs>. This is an exceptional case satisfying the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, under which the Laplacian instability of scalar perturbations can be avoided. In Sec. <ce:cross-ref refid="se0030" id="crf0120">3</ce:cross-ref>, we derive the background equations and stability conditions of tensor perturbations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>R</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math> by incorporating a perfect fluid. In Sec. <ce:cross-ref refid="se0060" id="crf0130">4</ce:cross-ref>, we proceed to the derivation of a second-order action of scalar perturbations and obtain conditions for the absence of ghosts and Laplacian instabilities in the scalar sector. In particular, we show that an effective cosmological equation of state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub></mml:math> needs to be in the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mn>6</mml:mn></mml:math> to ensure Laplacian stabilities of the perturbation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mi>δ</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:math>. Sec. <ce:cross-ref refid="se0090" id="crf0140">5</ce:cross-ref> is devoted to conclusions.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity</ce:section-title><ce:para id="pr0090">We first briefly revisit the cosmological stability in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity given by the action<ce:display><ce:formula id="fm0010"><ce:label>(2.1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mi mathvariant="script">S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>R</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>η</mml:mi><mml:mi>X</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>g</ce:italic> is a determinant of the metric tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>, <ce:italic>η</ce:italic> is a constant, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:mi>X</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi></mml:math> is a kinetic term of the scalar field <ce:italic>ϕ</ce:italic>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> are functions of <ce:italic>ϕ</ce:italic>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> is a GB term defined by<ce:display><ce:formula id="fm0020"><ce:label>(2.2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mi mathvariant="script">G</mml:mi><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub></mml:math> being the Ricci and Riemann tensors, respectively. For the matter action <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math>, we consider a perfect fluid minimally coupled to gravity.</ce:para><ce:para id="pr0100">The action <ce:cross-ref refid="fm0010" id="crf0150">(2.1)</ce:cross-ref> contains one scalar DOF <ce:italic>ϕ</ce:italic> besides the matter field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math>. If we consider Horndeski theories <ce:cross-ref refid="br1010" id="crf0160">[101]</ce:cross-ref> given by the action<ce:display><ce:formula id="fm0030"><ce:label>(2.3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mi mathvariant="script">S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo id="mmlbr0001">∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mspace width="0.2em"/><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>□</mml:mo><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>R</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>□</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">}</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="before">+</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>□</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mo>□</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">}</mml:mo></mml:mrow><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="before">+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> then the theory <ce:cross-ref refid="fm0010" id="crf0170">(2.1)</ce:cross-ref> can be accommodated by choosing the coupling functions <ce:cross-ref refid="br1030" id="crf0180">[103]</ce:cross-ref><ce:display><ce:formula id="fm0040"><ce:label>(2.4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub id="mmlbr0002"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>η</mml:mi><mml:mi>X</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">ln</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>7</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mi mathvariant="normal">ln</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="after">,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">ln</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="normal">ln</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where we use the notations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>∂</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:mi>X</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>∂</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:mi>ϕ</mml:mi></mml:math> for any arbitrary function <ce:italic>F</ce:italic>.</ce:para><ce:para id="pr0110">Let us consider a spatially flat FLRW background given by the line element <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:mi>a</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is a time-dependent scale factor. The perfect fluid has a density <ce:italic>ρ</ce:italic> and pressure <ce:italic>P</ce:italic>. The background equations as well as the perturbation equations in full Horndeski theories were derived in Refs. <ce:cross-refs refid="br1030 br1050 br1060 br1070" id="crs0130">[103,105–107]</ce:cross-refs>. On using the correspondence <ce:cross-ref refid="fm0040" id="crf0190">(2.4)</ce:cross-ref>, the background equations of motion in theories given by the action <ce:cross-ref refid="fm0010" id="crf0200">(2.1)</ce:cross-ref> are<ce:display><ce:formula id="fm0050"><ce:label>(2.5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mrow><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>ρ</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0060"><ce:label>(2.6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>P</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0070"><ce:label>(2.7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:mrow><mml:mi>η</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant="script">G</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0080"><ce:label>(2.8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:mi>H</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">/</mml:mo><mml:mi>a</mml:mi></mml:math> is the Hubble expansion rate, a dot represents the derivative with respect to <ce:italic>t</ce:italic>, and<ce:display><ce:formula id="fm0090"><ce:label>(2.9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0100"><ce:label>(2.10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0110"><ce:label>(2.11)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mrow><mml:mi mathvariant="script">G</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>24</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0120">In the presence of tensor perturbations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> with the perturbed line element <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:math>, the second-order action of traceless and divergence-free modes of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> was already derived in full Horndeski theories <ce:cross-refs refid="br1030 br1060 br1070" id="crs0140">[103,106,107]</ce:cross-refs>. In the current theory, the conditions for the absence of ghosts and Laplacian instabilities are<ce:display><ce:formula id="fm0120"><ce:label>(2.12)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0130"><ce:label>(2.13)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> are defined by Eqs. <ce:cross-ref refid="fm0090" id="crf0210">(2.9)</ce:cross-ref> and <ce:cross-ref refid="fm0100" id="crf0220">(2.10)</ce:cross-ref>, respectively. Note that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> determines the sign of a kinetic term of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math>, while <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> corresponds to the propagation speed squared of tensor perturbations.</ce:para><ce:para id="pr0130">For the scalar sector, we choose the perturbed line element <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:mi>α</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>B</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:math> in the flat gauge, where <ce:italic>α</ce:italic> and <ce:italic>B</ce:italic> are scalar metric perturbations. There is also a scalar-field perturbation <ce:italic>δϕ</ce:italic> besides the matter perturbation <ce:italic>δρ</ce:italic> and the fluid velocity potential <ce:italic>v</ce:italic>. After deriving the quadratic-order action of scalar perturbations, we can eliminate nondynamical variables <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, and <ce:italic>v</ce:italic> from the action. Then, we are left with the two dynamical perturbations <ce:italic>δϕ</ce:italic> and <ce:italic>δρ</ce:italic> in the second-order action. In the short-wavelength limit, there is neither ghost nor Laplacian instability for <ce:italic>δϕ</ce:italic> under the conditions <ce:cross-refs refid="br1030 br1060 br1070" id="crs0150">[103,106,107]</ce:cross-refs><ce:display><ce:formula id="fm0140"><ce:label>(2.14)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>96</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0150"><ce:label>(2.15)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>32</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>96</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math> corresponds to the propagation speed of <ce:italic>δϕ</ce:italic>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub></mml:math> is the cosmological effective equation of state defined by<ce:display><ce:formula id="fm0160"><ce:label>(2.16)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The stability conditions for <ce:italic>δρ</ce:italic> are given by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> is the matter sound speed squared.</ce:para><ce:para id="pr0140">Under the stability condition <ce:cross-ref refid="fm0120" id="crf0230">(2.12)</ce:cross-ref> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, the scalar no-ghost condition <ce:cross-ref refid="fm0140" id="crf0240">(2.14)</ce:cross-ref> is satisfied. Let us consider the case in which contributions of the scalar-GB coupling are suppressed, such that<ce:display><ce:formula id="fm0170"><ce:label>(2.17)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:mo stretchy="false">{</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo>,</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo>,</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">}</mml:mo><mml:mo>≪</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>≪</mml:mo><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Then, it follows that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>≃</mml:mo><mml:mn>1</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mn>2</mml:mn><mml:mi>η</mml:mi><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>≃</mml:mo><mml:mn>1</mml:mn></mml:math>. In such cases, provided that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, all the stability conditions are consistently satisfied. If the scalar-GB coupling contributes to the late-time cosmological dynamics, there is an observational bound on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> constrained from the GW170817 event together with the electromagnetic counterpart, i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si79.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>15</mml:mn></mml:mrow></mml:msup><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mn>7</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>16</mml:mn></mml:mrow></mml:msup></mml:math> <ce:cross-ref refid="br1080" id="crf0250">[108]</ce:cross-ref> for the redshift <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:mi>z</mml:mi><mml:mo>≤</mml:mo><mml:mn>0.009</mml:mn></mml:math>. This translates to the limit<ce:display><ce:formula id="fm0180"><ce:label>(2.18)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si81.svg"><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>≲</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>15</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> which gives a tight constraint on the amplitude of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. In this case, contributions of the scalar-GB coupling to the background Eqs. <ce:cross-ref refid="fm0050" id="crf0260">(2.5)</ce:cross-ref> and <ce:cross-ref refid="fm0060" id="crf0270">(2.6)</ce:cross-ref> are highly suppressed relative to the field density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si82.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and the matter density.</ce:para><ce:para id="pr0150">The bound <ce:cross-ref refid="fm0180" id="crf0280">(2.18)</ce:cross-ref> is not applied to early cosmological epochs including inflation, radiation, and matter eras. We note, however, that the dominance of the scalar-GB coupling to the background equations prevents the successful cosmic expansion history. This can also give rise to the violation of either of the stability conditions <ce:cross-ref refid="fm0120" id="crf0290">(2.12)</ce:cross-ref>-<ce:cross-ref refid="fm0150" id="crf0300">(2.15)</ce:cross-ref>. Provided the scalar-GB coupling is suppressed in such a way that inequalities <ce:cross-ref refid="fm0170" id="crf0310">(2.17)</ce:cross-ref> hold, the linear stabilities are ensured for both tensor and scalar perturbations.</ce:para></ce:section><ce:section id="se0030"><ce:label>3</ce:label><ce:section-title id="st0040"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity</ce:section-title><ce:para id="pr0160">We extend <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity to more general theories in which a canonical scalar field <ce:italic>ϕ</ce:italic> with a potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is coupled to the GB term of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. The action in such theories is given by<ce:display><ce:formula id="fm0190"><ce:label>(3.1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi mathvariant="script">S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>R</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>η</mml:mi><mml:mi>X</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where a matter field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math> is minimally coupled to gravity. It is more practical to introduce a scalar field <ce:italic>χ</ce:italic> and resort to the following action<ce:display><ce:formula id="fm0200"><ce:label>(3.2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:mi mathvariant="script">S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>R</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>η</mml:mi><mml:mi>X</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where<ce:display><ce:formula id="fm0210"><ce:label>(3.3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si85.svg"><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≡</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>χ</mml:mi><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> with the notation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si86.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>∂</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:mi>χ</mml:mi></mml:math>. Varying the action <ce:cross-ref refid="fm0200" id="crf0320">(3.2)</ce:cross-ref> with respect to <ce:italic>χ</ce:italic>, it follows that<ce:display><ce:formula id="fm0220"><ce:label>(3.4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>χ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> So long as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, we obtain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si89.svg"><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math>. In this case, the action <ce:cross-ref refid="fm0200" id="crf0330">(3.2)</ce:cross-ref> reduces to <ce:cross-ref refid="fm0190" id="crf0340">(3.1)</ce:cross-ref>. Thus, the equivalence of <ce:cross-ref refid="fm0200" id="crf0350">(3.2)</ce:cross-ref> with <ce:cross-ref refid="fm0190" id="crf0360">(3.1)</ce:cross-ref> holds for<ce:display><ce:formula id="fm0230"><ce:label>(3.5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si90.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> under which there is a new scalar DOF <ce:italic>χ</ce:italic> arising from the gravitational sector.</ce:para><ce:para id="pr0170">Theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> correspond to the coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:mi>f</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math>, in which case the cosmological stability conditions were already discussed in Sec. <ce:cross-ref refid="se0020" id="crf0370">2</ce:cross-ref>. In <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity, we do not have the additional scalar DOF <ce:italic>χ</ce:italic> arising from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math>, so the term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si92.svg"><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0220" id="crf0380">(3.4)</ce:cross-ref> does not have the meaning of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si93.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub></mml:math>. Thus, the action <ce:cross-ref refid="fm0200" id="crf0390">(3.2)</ce:cross-ref> with the new dynamical DOF <ce:italic>χ</ce:italic> does not reproduce the action <ce:cross-ref refid="fm0010" id="crf0400">(2.1)</ce:cross-ref> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity.</ce:para><ce:para id="pr0180">In the following, we will focus on theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, i.e., those containing the nonlinear dependence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> in <ce:italic>f</ce:italic>. For the matter field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math>, we incorporate a perfect fluid without a dynamical vector DOF. This matter sector is described by the Schutz-Sorkin action <ce:cross-refs refid="br0970 br0980 br0990" id="crs0160">[97–99]</ce:cross-refs><ce:display><ce:formula id="fm0240"><ce:label>(3.6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si94.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mspace width="0.2em"/><mml:mi>ρ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ℓ</mml:mi><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where the fluid density <ce:italic>ρ</ce:italic> is a function of its number density <ce:italic>n</ce:italic>. The vector field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math> is related to <ce:italic>n</ce:italic> according to the relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:mi>n</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si97.svg"><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mo stretchy="false">)</mml:mo></mml:math> is the fluid four velocity. A scalar quantity <ce:italic>ℓ</ce:italic> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math> is a Lagrange multiplier, with the notation of a partial derivative <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ℓ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>∂</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math>. Varying the matter action <ce:cross-ref refid="fm0240" id="crf0410">(3.6)</ce:cross-ref> with respect to <ce:italic>ℓ</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math>, respectively, we obtain<ce:display><ce:formula id="fm0250"><ce:label>(3.7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si99.svg"><mml:mrow><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0260"><ce:label>(3.8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si100.svg"><mml:mrow><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ℓ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>ρ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>n</mml:mi></mml:math>.</ce:para><ce:section id="se0040"><ce:label>3.1</ce:label><ce:section-title id="st0050">Background equations</ce:section-title><ce:para id="pr0190">We derive the background equations of motion on the spatially flat FLRW background given by the line element<ce:display><ce:formula id="fm0270"><ce:label>(3.9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si103.svg"><mml:mi>N</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is a lapse function. Since the fluid four velocity in its rest frame is given by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si104.svg"><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>, the vector field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math> has components <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.svg"><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>. From Eq. <ce:cross-ref refid="fm0250" id="crf0420">(3.7)</ce:cross-ref>, we obtain<ce:display><ce:formula id="fm0280"><ce:label>(3.10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si106.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mi>n</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mi mathvariant="normal">constant</mml:mi></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> which means that the total fluid number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si107.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is conserved. This translates to the differential equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si108.svg"><mml:mover accent="true"><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mi>n</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, which can be expressed as a form of the continuity equation<ce:display><ce:formula id="fm0290"><ce:label>(3.11)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si109.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>P</ce:italic> is a fluid pressure defined by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si110.svg"><mml:mi>P</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>n</mml:mi><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>ρ</mml:mi></mml:math>.</ce:para><ce:para id="pr0200">On the background <ce:cross-ref refid="fm0270" id="crf0430">(3.9)</ce:cross-ref>, the total action <ce:cross-ref refid="fm0200" id="crf0440">(3.2)</ce:cross-ref> is expressed in the form<ce:display><ce:formula id="fm0300"><ce:label>(3.12)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si111.svg"><mml:mi mathvariant="script">S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi>a</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>N</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>8</mml:mn><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>N</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>n</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mover accent="true"><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> From Eq. <ce:cross-ref refid="fm0260" id="crf0450">(3.8)</ce:cross-ref>, we obtain the following relation<ce:display><ce:formula id="fm0310"><ce:label>(3.13)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si112.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>N</mml:mi><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Varying the action <ce:cross-ref refid="fm0300" id="crf0460">(3.12)</ce:cross-ref> with respect to <ce:italic>N</ce:italic>, <ce:italic>a</ce:italic>, <ce:italic>ϕ</ce:italic>, <ce:italic>χ</ce:italic> respectively and setting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si113.svg"><mml:mi>N</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:math> at the end, we obtain the background equations of motion<ce:display><ce:formula id="fm0320"><ce:label>(3.14)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si114.svg"><mml:mrow><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>ρ</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0330"><ce:label>(3.15)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si115.svg"><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>P</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0340"><ce:label>(3.16)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si116.svg"><mml:mrow><mml:mi>η</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0350"><ce:label>(3.17)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si117.svg"><mml:mrow><mml:mi>χ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>24</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where<ce:display><ce:formula id="fm0360"><ce:label>(3.18)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si118.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0370"><ce:label>(3.19)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si119.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0210">We recall that the perfect fluid obeys the continuity Eq. <ce:cross-ref refid="fm0290" id="crf0470">(3.11)</ce:cross-ref>. We notice that Eqs. <ce:cross-ref refid="fm0320" id="crf0480">(3.14)</ce:cross-ref>-<ce:cross-ref refid="fm0340" id="crf0490">(3.16)</ce:cross-ref> are of similar forms to Eqs. <ce:cross-ref refid="fm0050" id="crf0500">(2.5)</ce:cross-ref>-<ce:cross-ref refid="fm0070" id="crf0510">(2.7)</ce:cross-ref> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity, but the expressions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> are different from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, respectively, because of the appearance of time derivatives of <ce:italic>χ</ce:italic>. These <ce:italic>χ</ce:italic> derivatives do not vanish for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, i.e., for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>. As we will show in Sec. <ce:cross-ref refid="se0060" id="crf0520">4</ce:cross-ref>, nonlinearities of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> in <ce:italic>f</ce:italic> are responsible for the appearance of a new scalar propagating DOF <ce:italic>δχ</ce:italic>.</ce:para></ce:section><ce:section id="se0050"><ce:label>3.2</ce:label><ce:section-title id="st0060">Stabilities in the tensor sector</ce:section-title><ce:para id="pr0220">We proceed to the derivation of stability conditions for tensor perturbations in theories given by the action <ce:cross-ref refid="fm0200" id="crf0530">(3.2)</ce:cross-ref>. The perturbed line element including the tensor perturbation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> is<ce:display><ce:formula id="fm0380"><ce:label>(3.20)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where we impose the traceless and divergence-free gauge conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si122.svg"><mml:msub><mml:mrow><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si123.svg"><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>. For the gravitational wave propagating along the <ce:italic>z</ce:italic> direction, nonvanishing components of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> are expressed in the form<ce:display><ce:formula id="fm0390"><ce:label>(3.21)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si124.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where the two polarized modes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si126.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> are functions of <ce:italic>t</ce:italic> and <ce:italic>z</ce:italic>.</ce:para><ce:para id="pr0230">The second-order action arising from the matter action <ce:cross-ref refid="fm0240" id="crf0540">(3.6)</ce:cross-ref> can be expressed as<ce:display><ce:formula id="fm0400"><ce:label>(3.22)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si127.svg"><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>P</mml:mi><mml:msubsup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>P</ce:italic> can be eliminated by using the background Eq. <ce:cross-ref refid="fm0330" id="crf0550">(3.15)</ce:cross-ref>. Expanding the total action <ce:cross-ref refid="fm0200" id="crf0560">(3.2)</ce:cross-ref> up to quadratic order in tensor perturbations and integrating it by parts, the resulting second-order action reduces to<ce:display><ce:formula id="fm0410"><ce:label>(3.23)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si128.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si129.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. We recall that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> are given by Eqs. <ce:cross-ref refid="fm0360" id="crf0570">(3.18)</ce:cross-ref> and <ce:cross-ref refid="fm0370" id="crf0580">(3.19)</ce:cross-ref>, respectively.</ce:para><ce:para id="pr0240">To avoid the ghost and Laplacian instabilities in the tensor sector, we require the two conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, which translate to<ce:display><ce:formula id="fm0420"><ce:label>(3.24)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0430"><ce:label>(3.25)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si133.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>8</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> In <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity without the scalar field <ce:italic>ϕ</ce:italic>, tensor stability conditions can be obtained by setting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si134.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si135.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> in Eqs. <ce:cross-ref refid="fm0420" id="crf0590">(3.24)</ce:cross-ref> and <ce:cross-ref refid="fm0430" id="crf0600">(3.25)</ce:cross-ref>.</ce:para><ce:para id="pr0250">We vary the action <ce:cross-ref refid="fm0410" id="crf0610">(3.23)</ce:cross-ref> with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si136.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:math>) in Fourier space with a comoving wavenumber <ce:bold><ce:italic>k</ce:italic></ce:bold>. Then, we obtain the tensor perturbation equation of motion<ce:display><ce:formula id="fm0440"><ce:label>(3.26)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si138.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si139.svg"><mml:mi>k</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi mathvariant="bold-italic">k</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:math>. Since <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:mi>ξ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub></mml:math>, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> dependence in <ce:italic>f</ce:italic> leads to the modified evolution of gravitational waves in comparison to GR. If the energy densities of <ce:italic>ϕ</ce:italic> and <ce:italic>χ</ce:italic> are relevant to the late-time cosmological dynamics after the matter dominance, the observational constraint on the tensor propagation speed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> arising from the GW170817 event <ce:cross-ref refid="br1080" id="crf0620">[108]</ce:cross-ref> (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si142.svg"><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mo>≲</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>15</mml:mn></mml:mrow></mml:msup></mml:math>) gives a tight bound on the scalar-GB coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. Such a stringent limit is not applied to the cosmological dynamics in the early Universe, but the conditions <ce:cross-ref refid="fm0420" id="crf0630">(3.24)</ce:cross-ref> and <ce:cross-ref refid="fm0430" id="crf0640">(3.25)</ce:cross-ref> need to be still satisfied.</ce:para></ce:section></ce:section><ce:section id="se0060"><ce:label>4</ce:label><ce:section-title id="st0070">Stabilities of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity in the scalar sector</ce:section-title><ce:para id="pr0260">In this section, we will derive conditions for the absence of scalar ghosts and Laplacian instabilities in theories given by the action <ce:cross-ref refid="fm0200" id="crf0650">(3.2)</ce:cross-ref>. A perturbed line element containing scalar perturbations <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>ζ</ce:italic>, and <ce:italic>E</ce:italic> is of the form<ce:display><ce:formula id="fm0450"><ce:label>(4.1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si143.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:mi>α</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>B</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:mi>ζ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi>E</mml:mi><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> For the scalar fields <ce:italic>ϕ</ce:italic> and <ce:italic>χ</ce:italic>, we consider perturbations <ce:italic>δϕ</ce:italic> and <ce:italic>δχ</ce:italic> on the background values <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si144.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si145.svg"><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, respectively, such that<ce:display><ce:formula id="fm0460"><ce:label>(4.2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where we will omit a bar from the background quantities in the following.</ce:para><ce:para id="pr0270">In the matter sector, the temporal and spatial components of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math> are decomposed into the background and perturbed parts as<ce:display><ce:formula id="fm0470"><ce:label>(4.3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si147.svg"><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>δ</mml:mi><mml:mi>J</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>j</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>δJ</ce:italic> and <ce:italic>δj</ce:italic> are scalar perturbations. In terms of the velocity potential <ce:italic>v</ce:italic>, the spatial component of fluid four velocity is expressed as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si148.svg"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>v</mml:mi></mml:math>. From Eq. <ce:cross-ref refid="fm0260" id="crf0660">(3.8)</ce:cross-ref>, the scalar quantity <ce:italic>ℓ</ce:italic> has a background part obeying the relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si149.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> besides a perturbation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mi>v</mml:mi></mml:math>. Then, we have<ce:display><ce:formula id="fm0480"><ce:label>(4.4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si151.svg"><mml:mi>ℓ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover><mml:mo movablelimits="false">∫</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mi mathvariant="normal">d</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mi>v</mml:mi><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Defining the matter density perturbation<ce:display><ce:formula id="fm0490"><ce:label>(4.5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si152.svg"><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi>δ</mml:mi><mml:mi>J</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mi>ζ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>E</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> the fluid number density <ce:italic>n</ce:italic> has a perturbation <ce:cross-refs refid="br1070 br1090" id="crs0170">[107,109]</ce:cross-refs><ce:display><ce:formula id="fm0500"><ce:label>(4.6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si153.svg"><mml:mi>δ</mml:mi><mml:mi>n</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>∂</mml:mo><mml:mi>χ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>∂</mml:mo><mml:mi>δ</mml:mi><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mi>ζ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>E</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ζ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>E</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mi>ζ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>E</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> up to second order. The matter sound speed squared is given by<ce:display><ce:formula id="fm0510"><ce:label>(4.7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si154.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>n</mml:mi><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Expanding <ce:cross-ref refid="fm0240" id="crf0670">(3.6)</ce:cross-ref> up to quadratic order in perturbations, we obtain the second-order matter action same as that derived in Refs. <ce:cross-refs refid="br1070 br1090" id="crs0180">[107,109]</ce:cross-refs>. Varying this matter action with respect to <ce:italic>δj</ce:italic> leads to<ce:display><ce:formula id="fm0520"><ce:label>(4.8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si155.svg"><mml:mo>∂</mml:mo><mml:mi>δ</mml:mi><mml:mi>j</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>v</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>∂</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> whose relation will be used to eliminate <ce:italic>δj</ce:italic>.</ce:para><ce:para id="pr0280">In the following, we choose the gauge<ce:display><ce:formula id="fm0530"><ce:label>(4.9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si156.svg"><mml:mi>E</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> under which a scalar quantity <ce:italic>ξ</ce:italic> associated with the spatial gauge transformation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si157.svg"><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi>ξ</mml:mi></mml:math> is fixed. A scalar quantity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> associated with the temporal part of the gauge transformation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si159.svg"><mml:mi>t</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>t</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> can be fixed by choosing a flat gauge (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>ζ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>) or a unitary gauge (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si161.svg"><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>). We do not specify the temporal gauge condition in deriving the second-order action, but we will do so at the end.</ce:para><ce:para id="pr0290">Expanding the total action <ce:cross-ref refid="fm0200" id="crf0680">(3.2)</ce:cross-ref> up to quadratic order in scalar perturbations and integrating it by parts, the resulting second-order action is given by<ce:display><ce:formula id="fm0540"><ce:label>(4.10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si162.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">flat</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where<ce:display><ce:formula id="fm0550"><ce:label>(4.11)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si163.svg"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">flat</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup id="mmlbr0003"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi>δ</mml:mi><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">}</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mi>α</mml:mi><mml:mspace linebreak="newline"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>16</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mi>δ</mml:mi><mml:msup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">{</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">}</mml:mo><mml:mi>α</mml:mi><mml:mspace linebreak="newline"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi>v</mml:mi><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>v</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mi>v</mml:mi><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mspace linebreak="newline"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mi>δ</mml:mi><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>α</mml:mi><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0560"><ce:label>(4.12)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si164.svg"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup id="mmlbr0004"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">{</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mi>α</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>v</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">}</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mspace linebreak="newline"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>ζ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">{</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mi>α</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">}</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>ζ</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> are given by Eqs. <ce:cross-ref refid="fm0360" id="crf0690">(3.18)</ce:cross-ref> and <ce:cross-ref refid="fm0370" id="crf0700">(3.19)</ce:cross-ref>, respectively, and<ce:display><ce:formula id="fm0570"><ce:label>(4.13)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si165.svg"><mml:msub id="mmlbr0005"><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>24</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>η</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>24</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0005" linebreakstyle="after">,</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>8</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>8</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0005" linebreakstyle="after">,</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>8</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>η</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>8</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Now, we switch to the Fourier space with a comoving wavenumber <ce:bold><ce:italic>k</ce:italic></ce:bold>. Varying the total action <ce:cross-ref refid="fm0540" id="crf0710">(4.10)</ce:cross-ref> with respect to <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, and <ce:italic>v</ce:italic>, respectively, we obtain<ce:display><ce:formula id="fm0580"><ce:label>(4.14)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si166.svg"><mml:msub id="mmlbr0006"><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mi>α</mml:mi><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0006" linebreakstyle="before">+</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mi>ζ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi>B</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0590"><ce:label>(4.15)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si167.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi>α</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>v</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0600"><ce:label>(4.16)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si168.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0300">In the following, we choose the flat gauge given by<ce:display><ce:formula id="fm0610"><ce:label>(4.17)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si169.svg"><mml:mi>ζ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> to obtain stability conditions for scalar perturbations. We will discuss the two cases: (A) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity and (B) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity in turn.</ce:para><ce:section id="se0070"><ce:label>4.1</ce:label><ce:section-title id="st0080"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity</ce:section-title><ce:para id="pr0310">In <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, we can construct gauge-invariant scalar perturbations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si170.svg"><mml:mi>δ</mml:mi><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">f</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mi>ζ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>H</mml:mi></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si171.svg"><mml:mi>δ</mml:mi><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">f</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mi>ζ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>H</mml:mi></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.svg"><mml:mi>δ</mml:mi><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">f</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mi>ζ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>H</mml:mi></mml:math>. For the gauge choice <ce:cross-ref refid="fm0610" id="crf0720">(4.17)</ce:cross-ref>, they reduce, respectively, to <ce:italic>δϕ</ce:italic>, <ce:italic>δχ</ce:italic>, and <ce:italic>δρ</ce:italic>, which correspond to the dynamical scalar DOFs. Note that the perturbation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si173.svg"><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>δ</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:math> arises from nonlinearities in the GB term. We solve Eqs. <ce:cross-ref refid="fm0580" id="crf0730">(4.14)</ce:cross-ref>-<ce:cross-ref refid="fm0600" id="crf0740">(4.16)</ce:cross-ref> for <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>v</ce:italic> and substitute them into Eq. <ce:cross-ref refid="fm0540" id="crf0750">(4.10)</ce:cross-ref>. Then, the resulting quadratic-order action in Fourier space is expressed in the form<ce:display><ce:formula id="fm0620"><ce:label>(4.18)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si174.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="bold-italic">K</mml:mi><mml:mover accent="true"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="bold-italic">G</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="bold-italic">M</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="bold-italic">B</mml:mi><mml:mover accent="true"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:bold><ce:italic>K</ce:italic></ce:bold>, <ce:bold><ce:italic>G</ce:italic></ce:bold>, <ce:bold><ce:italic>M</ce:italic></ce:bold>, <ce:bold><ce:italic>B</ce:italic></ce:bold> are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si175.svg"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math> matrices, and<ce:display><ce:formula id="fm0630"><ce:label>(4.19)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si176.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The leading-order contributions to <ce:bold><ce:italic>M</ce:italic></ce:bold> and <ce:bold><ce:italic>B</ce:italic></ce:bold> are of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si177.svg"><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math>. Taking the small-scale limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si178.svg"><mml:mi>k</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mo>∞</mml:mo></mml:math>, nonvanishing components of the symmetric matrices <ce:bold><ce:italic>K</ce:italic></ce:bold> and <ce:bold><ce:italic>G</ce:italic></ce:bold> are<ce:display><ce:formula id="fm0640"><ce:label>(4.20)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si179.svg"><mml:msub id="mmlbr0007"><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0007" linebreakstyle="after">,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>η</mml:mi><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> and<ce:display><ce:formula id="fm0650"><ce:label>(4.21)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si180.svg"><mml:msub id="mmlbr0008"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">[</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="after">,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">[</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="after">,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:mi>η</mml:mi><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">{</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">}</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> To derive these coefficients, we have absorbed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si181.svg"><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>-dependent terms present in <ce:bold><ce:italic>B</ce:italic></ce:bold> into the components of <ce:bold><ce:italic>G</ce:italic></ce:bold> and used the relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si182.svg"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>η</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:math>, and<ce:display><ce:formula id="fm0660"><ce:label>(4.22)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si183.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>H</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>H</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>η</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>H</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>H</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>H</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>H</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The scalar ghosts are absent under the following three conditions<ce:display><ce:formula id="fm0670"><ce:label>(4.23)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si184.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0680"><ce:label>(4.24)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si185.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0690"><ce:label>(4.25)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si186.svg"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">det</mml:mi></mml:mrow><mml:mspace width="0.2em"/><mml:mi mathvariant="bold-italic">K</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Under the no-ghost condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> of tensor perturbations, inequalities <ce:cross-ref refid="fm0670" id="crf0760">(4.23)</ce:cross-ref>-<ce:cross-ref refid="fm0690" id="crf0770">(4.25)</ce:cross-ref> are satisfied for<ce:display><ce:formula id="fm0700"><ce:label>(4.26)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si187.svg"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0710"><ce:label>(4.27)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si188.svg"><mml:mrow><mml:mi>η</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0320">In the limit of large <ce:italic>k</ce:italic>, dominant contributions to the second-order action <ce:cross-ref refid="fm0620" id="crf0780">(4.18)</ce:cross-ref> arise from <ce:bold><ce:italic>K</ce:italic></ce:bold> and <ce:bold><ce:italic>G</ce:italic></ce:bold>. Then, the dispersion relation can be expressed in the form<ce:display><ce:formula id="fm0720"><ce:label>(4.28)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si189.svg"><mml:mrow><mml:mi mathvariant="normal">det</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mi mathvariant="bold-italic">K</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="bold-italic">G</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si190.svg"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math> is the scalar propagation speed. Solving Eq. <ce:cross-ref refid="fm0720" id="crf0790">(4.28)</ce:cross-ref> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, we obtain the following three solutions<ce:display><ce:formula id="fm0730"><ce:label>(4.29)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si191.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0740"><ce:label>(4.30)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si192.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0750"><ce:label>(4.31)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si193.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> which correspond to the squared propagation speeds of <ce:italic>δϕ</ce:italic>, <ce:italic>δχ</ce:italic>, and <ce:italic>δρ</ce:italic>, respectively. The scalar perturbation <ce:italic>δϕ</ce:italic> has a luminal propagation speed, so it satisfies the Laplacian stability condition. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, the matter perturbation <ce:italic>δρ</ce:italic> is free from Laplacian instability. On using the background Eq. <ce:cross-ref refid="fm0330" id="crf0800">(3.15)</ce:cross-ref>, the sound speed squared <ce:cross-ref refid="fm0740" id="crf0810">(4.30)</ce:cross-ref> can be expressed as<ce:cross-ref refid="fn0010" id="crf0820"><ce:sup>1</ce:sup></ce:cross-ref><ce:footnote id="fn0010"><ce:label>1</ce:label><ce:note-para id="np0010">If we eliminate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> by using Eq. <ce:cross-ref refid="fm0330" id="crf0830">(3.15)</ce:cross-ref>, we can express Eq. <ce:cross-ref refid="fm0740" id="crf0840">(4.30)</ce:cross-ref> in the form<ce:display><ce:formula id="fm0760"><ce:label>(4.32)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si194.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mi>η</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> From this expression, it seems that the existence of the last term can lead to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si195.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> even in the decelerating Universe. In the absence of matter (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si196.svg"><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>P</mml:mi></mml:math>), this possibility was suggested in Ref. <ce:cross-ref refid="br0930" id="crf0850">[93]</ce:cross-ref>. Eliminating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> instead of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> from Eq. <ce:cross-ref refid="fm0740" id="crf0860">(4.30)</ce:cross-ref>, it is clear that this possibility is forbidden even in the presence of matter.</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0770"><ce:label>(4.33)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si197.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>4</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub></mml:math> is the effective equation of state defined by Eq. <ce:cross-ref refid="fm0160" id="crf0870">(2.16)</ce:cross-ref>. The Laplacian stability of <ce:italic>δχ</ce:italic> is ensured for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si195.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, i.e.,<ce:display><ce:formula id="fm0780"><ce:label>(4.34)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si198.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Since we need the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> for the absence of Laplacian instability in the tensor sector, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub></mml:math> must be in the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si199.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:math>. This translates to the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si200.svg"><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>¨</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">/</mml:mo><mml:mi>a</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, so the Laplacian stability of <ce:italic>δχ</ce:italic> requires that the Universe is accelerating. In decelerating cosmological epochs, the condition <ce:cross-ref refid="fm0780" id="crf0880">(4.34)</ce:cross-ref> is always violated for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>. During the radiation-dominated (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si201.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:math>) and matter-dominated (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si202.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>) eras, we have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si203.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>4</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si204.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:math>, respectively, which are both negative for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>.</ce:para><ce:para id="pr0330">We thus showed that, for scalar-GB couplings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> containing nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math>, <ce:italic>δχ</ce:italic> is prone to the Laplacian instability during the radiation and matter eras. Hence nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math> should not be present in decelerating cosmological epochs. Even if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si205.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> is positive in the inflationary epoch, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si205.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> changes its sign during the transition to a reheating epoch (in which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si206.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mn>0</mml:mn></mml:math> for a standard reheating scenario). During the epoch of late-time cosmic acceleration, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si205.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> can be positive, but it changes the sign as we go back to the matter era. Since <ce:italic>δχ</ce:italic> is coupled to <ce:italic>δϕ</ce:italic> and <ce:italic>δρ</ce:italic>, the instability of <ce:italic>δχ</ce:italic> leads to the growth of <ce:italic>δϕ</ce:italic> and <ce:italic>δρ</ce:italic> for perturbations deep inside the Hubble radius. This violates the successful background evolution during the decelerating cosmological epochs.</ce:para><ce:para id="pr0340">The squared propagation speeds <ce:cross-ref refid="fm0730" id="crf0890">(4.29)</ce:cross-ref>-<ce:cross-ref refid="fm0750" id="crf0900">(4.31)</ce:cross-ref> have been derived by choosing the flat gauge <ce:cross-ref refid="fm0610" id="crf0910">(4.17)</ce:cross-ref>, but they are independent of the gauge choices. Indeed, we will show in Appendix <ce:cross-ref refid="se0100" id="crf1290">A</ce:cross-ref> that the same values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si207.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si205.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si208.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> can be obtained by choosing the unitary gauge. We also note that the scalar propagation speed squared <ce:cross-ref refid="fm0150" id="crf0920">(2.15)</ce:cross-ref> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity is not equivalent to the value <ce:cross-ref refid="fm0730" id="crf0930">(4.29)</ce:cross-ref>. As we observe in Eq. <ce:cross-ref refid="fm0150" id="crf0940">(2.15)</ce:cross-ref>, the propagation of <ce:italic>ϕ</ce:italic> is affected by the coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> with the linear GB term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math>. In <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> theory with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, the new scalar field <ce:italic>χ</ce:italic> plays a role of the dynamical DOF arising from the nonlinear GB term. In this latter case, the propagation of the other field <ce:italic>ϕ</ce:italic> does not practically acquire the effect of a coupling with the GB term and hence <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si209.svg"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> reduces to the luminal value.</ce:para></ce:section><ce:section id="se0080"><ce:label>4.2</ce:label><ce:section-title id="st0090"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity</ce:section-title><ce:para id="pr0350">Finally, we also study the stability of scalar perturbations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity given by the action<ce:display><ce:formula id="fm0790"><ce:label>(4.35)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si210.svg"><mml:mi mathvariant="script">S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>R</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> In this case, there is no scalar field <ce:italic>ϕ</ce:italic> coupled to the GB term. The action <ce:cross-ref refid="fm0790" id="crf0950">(4.35)</ce:cross-ref> is equivalent to Eq. <ce:cross-ref refid="fm0200" id="crf0960">(3.2)</ce:cross-ref> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si211.svg"><mml:mi>ϕ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si212.svg"><mml:mi>X</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si213.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si214.svg"><mml:mi>U</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>χ</mml:mi><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si215.svg"><mml:mi>ξ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi>χ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. As shown in Ref. <ce:cross-ref refid="br1030" id="crf0970">[103]</ce:cross-ref>, this theory belongs to a subclass of Horndeski theories with one scalar DOF <ce:italic>χ</ce:italic> besides a matter fluid.</ce:para><ce:para id="pr0360">In <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity, the second-order action of scalar perturbations is obtained by setting <ce:italic>ϕ</ce:italic>, <ce:italic>δϕ</ce:italic>, and their derivatives 0 in Eqs. <ce:cross-ref refid="fm0550" id="crf0980">(4.11)</ce:cross-ref> and <ce:cross-ref refid="fm0560" id="crf0990">(4.12)</ce:cross-ref>. We choose the flat gauge <ce:cross-ref refid="fm0610" id="crf1000">(4.17)</ce:cross-ref> and eliminate <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>v</ce:italic> from the action by using Eqs. <ce:cross-ref refid="fm0580" id="crf1010">(4.14)</ce:cross-ref>-<ce:cross-ref refid="fm0600" id="crf1020">(4.16)</ce:cross-ref>. Then, the second-order scalar action reduces to the form <ce:cross-ref refid="fm0620" id="crf1030">(4.18)</ce:cross-ref> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si216.svg"><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:mn>2</mml:mn></mml:math> matrices <ce:bold><ce:italic>K</ce:italic></ce:bold>, <ce:bold><ce:italic>G</ce:italic></ce:bold>, <ce:bold><ce:italic>M</ce:italic></ce:bold>, <ce:bold><ce:italic>B</ce:italic></ce:bold> and two dynamical perturbations<ce:display><ce:formula id="fm0800"><ce:label>(4.36)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si217.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> In the small-scale limit, nonvanishing components of <ce:bold><ce:italic>K</ce:italic></ce:bold> and <ce:bold><ce:italic>G</ce:italic></ce:bold> are given by<ce:display><ce:formula id="fm0810"><ce:label>(4.37)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si218.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0820"><ce:label>(4.38)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si219.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">[</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> The no-ghost conditions correspond to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si220.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si221.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, which are satisfied for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>. The propagation speed squared for <ce:italic>δχ</ce:italic> is<ce:display><ce:formula id="fm0830"><ce:label>(4.39)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si222.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mrow><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where, in the last equality, we used the background Eq. <ce:cross-ref refid="fm0330" id="crf1040">(3.15)</ce:cross-ref> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si134.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>. The other matter propagation speed squared is given by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si223.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>. Since the last expression of Eq. <ce:cross-ref refid="fm0830" id="crf1050">(4.39)</ce:cross-ref> is of the same form as Eq. <ce:cross-ref refid="fm0770" id="crf1060">(4.33)</ce:cross-ref>, the Laplacian instability of <ce:italic>δχ</ce:italic> is present in decelerating cosmological epochs. In Ref. <ce:cross-ref refid="br1000" id="crf1070">[100]</ce:cross-ref>, violent growth of matter perturbations was found during the radiation and matter eras for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> models of late-time cosmic acceleration. This is attributed to the Laplacian instability of <ce:italic>δχ</ce:italic> coupled to <ce:italic>δρ</ce:italic>, which inevitably occurs for nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>.</ce:para></ce:section></ce:section><ce:section id="se0090" role="conclusion"><ce:label>5</ce:label><ce:section-title id="st0100">Conclusions</ce:section-title><ce:para id="pr0370">In this letter, we studied the stability of cosmological perturbations on the spatially flat FLRW background in scalar-GB theories given by the action <ce:cross-ref refid="fm0190" id="crf1080">(3.1)</ce:cross-ref>. Provided that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>, the action <ce:cross-ref refid="fm0190" id="crf1090">(3.1)</ce:cross-ref> is equivalent to <ce:cross-ref refid="fm0200" id="crf1100">(3.2)</ce:cross-ref> with a new scalar DOF <ce:italic>χ</ce:italic> arising from nonlinear GB terms. Theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> correspond to a linear GB term coupled to a scalar field <ce:italic>ϕ</ce:italic> of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math>, which belongs to a subclass of Horndeski theories. To make a comparison with the scalar-GB coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> containing nonlinear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mi mathvariant="script">G</mml:mi></mml:math>, we first revisited stabilities of cosmological perturbations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math> gravity in Sec. <ce:cross-ref refid="se0020" id="crf1110">2</ce:cross-ref>. In this latter theory, provided that the scalar-GB coupling is subdominant to the background equations of motion, the stability conditions of tensor and scalar perturbations can be consistently satisfied.</ce:para><ce:para id="pr0380">In Sec. <ce:cross-ref refid="se0030" id="crf1120">3</ce:cross-ref>, we derived the background equations and stability conditions of tensor perturbations for the scalar-GB coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>. Besides a canonical scalar field <ce:italic>ϕ</ce:italic> with the kinetic term <ce:italic>ηX</ce:italic> and the potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, we incorporate a perfect fluid given by the Schutz-Sorkin action <ce:cross-ref refid="fm0240" id="crf1130">(3.6)</ce:cross-ref>. The absence of ghosts and Laplacian instabilities requires that the quantities <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> defined by Eqs. <ce:cross-ref refid="fm0360" id="crf1140">(3.18)</ce:cross-ref> and <ce:cross-ref refid="fm0370" id="crf1150">(3.19)</ce:cross-ref> are both positive. In terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, the background equations of motion in the gravitational sector can be expressed in a simple manner as Eqs. <ce:cross-ref refid="fm0320" id="crf1160">(3.14)</ce:cross-ref> and <ce:cross-ref refid="fm0330" id="crf1170">(3.15)</ce:cross-ref>, where the latter is used to simplify a scalar sound speed later.</ce:para><ce:para id="pr0390">In Sec. <ce:cross-ref refid="se0060" id="crf1180">4</ce:cross-ref>, we expanded the action in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math> up to quadratic order in scalar perturbations. After eliminating nondynamical variables <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, and <ce:italic>v</ce:italic>, the second-order action is of the form <ce:cross-ref refid="fm0620" id="crf1190">(4.18)</ce:cross-ref> with three dynamical perturbations <ce:cross-ref refid="fm0630" id="crf1200">(4.19)</ce:cross-ref>. With the no-ghost condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> of tensor perturbations, the scalar ghosts are absent for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>. The sound speeds of perturbations <ce:italic>δϕ</ce:italic> and <ce:italic>δρ</ce:italic> have the standard values 1 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si224.svg"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math>, respectively. However, the squared propagation speed of <ce:italic>δχ</ce:italic>, which arises from nonlinear GB functions in <ce:italic>f</ce:italic>, has a nontrivial value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si225.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:math>. Since the positivity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si205.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> requires that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mn>6</mml:mn></mml:math>, we have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si199.svg"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">eff</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:math> under the absence of Laplacian instability in the tensor sector (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>). This means that the scalar perturbation associated with nonlinearities of the GB term is subject to Laplacian instability during decelerating cosmological epochs including radiation and matter eras. The same property also holds for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>.</ce:para><ce:para id="pr0400">We thus showed that a canonical scalar field <ce:italic>ϕ</ce:italic> coupled to a nonlinear GB term does not modify the property of negative values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si205.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> in the decelerating Universe. During inflation or the epoch of late-time cosmic acceleration, it is possible to avoid Laplacian instability of the perturbation <ce:italic>δχ</ce:italic> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>. However, in the subsequent reheating period after inflation or in the preceding matter era before dark energy dominance, the Laplacian instability inevitably emerges to violate the successful background cosmological evolution. We have shown this for a canonical scalar field <ce:italic>ϕ</ce:italic>, but it may be interesting to see whether the same property persists for the scalar field <ce:italic>ϕ</ce:italic> arising in Horndeski theories and its extensions like DHOST theories <ce:cross-refs refid="br1100 br1110" id="crs0190">[110,111]</ce:cross-refs>. While we focused on the analysis on the FLRW background, it will be also of interest to study whether some instabilities are present for perturbations on a static and spherically symmetric background in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math>. The latter is important for the construction of stable hairy black hole or neutron star solutions in theories beyond the scalar-GB coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:math>. These issues are left for future works.</ce:para> </ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0150">Declaration of Competing Interest</ce:section-title><ce:para id="pr0450">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0110">Acknowledgements</ce:section-title><ce:para id="pr0410">ST is supported by the Grant-in-Aid for Scientific Research Fund of the <ce:grant-sponsor id="gsp0010" sponsor-id="https://doi.org/10.13039/501100001691">JSPS</ce:grant-sponsor> Nos. <ce:grant-number refid="gsp0010">19K03854</ce:grant-number> and <ce:grant-number refid="gsp0010">22K03642</ce:grant-number>.</ce:para></ce:acknowledgment><ce:appendices><ce:section id="se0100"><ce:label>Appendix A</ce:label><ce:section-title id="st0120">Stability conditions in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity in unitary gauge</ce:section-title><ce:para id="pr0420">In this Appendix, we derive stability conditions of scalar perturbations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity by choosing the unitary gauge<ce:display><ce:formula id="fm0840"><ce:label>(A.1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si226.svg"><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Then, the gauge-invariant perturbations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si227.svg"><mml:mi mathvariant="script">R</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>ζ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>H</mml:mi><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si228.svg"><mml:mi>δ</mml:mi><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">u</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si229.svg"><mml:mi>δ</mml:mi><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">u</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mspace width="0.2em"/><mml:mi>δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:math> reduce, respectively, to <ce:italic>ζ</ce:italic>, <ce:italic>δχ</ce:italic>, and <ce:italic>δρ</ce:italic>. After the elimination of nondynamical variables <ce:italic>α</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>v</ce:italic> from Eq. <ce:cross-ref refid="fm0540" id="crf1210">(4.10)</ce:cross-ref>, the second-order action reduces to the form <ce:cross-ref refid="fm0620" id="crf1220">(4.18)</ce:cross-ref> with the dynamical perturbations<ce:display><ce:formula id="fm0850"><ce:label>(A.2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si230.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>ζ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mi>χ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mi>ρ</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where nonvanishing matrix components of <ce:bold><ce:italic>K</ce:italic></ce:bold> and <ce:bold><ce:italic>G</ce:italic></ce:bold> are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si231.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si232.svg"><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub></mml:math>. In the short-wavelength limit, the ghosts are absent for<ce:display><ce:formula id="fm0860"><ce:label>(A.3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si184.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0870"><ce:label>(A.4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si234.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0880"><ce:label>(A.5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si235.svg"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">det</mml:mi></mml:mrow><mml:mspace width="0.2em"/><mml:mi mathvariant="bold-italic">K</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Under the tensor no-ghost condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>, inequalities <ce:cross-ref refid="fm0860" id="crf1230">(A.3)</ce:cross-ref>-<ce:cross-ref refid="fm0880" id="crf1240">(A.5)</ce:cross-ref> are satisfied for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>P</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>. These conditions are the same as those derived by choosing the flat gauge.</ce:para><ce:para id="pr0430">The scalar propagation speed squared <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> can be derived by solving the dispersion relation <ce:cross-ref refid="fm0720" id="crf1250">(4.28)</ce:cross-ref>. On using the background Eq. <ce:cross-ref refid="fm0330" id="crf1260">(3.15)</ce:cross-ref>, we obtain the three values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> exactly the same as Eqs. <ce:cross-ref refid="fm0730" id="crf1270">(4.29)</ce:cross-ref>-<ce:cross-ref refid="fm0750" id="crf1280">(4.31)</ce:cross-ref>. 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xml:lang="en"><item-info><jid>NUPHB</jid><aid>116106</aid><ce:article-number>116106</ce:article-number><ce:pii>S0550-3213(23)00035-4</ce:pii><ce:doi>10.1016/j.nuclphysb.2023.116106</ce:doi><ce:copyright year="2023" type="other">The Author(s)</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>High Energy Physics – Phenomenology</ce:text></ce:doctopic></ce:doctopics></item-info><head><ce:title id="ti0010">The formal seesaw mechanism of Majorana neutrinos with unbroken gauge symmetry</ce:title><ce:author-group id="ag0010"><ce:author id="au0010" author-id="S0550321323000354-c4240942af9bb8ef222f4164c8aa793a"><ce:given-name>Zhi-zhong</ce:given-name><ce:surname>Xing</ce:surname><ce:contributor-role role="http://credit.niso.org/contributor-roles/conceptualization">Conceptualization</ce:contributor-role><ce:contributor-role role="http://credit.niso.org/contributor-roles/investigation">Investigation</ce:contributor-role><ce:contributor-role role="http://credit.niso.org/contributor-roles/methodology">Methodology</ce:contributor-role><ce:contributor-role role="http://credit.niso.org/contributor-roles/writing-original-draft">Writing – original draft</ce:contributor-role><ce:contributor-role role="http://credit.niso.org/contributor-roles/writing-review-editing">Writing – review & editing</ce:contributor-role><ce:cross-ref refid="aff0010" id="crf0010"><ce:sup>a</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0020" id="crf0020"><ce:sup>b</ce:sup></ce:cross-ref><ce:cross-ref refid="cr0010" id="crf0030"><ce:sup>⁎</ce:sup></ce:cross-ref><ce:e-address type="email" xlink:href="mailto:xingzz@ihep.ac.cn" id="ea0010">xingzz@ihep.ac.cn</ce:e-address></ce:author><ce:affiliation id="aff0010" affiliation-id="S0550321323000354-bf3e945671ea7f562868e9da8dd114da"><ce:label>a</ce:label><ce:textfn>Institute of High Energy Physics and School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China</ce:textfn><sa:affiliation><sa:organization>Institute of High Energy Physics</sa:organization><sa:organization>School of Physical Sciences</sa:organization><sa:organization>University of Chinese Academy of Sciences</sa:organization><sa:city>Beijing</sa:city><sa:postal-code>100049</sa:postal-code><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0005">Institute of High Energy Physics and School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0020" affiliation-id="S0550321323000354-45299efea279d8f90d358ea4c3f8be3e"><ce:label>b</ce:label><ce:textfn>Center of High Energy Physics, Peking University, Beijing 100871, China</ce:textfn><sa:affiliation><sa:organization>Center of High Energy Physics</sa:organization><sa:organization>Peking University</sa:organization><sa:city>Beijing</sa:city><sa:postal-code>100871</sa:postal-code><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0010">Center of High Energy Physics, Peking University, Beijing 100871, China</ce:source-text></ce:affiliation><ce:correspondence id="cr0010"><ce:label>⁎</ce:label><ce:text>Correspondence to: Institute of High Energy Physics and School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China.</ce:text><sa:affiliation><sa:organization>Institute of High Energy Physics</sa:organization><sa:organization>School of Physical Sciences</sa:organization><sa:organization>University of Chinese Academy of Sciences</sa:organization><sa:city>Beijing</sa:city><sa:postal-code>100049</sa:postal-code><sa:country>China</sa:country></sa:affiliation></ce:correspondence></ce:author-group><ce:date-received day="28" month="1" year="2023"/><ce:date-revised day="30" month="1" year="2023"/><ce:date-accepted day="1" month="2" year="2023"/><ce:miscellaneous id="ms0010">Editor: Tommy Ohlsson</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0010">We reformulate the canonical seesaw mechanism in the case that the electroweak gauge symmetry is unbroken, and show that it can <ce:italic>formally</ce:italic> work and allow us to derive an exact seesaw formula for the light and heavy Majorana neutrinos. We elucidate the reason why there is a mismatch between the mass eigenstates of heavy Majorana neutrinos associated with thermal leptogenesis and those associated with the seesaw framework, and establish the exact and explicit relations between the <ce:italic>original</ce:italic> and <ce:italic>derivational</ce:italic> seesaw parameters by using an Euler-like parametrization of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> active-sterile flavor mixing matrix.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0170">Data availability</ce:section-title><ce:para id="pr0260">No data was used for the research described in the article.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010"><ce:label>1</ce:label><ce:section-title id="st0020">Motivation</ce:section-title><ce:para id="pr0010">Among all the proposed mechanisms toward deeply understanding the true origin of tiny masses of the three known neutrinos <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>), whose flavor eigenstates are commonly denoted as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:mi>α</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mi>μ</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi></mml:math>), the canonical seesaw mechanism <ce:cross-refs refid="br0010 br0020 br0030 br0040 br0050" id="crs0010">[1–5]</ce:cross-refs> stands out as being most economical and most natural. The simplicity of this mechanism lies in two aspects: (a) it just takes into account the right-handed neutrino fields <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math>, the chiral counterparts of the left-handed neutrino fields <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:mi>α</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mi>μ</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi></mml:math>), which were originally ignored from the particle content of the standard model (SM) <ce:cross-ref refid="br0060" id="crf0040">[6]</ce:cross-ref>; (b) it simply allows for lepton number violation or the Majorana nature of massive neutrinos <ce:cross-ref refid="br0070" id="crf0050">[7]</ce:cross-ref>, which is completely harmless to the theoretical framework of the SM itself. The naturalness of this mechanism is reflected in its attributing the small masses of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> to the existence of three heavy Majorana neutrinos <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>), whose masses are expected to be far above the fulcrum of the seesaw — presumably the electroweak symmetry breaking scale of the SM characterized by the vacuum expectation value of the Higgs field. On the other hand, the seesaw mechanism offers a big bonus to cosmology: the CP-violating and out-of-equilibrium decays of heavy Majorana neutrinos may give rise to a net lepton-antilepton asymmetry in the early Universe, and such a <ce:italic>leptogenesis</ce:italic> mechanism <ce:cross-ref refid="br0080" id="crf0060">[8]</ce:cross-ref> can finally lead to <ce:italic>baryogenesis</ce:italic> as a natural interpretation of the observed baryon-antibaryon asymmetry in today's Universe <ce:cross-ref refid="br0090" id="crf0070">[9]</ce:cross-ref>. In this sense the seesaw mechanism is the very <ce:italic>stone</ce:italic> that can kill two fundamental <ce:italic>birds</ce:italic> in particle physics and cosmology.</ce:para><ce:para id="pr0020">Note that the seesaw mechanism is expected to take effect at a superhigh energy scale Λ which is essentially of the order of the heavy Majorana neutrino masses. But the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:mi mathvariant="normal">SU</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mi mathvariant="normal">U</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Y</mml:mi></mml:mrow></mml:msub></mml:math> electroweak gauge symmetry has been unbroken until the Higgs field develops a nonzero vacuum expectation value <ce:italic>v</ce:italic> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mspace width="0.25em"/><mml:mtext>GeV</mml:mtext></mml:math>. In this situation the three active neutrinos are actually impossible to acquire their <ce:italic>true</ce:italic> masses of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> at the seesaw scale Λ due to the absence of a <ce:italic>real</ce:italic> fulcrum of the seesaw. On the other hand, thermal leptogenesis can be realized via the lepton-number-violating decays of heavy Majorana neutrinos into the leptonic and Higgs doublets at Λ. So we are well motivated to ask a conceptually important question: how can the seesaw mechanism <ce:italic>formally</ce:italic> survive with the unbroken electroweak gauge symmetry and work together with the leptogenesis mechanism? If the answer to this question is affirmative, we wonder whether the mass eigenstates of heavy Majorana neutrinos associated with thermal leptogenesis are exactly the same as those associated with the seesaw mechanism itself.<ce:cross-ref refid="fn0010" id="crf0080"><ce:sup>1</ce:sup></ce:cross-ref><ce:footnote id="fn0010"><ce:label>1</ce:label><ce:note-para id="np0010">A mismatch of this kind has been observed and discussed in the seesaw framework <ce:italic>after</ce:italic> spontaneous electroweak symmetry breaking and in an <ce:italic>approximate</ce:italic> way (see, e.g., Refs. <ce:cross-refs refid="br0100 br0110 br0120 br0130 br0140" id="crs0020">[10–14]</ce:cross-refs>). Here we shall take a new look at it <ce:italic>before</ce:italic> electroweak symmetry breaking and in an <ce:italic>exact</ce:italic> way at the tree level.</ce:note-para></ce:footnote> In case that there exists a mismatch between these two sets of mass bases, then the question becomes how small this mismatch is likely to be.</ce:para><ce:para id="pr0030">To answer the above questions and clarify some conceptual ambiguities that have never been taken seriously, we are going to study how to make the seesaw mechanism formally work before spontaneous electroweak symmetry breaking. We show that an exact seesaw relation between the light and heavy Majorana neutrinos can be established far above the electroweak scale, and it becomes the realistic seesaw relation after the Higgs field develops its vacuum expectation value. In this way it is straightforward to elucidate the reason why there is a mismatch between the mass eigenstates of heavy Majorana neutrinos associated with thermal leptogenesis and those associated with the seesaw mechanism. With the help of a full Euler-like parametrization of the flavor structure in the seesaw framework, we illuminate such a mismatch in a more specific way. The exact and explicit relations between the <ce:italic>original</ce:italic> and <ce:italic>derivational</ce:italic> parameters of massive Majorana neutrinos are obtained as a by-product, and they are expected to be useful in determining or constraining some of the original seesaw parameters from the low-energy neutrino experiments.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">A formal seesaw mechanism?</ce:section-title><ce:section id="se0030"><ce:label>2.1</ce:label><ce:section-title id="st0040">The leptonic Yukawa interactions</ce:section-title><ce:para id="pr0040">Let us begin with the gauge-invariant leptonic Yukawa interactions and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mi mathvariant="normal">SU</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math>-singlet Majorana neutrino mass term of the canonical seesaw mechanism at Λ<ce:cross-ref refid="fn0020" id="crf0090"><ce:sup>2</ce:sup></ce:cross-ref><ce:footnote id="fn0020"><ce:label>2</ce:label><ce:note-para id="np0020">Throughout this paper, our discussions are subject to the minimal extension of the SM with three right-handed neutrino fields and lepton number violation at <ce:italic>zero</ce:italic> temperature, so as to make our key point clear and avoid possible complications (e.g., thermal corrections to the masses of heavy Majorana neutrinos <ce:cross-ref refid="br0140" id="crf0100">[14]</ce:cross-ref>).</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> denotes the leptonic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mi mathvariant="normal">SU</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> doublet of the SM with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> standing respectively for the column vectors of the left-handed neutrino and charged lepton fields, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover><mml:mo>≡</mml:mo><mml:mi mathvariant="normal">i</mml:mi><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:mi>H</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mtd><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> being the Higgs doublet of the SM and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> being the second Pauli matrix, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> stand respectively for the column vectors of the right-handed charged lepton and neutrino fields which are the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mi mathvariant="normal">SU</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> singlets, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo>≡</mml:mo><mml:mi mathvariant="script">C</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:mi mathvariant="script">C</mml:mi></mml:math> being the charge-conjugation matrix and satisfying <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msup><mml:mrow><mml:mi mathvariant="script">C</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="script">C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="script">C</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="script">C</mml:mi></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> represent the respective Yukawa coupling matrices of charged leptons and neutrinos, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math> is the symmetric right-handed neutrino mass matrix. In Eq. <ce:cross-ref refid="fm0010" id="crf0110">(1)</ce:cross-ref> the hypercharges of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math>, <ce:italic>H</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover></mml:math> are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math>, −1, 0, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math>, respectively. Since <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math> is a Lorentz scalar and can be transformed into<ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msubsup><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo>≡</mml:mo><mml:mi mathvariant="script">C</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> is the charge-conjugated counterpart of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math>, one may easily rewrite Eq. <ce:cross-ref refid="fm0010" id="crf0120">(1)</ce:cross-ref> as<ce:display><ce:formula id="fm0030"><ce:label>(3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true" id="mmlbr0001"><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">]</mml:mo></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="before">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> This expression is highly nontrivial in the sense that it clearly shows a direct correlation between the left- and right-handed neutrino fields via their Yukawa couplings to the neutral component of the Higgs doublet even though the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:mi mathvariant="normal">SU</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mi mathvariant="normal">U</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Y</mml:mi></mml:mrow></mml:msub></mml:math> gauge symmetry is perfect at the seesaw scale Λ. In this situation the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math> Yukawa coupling matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> can be regarded as a “virtual” fulcrum of the seesaw before spontaneous electroweak symmetry breaking.</ce:para><ce:para id="pr0050">Note that both the scalar field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> and its charge-conjugated counterpart <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> have the mass dimension and act like two complex numbers in Eq. <ce:cross-ref refid="fm0030" id="crf0130">(3)</ce:cross-ref>. But of course they possess the respective hypercharges <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msup></mml:math> do. After spontaneous symmetry breaking <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> will acquire the same vacuum expectation value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mi>v</mml:mi><mml:mo>≃</mml:mo><mml:mn>246</mml:mn><mml:mspace width="0.25em"/><mml:mtext>GeV</mml:mtext></mml:math>, together with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, as in the SM. Then the formal seesaw will acquire a real fulcrum which allows one to naturally attribute the smallness of three active Majorana neutrino masses to the existence of three heavy Majorana neutrinos, as can be seen later on.</ce:para></ce:section><ce:section id="se0040"><ce:label>2.2</ce:label><ce:section-title id="st0050">The leptogenesis-associated basis</ce:section-title><ce:para id="pr0060">Now that all the SM particles are exactly massless in the early Universe when the temperature is far above the electroweak scale, a realization of thermal leptogenesis at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>≫</mml:mo><mml:mi>v</mml:mi></mml:math> only needs to calculate the lepton-number-violating decays of heavy Majorana neutrinos into the leptonic doublet and the Higgs doublet at the one-loop level by simply starting from Eq. <ce:cross-ref refid="fm0010" id="crf0140">(1)</ce:cross-ref> instead of Eq. <ce:cross-ref refid="fm0030" id="crf0150">(3)</ce:cross-ref> (see, e.g., Refs. <ce:cross-refs refid="br0080 br0150 br0160 br0170 br0180" id="crs0030">[8,15–18]</ce:cross-refs>). In this case the column vector of the mass eigenstates of three heavy Majorana neutrinos, denoted as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:msup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math>, can easily be obtained by making the Autonne-Takagi transformation <ce:cross-refs refid="br0190 br0200" id="crs0040">[19,20]</ce:cross-refs> as follows:<ce:display><ce:formula id="fm0040"><ce:label>(4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> is a unitary matrix, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mrow><mml:mi mathvariant="normal">Diag</mml:mi></mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">}</mml:mo></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> being the masses of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>). As a result, the Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0010" id="crf0160">(1)</ce:cross-ref> becomes<ce:display><ce:formula id="fm0050"><ce:label>(5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:math> is defined for the sake of simplicity. The rates of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> decaying into <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> and <ce:italic>H</ce:italic> or their CP-conjugated states are therefore determined by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>, so are the corresponding CP-violating asymmetries associated closely with thermal leptogenesis <ce:cross-refs refid="br0150 br0160 br0170 br0180" id="crs0050">[15–18]</ce:cross-refs>.<ce:cross-ref refid="fn0030" id="crf0170"><ce:sup>3</ce:sup></ce:cross-ref><ce:footnote id="fn0030"><ce:label>3</ce:label><ce:note-para id="np0030">Here we have used some <ce:italic>calligraphic</ce:italic> characters to denote the relevant physical quantities in the basis where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math> is diagonalized by the unitary transformation made in Eq. <ce:cross-ref refid="fm0040" id="crf0180">(4)</ce:cross-ref>. This basis is associated with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> decays and thermal leptogenesis, and it is conceptually different from the basis taken for the seesaw mechanism as can be seen below.</ce:note-para></ce:footnote> To be more specific, the flavor-dependent CP-violating asymmetries of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> decays are given by<ce:display><ce:formula id="fm0060"><ce:label>(6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mrow><mml:mtable align="axis -1" displaystyle="true" columnspacing="0.2em"><mml:mtr><mml:mtd columnalign="right"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo></mml:mtd><mml:mtd columnalign="left"><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mo>=</mml:mo></mml:mtd><mml:mtd columnalign="left"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>≠</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mi mathvariant="normal">Im</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mi>ζ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></ce:formula></ce:display> where the Latin and Greek subscripts run respectively over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mi>μ</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> are defined, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi mathvariant="normal">ln</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:mi>ζ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:math> are the loop functions. A net lepton-antilepton asymmetry can therefore result from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:math> in the early Universe, and later on it can be partly converted into a net baryon-antibaryon asymmetry via the sphaleron interactions (see Ref. <ce:cross-ref refid="br0210" id="crf0190">[21]</ce:cross-ref> for a recent review).</ce:para><ce:para id="pr0070">At this point it is worth remarking that the right-handed neutrino fields <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math> have zero weak isospin and hypercharge, and hence they have no coupling with the charged and neutral gauge bosons of the SM. As a consequence, the mass eigenstates of heavy Majorana neutrinos obtained from Eq. <ce:cross-ref refid="fm0040" id="crf0200">(4)</ce:cross-ref> do not participate in the weak charged-current interactions of the SM,<ce:display><ce:formula id="fm0070"><ce:label>(7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cc</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>g</ce:italic> denotes the weak gauge coupling constant, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> represent the first and second Pauli matrices, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msubsup></mml:math> are two of the original <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mi mathvariant="normal">SU</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> gauge fields, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msubsup><mml:mo>≡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>∓</mml:mo><mml:mi mathvariant="normal">i</mml:mi><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math> stand for the fields of the physical charged gauge bosons <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msup></mml:math>. But in the seesaw framework we shall see that the expression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cc</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0070" id="crf0210">(7)</ce:cross-ref> will get modified, and the corresponding mass eigenstates of three heavy Majorana neutrinos can definitely take part in the weak charged-current interactions.</ce:para></ce:section><ce:section id="se0050"><ce:label>2.3</ce:label><ce:section-title id="st0060">The seesaw-associated basis</ce:section-title><ce:para id="pr0080">We proceed to show that the canonical seesaw mechanism can “formally” work before spontaneous electroweak symmetry breaking but the corresponding mass eigenstates of three heavy Majorana neutrinos are not exactly the same as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>) obtained above for the neutrino decays and thermal leptogenesis. To clarify this important point, let us diagonalize the symmetric <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> matrix in Eq. <ce:cross-ref refid="fm0030" id="crf0220">(3)</ce:cross-ref> in the following Autonne-Takagi way:<ce:display><ce:formula id="fm0080"><ce:label>(8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">U</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mi mathvariant="double-struck">U</mml:mi></mml:math> is a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> unitary matrix, and the diagonal and real matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> are defined as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mrow><mml:mi mathvariant="normal">Diag</mml:mi></mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">{</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">}</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mrow><mml:mi mathvariant="normal">Diag</mml:mi></mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">{</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">}</mml:mo></mml:math>. Meanwhile, the column vectors of left- and right-handed neutrino fields <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">]</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">]</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> undergo the transformations<ce:display><ce:formula id="fm0090"><ce:label>(9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:mrow><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo stretchy="false">⟶</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo stretchy="false">⟶</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">U</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> such that the Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0030" id="crf0230">(3)</ce:cross-ref> keeps unchanged and thus its <ce:italic>overall</ce:italic> gauge symmetry is unbroken. Now that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> is dimensionless and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> has the same mass dimension as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math>, one may argue that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si79.svg"><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> should be the “working” or “virtual” mass parameters of three light Majorana neutrinos as the electroweak gauge symmetry is unbroken at the seesaw scale Λ. In comparison, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> are essentially the true masses of three heavy Majorana neutrinos in the existence of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si81.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mo>⁎</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>-mediated neutrino Yukawa interactions. Along this line of thought, we find that it is useful to decompose <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mi mathvariant="double-struck">U</mml:mi></mml:math> into the product of three matrices,<ce:display><ce:formula id="fm0100"><ce:label>(10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si82.svg"><mml:mrow><mml:mi mathvariant="double-struck">U</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>I</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>A</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>R</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mi>S</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>B</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:mi>I</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math> unitary matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> has been defined in Eq. <ce:cross-ref refid="fm0040" id="crf0240">(4)</ce:cross-ref> to primarily describe flavor mixing in the sterile (heavy) neutrino sector, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> denotes the other <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math> unitary matrix that is mainly responsible for flavor mixing in the active (light) neutrino sector, while the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math> matrices <ce:italic>A</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>R</ce:italic> and <ce:italic>S</ce:italic> signify the interplay between these two sectors <ce:cross-refs refid="br0220 br0230 br0240" id="crs0060">[22–24]</ce:cross-refs>. The unitarity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mi mathvariant="double-struck">U</mml:mi></mml:math> assures<ce:display><ce:formula id="fm0110"><ce:label>(11)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:mi id="mmlbr0002">A</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>B</mml:mi><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi><mml:mspace width="0.25em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="after">,</mml:mo><mml:mi>A</mml:mi><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mi>B</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn mathvariant="bold">0</mml:mn><mml:mspace width="0.25em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="after">,</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mi>A</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mi>B</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> On the other hand, the arbitrary charged-lepton Yukawa coupling matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0030" id="crf0250">(3)</ce:cross-ref> can be diagonalized by a bi-unitary transformation:<ce:display><ce:formula id="fm0120"><ce:label>(12)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si85.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si86.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:math> are unitary, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mrow><mml:mi mathvariant="normal">Diag</mml:mi></mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">{</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">}</mml:mo></mml:math> stands for the “working” or “virtual” masses of three charged leptons <ce:italic>before</ce:italic> spontaneous electroweak symmetry breaking,<ce:cross-ref refid="fn0040" id="crf0260"><ce:sup>4</ce:sup></ce:cross-ref><ce:footnote id="fn0040"><ce:label>4</ce:label><ce:note-para id="np0040">Note that the scalar field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> in Eq. <ce:cross-ref refid="fm0120" id="crf0270">(12)</ce:cross-ref> carries a hypercharge, and hence <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si89.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:math> cannot be simply understood as a diagonal “mass” matrix. The physical meaning of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si89.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:math> is actually vague in our calculations which are mathematically exact and clear, so is the physical meaning of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0080" id="crf0280">(8)</ce:cross-ref>. But this vagueness will automatically disappear after spontaneous electroweak symmetry breaking, as can be subsequently seen.</ce:note-para></ce:footnote> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si90.svg"><mml:msup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>e</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>μ</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>τ</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> is defined as the column vector of the mass eigenstates of three charged leptons versus the column vector of their flavor eigenstates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:mi>l</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math>. Substituting Eqs. <ce:cross-ref refid="fm0080" id="crf0290">(8)</ce:cross-ref>—<ce:cross-ref refid="fm0100" id="crf0300">(10)</ce:cross-ref> and <ce:cross-ref refid="fm0120" id="crf0310">(12)</ce:cross-ref> into Eq. <ce:cross-ref refid="fm0030" id="crf0320">(3)</ce:cross-ref>, we immediately arrive at<ce:display><ce:formula id="fm0130"><ce:label>(13)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si92.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si93.svg"><mml:msup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> denotes the column vector of the <ce:italic>working</ce:italic> mass eigenstates of three light Majorana neutrinos far above the electroweak scale, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si94.svg"><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> stands for the column vectors of the mass eigenstates of three heavy Majorana neutrinos relevant to the seesaw mechanism at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>≫</mml:mo><mml:mi>v</mml:mi></mml:math>. In this case the flavor eigenstates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math> can be expressed in terms of the mass eigenstates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> or their charge-conjugated states as follows:<ce:display><ce:formula id="fm0140"><ce:label>(14)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si97.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>U</mml:mi><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:mi>U</mml:mi><mml:mo>≡</mml:mo><mml:mi>A</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si99.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>≡</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mi>B</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si100.svg"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>≡</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mi>S</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> are defined. Taking account of the Majorana property of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg"><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> <ce:cross-ref refid="br0070" id="crf0330">[7]</ce:cross-ref> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>), one simply obtains <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si103.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si104.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math>. One may then substitute the expression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.svg"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0120" id="crf0340">(12)</ce:cross-ref> and that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0140" id="crf0350">(14)</ce:cross-ref> into the standard form of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cc</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0070" id="crf0360">(7)</ce:cross-ref> and get at<ce:display><ce:formula id="fm0150"><ce:label>(15)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si106.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cc</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>e</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>μ</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>τ</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si107.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:mi>U</mml:mi></mml:math> is just the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix <ce:cross-refs refid="br0250 br0260 br0270" id="crs0070">[25–27]</ce:cross-refs> used to describe the flavor oscillations of three active neutrinos, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si108.svg"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:mi>R</mml:mi></mml:math> is an analogue of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si109.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub></mml:math> in the seesaw mechanism which characterizes the strengths of weak charged-current interactions for three heavy Majorana neutrinos.</ce:para><ce:para id="pr0090">Without loss of generality, one may choose a convenient flavor basis in which the mass eigenstates of three charged leptons are identified with their corresponding flavor eigenstates (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si110.svg"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math>, or equivalently <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si111.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi></mml:math>). In this case we are simply left with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si112.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>U</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si113.svg"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>R</mml:mi></mml:math>, namely the effects of lepton flavor mixing originate purely from the active and sterile Majorana neutrino sectors and from the interplay between these two sectors. We shall take advantage of this flavor basis in the following discussions unless otherwise specified.</ce:para></ce:section><ce:section id="se0060"><ce:label>2.4</ce:label><ce:section-title id="st0070">Mismatch between the two bases</ce:section-title><ce:para id="pr0100">Before discussing a mismatch between the mass eigenstates of heavy Majorana neutrinos associated with thermal leptogenesis and those associated with the seesaw mechanism, let us take a look at the flavor structures of active and sterile neutrinos in the case that the electroweak gauge symmetry is unbroken at Λ. First of all, a combination of Eqs. <ce:cross-ref refid="fm0080" id="crf0370">(8)</ce:cross-ref> and <ce:cross-ref refid="fm0100" id="crf0380">(10)</ce:cross-ref> allows us to immediately derive the exact seesaw relation between the working masses of three light Majorana neutrinos and the real masses of three heavy Majorana neutrinos:<ce:display><ce:formula id="fm0160"><ce:label>(16)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si114.svg"><mml:mrow><mml:mi>U</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn mathvariant="bold">0</mml:mn><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> in which <ce:italic>U</ce:italic> and <ce:italic>R</ce:italic> are also correlated with each other via the unitarity condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si115.svg"><mml:mi>U</mml:mi><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi></mml:math>. Note that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si116.svg"><mml:mi>U</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>A</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> holds, where the unitary matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is primarily responsible for flavor mixing of the three active neutrinos. So we find it useful to rewrite Eq. <ce:cross-ref refid="fm0160" id="crf0390">(16)</ce:cross-ref> as<ce:display><ce:formula id="fm0170"><ce:label>(17)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si117.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi mathvariant="normal">i</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi mathvariant="normal">i</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> whose left- and right-hand sides are composed of the <ce:italic>derivational</ce:italic> and <ce:italic>original</ce:italic> seesaw parameters, respectively. This point will become more obvious when a complete Euler-like parametrization of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> unitary matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mi mathvariant="double-struck">U</mml:mi></mml:math> in Eq. <ce:cross-ref refid="fm0100" id="crf0400">(10)</ce:cross-ref> is adopted, as can be seen in section <ce:cross-ref refid="se0080" id="crf0410">3</ce:cross-ref>. Needless to say, the active-sterile flavor mixing matrix <ce:italic>R</ce:italic> essentially plays the role of the neutrino Yukawa coupling matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> in the canonical seesaw framework,<ce:display><ce:formula id="fm0180"><ce:label>(18)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si118.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi>I</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>;</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> and the right-handed Majorana neutrino mass matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math> can be reconstructed into the form<ce:display><ce:formula id="fm0190"><ce:label>(19)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si119.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Note that all the quantities in Eqs. <ce:cross-ref refid="fm0180" id="crf0420">(18)</ce:cross-ref> and <ce:cross-ref refid="fm0190" id="crf0430">(19)</ce:cross-ref> belong to the <ce:italic>original</ce:italic> seesaw parameters in the sense that they have nothing to do with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> — the working masses and the primary flavor mixing matrix of three light Majorana neutrinos which are <ce:italic>derived</ce:italic> from the seesaw mechanism.</ce:para><ce:para id="pr0110">Now we turn to an unavoidable mismatch between the mass eigenstates of three heavy Majorana neutrinos associated with the seesaw and leptogenesis mechanisms. Eq. <ce:cross-ref refid="fm0140" id="crf0440">(14)</ce:cross-ref> tells us that the mass eigenstates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> in the seesaw basis can be expressed as<ce:display><ce:formula id="fm0200"><ce:label>(20)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where Eq. <ce:cross-ref refid="fm0040" id="crf0450">(4)</ce:cross-ref> has been used to link <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math>. To be more explicit, Eq. <ce:cross-ref refid="fm0200" id="crf0460">(20)</ce:cross-ref> means<ce:display><ce:formula id="fm0210"><ce:label>(21)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si122.svg"><mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> from which the differences between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> in the seesaw basis and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>) in the thermal leptogenesis basis can be clearly seen. Similarly, a combination of Eqs. <ce:cross-ref refid="fm0040" id="crf0470">(4)</ce:cross-ref> and <ce:cross-ref refid="fm0190" id="crf0480">(19)</ce:cross-ref> leads us to<ce:display><ce:formula id="fm0220"><ce:label>(22)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si123.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>B</mml:mi><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> from which one may easily see the difference between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si124.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>. Although <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub></mml:math>) would exactly coincide with each other if the Yukawa coupling matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> (or equivalently, <ce:italic>R</ce:italic> or <ce:italic>S</ce:italic>) were switched off, such a coincidence would make no sense because both the seesaw and leptogenesis mechanisms would fail in this special case. In the presence of the neutrino Yukawa interactions, thermal leptogenesis may take effect via the CP-violating and out-of-equilibrium decays of heavy Majorana neutrinos into the leptonic and Higgs doublets, while the seesaw mechanism can “formally” work with the help of an interplay between the active and sterile neutrino fields coupled only to the neutral component of the Higgs doublet. That is the key reason why there is an inevitable mismatch between the seesaw- and leptogenesis-associated bases for heavy Majorana neutrinos before spontaneous electroweak symmetry breaking.</ce:para></ce:section><ce:section id="se0070"><ce:label>2.5</ce:label><ce:section-title id="st0080">After gauge symmetry breaking</ce:section-title><ce:para id="pr0120">So far we have made some proper transformations of the charged lepton and neutrino fields in the flavor space to obtain their respective working or true mass eigenstates. All such unitary flavor basis transformations are completely reversible, and hence they do not affect the gauge invariance of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub></mml:math> at the seesaw scale. As already shown in Eqs. <ce:cross-ref refid="fm0200" id="crf0490">(20)</ce:cross-ref> and <ce:cross-ref refid="fm0220" id="crf0500">(22)</ce:cross-ref>, a seeable mismatch between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> or between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si124.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> results from the fact that the working seesaw mechanism itself is only associated with the neutral component of the Higgs doublet while the heavy Majorana neutrino decays and thermal leptogenesis at the seesaw scale Λ are associated with the whole Higgs doublet. This unavoidable mismatch deserves to be conceptually clarified as we have done, because it is an intrinsic issue of the seesaw and leptogenesis mechanisms.</ce:para><ce:para id="pr0130">It is now straightforward to prove that the <ce:italic>formal</ce:italic> seesaw mechanism far above the electroweak scale will become <ce:italic>real</ce:italic> after the Higgs potential of the SM is minimized at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si126.svg"><mml:mo stretchy="false">〈</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo>≡</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math> with a special direction characterized by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si127.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si128.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math>, by which the electroweak gauge symmetry is spontaneously broken and thus all the particles coupled to the Higgs field acquire their nonzero masses. In this case the Lagrangian in Eq. <ce:cross-ref refid="fm0030" id="crf0510">(3)</ce:cross-ref> can be simplified to a more popular form,<ce:display><ce:formula id="fm0230"><ce:label>(23)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si129.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">]</mml:mo></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mi>v</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math> denotes the charged lepton mass matrix, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>v</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math> is usually referred to as the Dirac neutrino mass matrix. The expression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub></mml:math> in terms of the seesaw parameters can be directly read off from Eq. <ce:cross-ref refid="fm0180" id="crf0520">(18)</ce:cross-ref>, namely<ce:display><ce:formula id="fm0240"><ce:label>(24)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si133.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi>I</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> We find that the exact seesaw formula obtained in Eq. <ce:cross-ref refid="fm0160" id="crf0530">(16)</ce:cross-ref> and the analytical results obtained in Eqs. <ce:cross-ref refid="fm0190" id="crf0540">(19)</ce:cross-ref>—<ce:cross-ref refid="fm0220" id="crf0550">(22)</ce:cross-ref> formally keep unchanged after spontaneous gauge symmetry breaking, but they are now subject to the electroweak scale. In other words, the electroweak symmetry breaking itself does not really affect the flavor structures of the seesaw mechanism. This observation implies that it is possible to determine or constrain some of the original seesaw-associated flavor parameters in some low-energy neutrino experiments, after the radiative corrections to such parameters are properly taken into account with the help of the relevant renormaliztion-group equations (RGEs) between a superhigh seesaw scale and the electroweak scale <ce:cross-ref refid="br0280" id="crf0560">[28]</ce:cross-ref>.</ce:para><ce:para id="pr0140">Note that the exact seesaw formula obtained in Eq. <ce:cross-ref refid="fm0160" id="crf0570">(16)</ce:cross-ref> can be simplified to the more popular form in the leading-order approximations of Eqs. <ce:cross-ref refid="fm0190" id="crf0580">(19)</ce:cross-ref> and <ce:cross-ref refid="fm0240" id="crf0590">(24)</ce:cross-ref>. That is, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si134.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si135.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math>, so the <ce:italic>effective</ce:italic> mass matrix for three active Majorana neutrinos is given by<ce:display><ce:formula id="fm0250"><ce:label>(25)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si136.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:mo>≃</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"><mml:mi>A</mml:mi><mml:mo>≃</mml:mo><mml:mi>B</mml:mi><mml:mo>≃</mml:mo><mml:mi>I</mml:mi></mml:math> has been assumed (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si138.svg"><mml:mi>U</mml:mi><mml:mo>≃</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> holds in the neglect of the non-unitary effects characterized by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si139.svg"><mml:mi>A</mml:mi><mml:mo>≠</mml:mo><mml:mi>I</mml:mi></mml:math>). In this approximation the effective Majorana mass term for three active neutrinos at low energies turns out to be<ce:display><ce:formula id="fm0260"><ce:label>(26)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where the column vector of the light neutrino mass eigenstates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> has already been defined below Eq. <ce:cross-ref refid="fm0130" id="crf0600">(13)</ce:cross-ref>, and the physical meaning of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> as the diagonal Majorana neutrino mass matrix becomes definite and obvious.</ce:para></ce:section></ce:section><ce:section id="se0080"><ce:label>3</ce:label><ce:section-title id="st0090">How small is the mismatch?</ce:section-title><ce:section id="se0090"><ce:label>3.1</ce:label><ce:section-title id="st0100">An Euler-like parametrization</ce:section-title><ce:para id="pr0150">To clearly see how small the difference between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub></mml:math>) is expected to be, let us follow Refs. <ce:cross-refs refid="br0220 br0230 br0240" id="crs0080">[22–24]</ce:cross-refs> to make an Euler-like parametrization of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> unitary matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mi mathvariant="double-struck">U</mml:mi></mml:math> in Eq. <ce:cross-ref refid="fm0100" id="crf0610">(10)</ce:cross-ref>. First of all we introduce fifteen <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> Euler-like unitary matrices of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si142.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mn>6</mml:mn></mml:math>): its <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si143.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si144.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> entries are both identical to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si145.svg"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mi mathvariant="normal">cos</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> being a flavor mixing angle and lying in the first quadrant, its other four diagonal elements are all equal to one, its <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si147.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si148.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> entries are given respectively by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si149.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">i</mml:mi><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mi mathvariant="normal">sin</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">i</mml:mi><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mi mathvariant="normal">sin</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si151.svg"><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> being a CP-violating phase, and its other off-diagonal elements are all equal to zero. These matrices are then grouped in the following way to respectively describe the <ce:italic>active</ce:italic> flavor sector, the <ce:italic>sterile</ce:italic> flavor sector and the <ce:italic>interplay</ce:italic> between these two sectors:<ce:display><ce:formula id="fm0270"><ce:label>(27)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si152.svg"><mml:mrow><mml:mo stretchy="true" id="mmlbr0003">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:mi>I</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="true" linebreak="newline" indentalign="id" indenttarget="mmlbr0003" linebreakstyle="before">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>I</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>56</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>46</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>45</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="true" linebreak="newline" indentalign="id" indenttarget="mmlbr0003" linebreakstyle="before">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>A</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>R</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mi>S</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>B</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where the pattern of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is quite similar to the standard parametrization of a unitary PMNS matrix as advocated by the Particle Data Group <ce:cross-ref refid="br0090" id="crf0620">[9]</ce:cross-ref>,<ce:cross-ref refid="fn0050" id="crf0630"><ce:sup>5</ce:sup></ce:cross-ref><ce:footnote id="fn0050"><ce:label>5</ce:label><ce:note-para id="np0050">When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is applied to the phenomenology of neutrino physics in the basis of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si111.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi></mml:math>, it is the phase parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si153.svg"><mml:mi>δ</mml:mi><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub></mml:math> that characterizes the strength of CP violation in neutrino oscillations.</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0280"><ce:label>(28)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si154.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> and the expression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> can be directly read off from that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> with the subscript replacements <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si155.svg"><mml:mn>12</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>45</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si156.svg"><mml:mn>13</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>46</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si157.svg"><mml:mn>23</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>56</mml:mn></mml:math> for the three rotation angles and three CP-violating phases. The explicit expressions of <ce:italic>A</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>R</ce:italic> and <ce:italic>S</ce:italic> in terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si159.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>j</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:math>) are rather lengthy, and hence they are listed in Eqs. <ce:cross-ref refid="fm0360" id="crf0640">(A.1)</ce:cross-ref> and <ce:cross-ref refid="fm0370" id="crf0650">(A.2)</ce:cross-ref> in Appendix <ce:cross-ref refid="se0130" id="crf0660">A</ce:cross-ref> for the same of simplicity. Among the four active-sterile flavor mixing matrices, only <ce:italic>A</ce:italic> and <ce:italic>R</ce:italic> affect the physical processes in which the light and heavy Majorana neutrinos take part, as can be seen from Eq. <ce:cross-ref refid="fm0150" id="crf0670">(15)</ce:cross-ref>. As both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si116.svg"><mml:mi>U</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>A</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> and <ce:italic>R</ce:italic> appear in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cc</mml:mi></mml:mrow></mml:msub></mml:math> in the chosen flavor basis (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si111.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi></mml:math>), three of the nice CP-violating phases (or their combinations) of <ce:italic>A</ce:italic> and <ce:italic>R</ce:italic> can always be rotated away by properly redefining the phases of three charged lepton fields <ce:cross-refs refid="br0290 br0300" id="crs0090">[29,30]</ce:cross-refs>.</ce:para><ce:para id="pr0160">The PMNS matrix <ce:italic>U</ce:italic> is obviously non-unitary because of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si161.svg"><mml:mi>U</mml:mi><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>A</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo>≠</mml:mo><mml:mi>I</mml:mi></mml:math>, but its deviation from exact unitarity (i.e., from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math>) is found to be very small. A detailed and careful analysis of currently available electroweak precision measurements and neutrino oscillation data has put a stringent constraint on the non-unitarity of <ce:italic>U</ce:italic> — the latter is below or far below <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si162.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> <ce:cross-refs refid="br0310 br0320 br0330 br0340 br0350" id="crs0100">[31–35]</ce:cross-refs>. This result implies that the deviation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si163.svg"><mml:mi>A</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup></mml:math> from <ce:italic>I</ce:italic> ought to be smaller than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si162.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>, and thus the nine active-sterile flavor mixing angles in <ce:italic>R</ce:italic> should be smaller than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si164.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>. The advantage of such a phenomenological observation is that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si138.svg"><mml:mi>U</mml:mi><mml:mo>≃</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> can be a quite reliable approximation in most cases, but its disadvantage is that an experimental exploration of the seesaw-induced non-unitary effects of <ce:italic>U</ce:italic> at low energies will be rather challenging.</ce:para></ce:section><ce:section id="se0100"><ce:label>3.2</ce:label><ce:section-title id="st0110">Smallness of the mismatch</ce:section-title><ce:para id="pr0170">Eq. <ce:cross-ref refid="fm0200" id="crf0680">(20)</ce:cross-ref> tells us that a difference between the mass eigenstates of three heavy Majorana neutrinos associated with the seesaw mechanism (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math>) and those associated with thermal leptogenesis (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math>) is mainly characterized by the deviation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si165.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> from the identity matrix <ce:italic>I</ce:italic>. With the help of Eq. <ce:cross-ref refid="fm0370" id="crf0690">(A.2)</ce:cross-ref>, we arrive at<ce:display><ce:formula id="fm0290"><ce:label>(29)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si166.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="newline" indentalign="left" linebreakstyle="before" id="mmlbr0004">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0004" linebreakstyle="before">≃</mml:mo><mml:mi>I</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mspace width="0.25em"/><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/></mml:mtd><mml:mtd columnalign="center"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si167.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">i</mml:mi><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mi mathvariant="normal">tan</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> is defined, and all the terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si168.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math> or smaller have been omitted from the second equation as an excellent approximation due to the smallness of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>j</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:math>). We see that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si165.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> is also a lower triangular matrix like <ce:italic>B</ce:italic> itself. On the other hand, the factor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si169.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> appearing in Eq. <ce:cross-ref refid="fm0200" id="crf0700">(20)</ce:cross-ref> can be explicitly expressed as follows:<ce:display><ce:formula id="fm0300"><ce:label>(30)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si170.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where Eq. <ce:cross-ref refid="fm0370" id="crf0710">(A.2)</ce:cross-ref> has been used, and the terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si171.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math> or smaller have been omitted from the second equation as a very good approximation. Now we conclude that the heavy Majorana neutrino mass basis <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> is identical to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> up to the accuracy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math>, but it contains a small contribution of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si173.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math> from the light Majorana neutrino mass basis <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si174.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:math> in the seesaw framework. Since the magnitudes of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>j</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:math>) are highly suppressed in a realistic seesaw model with little fine-tuning, the mismatch between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> is expected to be negligible in most cases.</ce:para><ce:para id="pr0180">Let us proceed to examine how small the difference between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0210" id="crf0720">(21)</ce:cross-ref> can be. First of all, Eq. <ce:cross-ref refid="fm0370" id="crf0730">(A.2)</ce:cross-ref> allows us to make the approximation<ce:display><ce:formula id="fm0310"><ce:label>(31)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si175.svg"><mml:mrow><mml:mi>B</mml:mi><mml:mo>≃</mml:mo><mml:mi>I</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mspace width="0.25em"/><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.25em"/></mml:mtd><mml:mtd columnalign="center"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where the terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si168.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math> or smaller have been omitted. Secondly, we obtain<ce:display><ce:formula id="fm0320"><ce:label>(32)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si176.svg"><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo linebreak="newline" indentalign="left" linebreakstyle="before" id="mmlbr0005">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0005" linebreakstyle="before">≃</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> from Eq. <ce:cross-ref refid="fm0360" id="crf0740">(A.1)</ce:cross-ref>, where the terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si171.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math> or smaller have been neglected in the second equation as a reasonably good approximation. The exact expression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si177.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi></mml:math> can be directly read off from that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si178.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> with the help of Eq. <ce:cross-ref refid="fm0320" id="crf0750">(32)</ce:cross-ref> by making the subscript replacements <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si179.svg"><mml:mn>15</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>24</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si180.svg"><mml:mn>16</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>34</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si181.svg"><mml:mn>26</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>35</mml:mn></mml:math>, so can its approximate expression. As a consequence,<ce:display><ce:formula id="fm0330"><ce:label>(33)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si182.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo>≃</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mspace width="0.25em"/><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mspace width="0.25em"/><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.25em"/></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:math></ce:formula></ce:display> holds in the same approximation as made above. This result implies that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> are identical to each other up to the accuracy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math>, simply because on the right-hand side of Eq. <ce:cross-ref refid="fm0220" id="crf0760">(22)</ce:cross-ref> the second term is suppressed in magnitude to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si168.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math> as compared with the first term.</ce:para><ce:para id="pr0190">It is worth remarking that our above analytical approximations are more or less subject to the canonical seesaw mechanism at an energy scale far above the electroweak scale, and thus the mismatch between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub></mml:math>) is very small. This situation will change when the low-scale seesaw and leptogenesis scenarios, in which a mismatch between the two sets of mass bases for heavy Majorana neutrinos is crucial, are taken into account (see, e.g., Refs. <ce:cross-refs refid="br0110 br0120" id="crs0110">[11,12]</ce:cross-refs>).</ce:para></ce:section><ce:section id="se0110"><ce:label>3.3</ce:label><ce:section-title id="st0120">Determination of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></ce:section-title><ce:para id="pr0200">As already shown in Eq. <ce:cross-ref refid="fm0170" id="crf0770">(17)</ce:cross-ref>, the nine effective flavor parameters of three light Majorana neutrinos in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> (i.e., three effective masses, three flavor mixing angles and three CP-violating phases) can be expressed in terms of the eighteen seesaw parameters hidden in <ce:italic>A</ce:italic>, <ce:italic>R</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> (i.e., three heavy Majorana neutrino masses, nine active-sterile flavor mixing angles and six CP-violating phases). It is obvious that all the derivational seesaw parameters on the left-hand side of Eq. <ce:cross-ref refid="fm0170" id="crf0780">(17)</ce:cross-ref> would vanish if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si183.svg"><mml:mi>R</mml:mi><mml:mo>∝</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> were switched off. So this equation provides an unambiguous way to determine the light degrees of freedom from the heavy degrees of freedom in the seesaw framework.</ce:para><ce:para id="pr0210">To be more specific, the six independent elements of the effective Majorana neutrino mass matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si184.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup></mml:math> are given as follows:<ce:display><ce:formula id="fm0340"><ce:label>(34)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si185.svg"><mml:msub id="mmlbr0006"><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub id="mmlbr0007"><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0006" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0007" linebreakstyle="before">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0006" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0007" linebreakstyle="before">−</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0006" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0006" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0007" linebreakstyle="before">−</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0006" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> On the other hand, Eq. <ce:cross-ref refid="fm0170" id="crf0790">(17)</ce:cross-ref> tells us that these six matrix elements can originally be determined by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si186.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> thanks to the exact seesaw relation bridging the big gap between the light and heavy Majorana neutrinos. With the help of the explicit expression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si187.svg"><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi></mml:math> given in Eq. <ce:cross-ref refid="fm0320" id="crf0800">(32)</ce:cross-ref>, it is straightforward to obtain the expressions for the elements of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si188.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> in terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si151.svg"><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>j</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:math>). Instead of presenting the exact analytical results, which are rather lengthy and hence less instructive, here we make the leading-order approximations for the expressions of <ce:italic>A</ce:italic> and <ce:italic>R</ce:italic> given in Eq. <ce:cross-ref refid="fm0360" id="crf0810">(A.1)</ce:cross-ref> and then arrive at<ce:display><ce:formula id="fm0350"><ce:label>(35)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si189.svg"><mml:msub id="mmlbr0008"><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Let us emphasize that there appear nine CP-violating phases in Eq. <ce:cross-ref refid="fm0350" id="crf0820">(35)</ce:cross-ref>, but three of them (or their combinations) are redundant and can always be removed by rephasing the charged lepton fields in a proper way.<ce:cross-ref refid="fn0060" id="crf0830"><ce:sup>6</ce:sup></ce:cross-ref><ce:footnote id="fn0060"><ce:label>6</ce:label><ce:note-para id="np0060">A straightforward way to remove the three redundant phase parameters of <ce:italic>A</ce:italic> and <ce:italic>R</ce:italic> is just to switch off three of the nine phases in the nine active-sterile flavor mixing matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>j</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:math>) in Eq. <ce:cross-ref refid="fm0270" id="crf0840">(27)</ce:cross-ref> from the very beginning. As there are many options in doing so, we do not go into details here.</ce:note-para></ce:footnote> A combination of Eqs. <ce:cross-ref refid="fm0340" id="crf0850">(34)</ce:cross-ref> and <ce:cross-ref refid="fm0350" id="crf0860">(35)</ce:cross-ref> allows us to establish the direct relations between the nine derivational and eighteen original seesaw parameters. So the former can in principle be determined from the latter for a given seesaw model (a top-down approach), and the latter may be partly probed or constrained from the former with the help of some low-energy neutrino experiments (a bottom-up approach). A careful and detailed analysis of the parameter space along this line of thought will be made elsewhere.</ce:para></ce:section></ce:section><ce:section id="se0120"><ce:label>4</ce:label><ce:section-title id="st0130">Summary</ce:section-title><ce:para id="pr0220">We have reformulated the canonical seesaw mechanism by considering the fact that the electroweak gauge symmetry is unbroken at the seesaw scale characterized by the masses of heavy Majorana neutrinos, and shown that it can <ce:italic>formally</ce:italic> work and allow us to derive an exact seesaw relation between the active (light) and sterile (heavy) Majorana neutrinos. In this way we have elucidated the reason why there is an unavoidable mismatch between the mass eigenstates of heavy Majorana neutrinos associated with the seesaw and thermal leptogenesis mechanisms. The smallness of this mismatch has been discussed with the help of a complete Euler-like parametrization of the flavor structure in the seesaw framework, and the exact and explicit relations between the <ce:italic>original</ce:italic> and <ce:italic>derivational</ce:italic> seesaw parameters have been established as a by-product.</ce:para><ce:para id="pr0230">We hope that this work may help clarify some conceptual ambiguities associated with the validity of the seesaw mechanism before and after spontaneous electroweak symmetry breaking, because such ambiguities have never been taken serious in the literature. It should also be helpful to clarify the ambiguities associated with the RGE evolution between the “virtual” flavor parameters of Majorana neutrinos at the seesaw scale and those “real” ones at the electroweak scale, which is crucial to bridge the gap between a well-motivated UV-complete flavor theory including the seesaw mechanism and all the possible low-energy flavor experiments.</ce:para> </ce:section><ce:section id="se0140"><ce:section-title id="st0180">CRediT authorship contribution statement</ce:section-title><ce:para id="pr0270"><ce:bold>Zhi-zhong Xing:</ce:bold> Conceptualization, Investigation, Methodology, Writing – original draft, Writing – review & editing.</ce:para></ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0190">Declaration of Competing Interest</ce:section-title><ce:para id="pr0280">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0140">Acknowledgements</ce:section-title><ce:para id="pr0240">I am greatly indebted to Di Zhang and Shun Zhou for numerous helpful discussions and comments about this paper, a work dedicated to the 50th birthday of my home institute, the Institute of High Energy Physics, which was founded on 1 February 1973. I would also like to thank Marco Drewes for very useful discussions and comments. My research is supported in part by the <ce:grant-sponsor id="gsp0010" sponsor-id="https://doi.org/10.13039/501100001809">National Natural Science Foundation of China</ce:grant-sponsor> under grant No. <ce:grant-number refid="gsp0010">12075254</ce:grant-number> and grant No. <ce:grant-number refid="gsp0010">11835013</ce:grant-number>.</ce:para></ce:acknowledgment><ce:appendices><ce:section id="se0130"><ce:label>Appendix A</ce:label><ce:section-title id="st0150">The expressions of <ce:italic>A</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>R</ce:italic> and <ce:italic>S</ce:italic></ce:section-title><ce:para id="pr0250">Given the Euler-like parametrization of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> unitary flavor mixing matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mi mathvariant="double-struck">U</mml:mi></mml:math> decomposed in Eq. <ce:cross-ref refid="fm0270" id="crf0870">(27)</ce:cross-ref>, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math> active-sterile flavor mixing matrices <ce:italic>A</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>R</ce:italic> and <ce:italic>S</ce:italic> depend on the same nine rotation angles <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> and nine phase angles <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si151.svg"><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>j</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:math>). To be explicit <ce:cross-refs refid="br0220 br0230" id="crs0120">[22,23]</ce:cross-refs>,<ce:display><ce:formula id="fm0360"><ce:label>(A.1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si190.svg"><mml:mi id="mmlbr0009">A</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd 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stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> We see that both <ce:italic>A</ce:italic> and <ce:italic>B</ce:italic> are the lower triangular matrices, and the expression of <ce:italic>B</ce:italic> can be read off from that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si192.svg"><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> with the subscript replacements <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si179.svg"><mml:mn>15</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>24</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si180.svg"><mml:mn>16</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>34</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si181.svg"><mml:mn>26</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>35</mml:mn></mml:math>. The expression of <ce:italic>S</ce:italic> can be similarly obtained from that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si193.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> with the same subscript replacements <ce:cross-ref refid="br0240" id="crf0880">[24]</ce:cross-ref>. Note, however, that <ce:italic>B</ce:italic> and <ce:italic>S</ce:italic> do not affect any physical processes in the seesaw mechanism.</ce:para></ce:section></ce:appendices></body><tail><ce:bibliography id="bl0010"><ce:section-title id="st0160">References</ce:section-title><ce:bibliography-sec id="bs0010"><ce:bib-reference id="br0010"><ce:label>[1]</ce:label><sb:reference id="bibADD2B090BFBF49417BB3AB7A01295D54s1"><sb:contribution><sb:authors><sb:author><ce:given-name>P.</ce:given-name><ce:surname>Minkowski</ce:surname></sb:author></sb:authors><sb:title><sb:maintitle><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si194.svg"><mml:mi>μ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>e</mml:mi><mml:mi>γ</mml:mi></mml:math> at a rate of one out of 10<ce:sup>9</ce:sup> muon decays?</sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Phys. Lett. B</sb:maintitle></sb:title><sb:volume-nr>67</sb:volume-nr></sb:series><sb:date>1977</sb:date></sb:issue><sb:pages><sb:first-page>421</sb:first-page></sb:pages></sb:host></sb:reference><ce:source-text id="srct0015">P. Minkowski, “μ→eγ at a rate of one out of 109 muon decays?” Phys. Lett. 67B (1977) 421.</ce:source-text></ce:bib-reference><ce:bib-reference id="br0020"><ce:label>[2]</ce:label><sb:reference id="bib5BF9E2DBBCA42A64CDAE66A10E170751s1"><sb:contribution><sb:authors><sb:author><ce:given-name>T.</ce:given-name><ce:surname>Yanagida</ce:surname></sb:author></sb:authors><sb:title><sb:maintitle>Horizontal gauge symmetry and masses of neutrinos</sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Conf. Proc. C</sb:maintitle></sb:title><sb:volume-nr>7902131</sb:volume-nr></sb:series><sb:date>1979</sb:date></sb:issue><sb:pages><sb:first-page>95</sb:first-page></sb:pages></sb:host></sb:reference><ce:source-text id="srct0020">T. Yanagida, “Horizontal gauge symmetry and masses of neutrinos,” Conf. Proc. C 7902131 (1979) 95.</ce:source-text></ce:bib-reference><ce:bib-reference id="br0030"><ce:label>[3]</ce:label><sb:reference id="bib882BADE8E896CBC1DEA7386BD15AA05Es1"><sb:contribution><sb:authors><sb:author><ce:given-name>M.</ce:given-name><ce:surname>Gell-Mann</ce:surname></sb:author><sb:author><ce:given-name>P.</ce:given-name><ce:surname>Ramond</ce:surname></sb:author><sb:author><ce:given-name>R.</ce:given-name><ce:surname>Slansky</ce:surname></sb:author></sb:authors><sb:title><sb:maintitle>Complex spinors and unified theories</sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Conf. Proc. 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B 824 (2022), 136797 [arXiv:2109.13622 [hep-ph]].</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> \ No newline at end of file +<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd"><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="fla" xml:lang="en"><item-info><jid>NUPHB</jid><aid>116106</aid><ce:article-number>116106</ce:article-number><ce:pii>S0550-3213(23)00035-4</ce:pii><ce:doi>10.1016/j.nuclphysb.2023.116106</ce:doi><ce:copyright year="2023" type="other">The Author(s)</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>High Energy Physics – Phenomenology</ce:text></ce:doctopic></ce:doctopics></item-info><head><ce:title id="ti0010">The formal seesaw mechanism of Majorana neutrinos with unbroken gauge symmetry</ce:title><ce:author-group id="ag0010"><ce:author id="au0010" author-id="S0550321323000354-c4240942af9bb8ef222f4164c8aa793a"><ce:given-name>Zhi-zhong</ce:given-name><ce:surname>Xing</ce:surname><ce:contributor-role role="http://credit.niso.org/contributor-roles/conceptualization">Conceptualization</ce:contributor-role><ce:contributor-role role="http://credit.niso.org/contributor-roles/investigation">Investigation</ce:contributor-role><ce:contributor-role role="http://credit.niso.org/contributor-roles/methodology">Methodology</ce:contributor-role><ce:contributor-role role="http://credit.niso.org/contributor-roles/writing-original-draft">Writing – original draft</ce:contributor-role><ce:contributor-role role="http://credit.niso.org/contributor-roles/writing-review-editing">Writing – review & editing</ce:contributor-role><ce:cross-ref refid="aff0010" id="crf0010"><ce:sup>a</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0020" id="crf0020"><ce:sup>b</ce:sup></ce:cross-ref><ce:cross-ref refid="cr0010" id="crf0030"><ce:sup>⁎</ce:sup></ce:cross-ref><ce:e-address type="email" xlink:href="mailto:xingzz@ihep.ac.cn" id="ea0010">xingzz@ihep.ac.cn</ce:e-address></ce:author><ce:affiliation id="aff0010" affiliation-id="S0550321323000354-bf3e945671ea7f562868e9da8dd114da"><ce:label>a</ce:label><ce:textfn>Institute of High Energy Physics and School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China</ce:textfn><sa:affiliation><sa:organization>Institute of High Energy Physics</sa:organization><sa:organization>School of Physical Sciences</sa:organization><sa:organization>University of Chinese Academy of Sciences</sa:organization><sa:city>Beijing</sa:city><sa:postal-code>100049</sa:postal-code><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0005">Institute of High Energy Physics and School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0020" affiliation-id="S0550321323000354-45299efea279d8f90d358ea4c3f8be3e"><ce:label>b</ce:label><ce:textfn>Center of High Energy Physics, Peking University, Beijing 100871, China</ce:textfn><sa:affiliation><sa:organization>Center of High Energy Physics</sa:organization><sa:organization>Peking University</sa:organization><sa:city>Beijing</sa:city><sa:postal-code>100871</sa:postal-code><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0010">Center of High Energy Physics, Peking University, Beijing 100871, China</ce:source-text></ce:affiliation><ce:correspondence id="cr0010"><ce:label>⁎</ce:label><ce:text>Correspondence to: Institute of High Energy Physics and School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China.</ce:text><sa:affiliation><sa:organization>Institute of High Energy Physics</sa:organization><sa:organization>School of Physical Sciences</sa:organization><sa:organization>University of Chinese Academy of Sciences</sa:organization><sa:city>Beijing</sa:city><sa:postal-code>100049</sa:postal-code><sa:country>China</sa:country></sa:affiliation></ce:correspondence></ce:author-group><ce:date-received day="28" month="1" year="2023"/><ce:date-revised day="30" month="1" year="2023"/><ce:date-accepted day="1" month="2" year="2023"/><ce:miscellaneous id="ms0010">Editor: Tommy Ohlsson</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0010">We reformulate the canonical seesaw mechanism in the case that the electroweak gauge symmetry is unbroken, and show that it can <ce:italic>formally</ce:italic> work and allow us to derive an exact seesaw formula for the light and heavy Majorana neutrinos. We elucidate the reason why there is a mismatch between the mass eigenstates of heavy Majorana neutrinos associated with thermal leptogenesis and those associated with the seesaw framework, and establish the exact and explicit relations between the <ce:italic>original</ce:italic> and <ce:italic>derivational</ce:italic> seesaw parameters by using an Euler-like parametrization of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> active-sterile flavor mixing matrix.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0170">Data availability</ce:section-title><ce:para id="pr0260">No data was used for the research described in the article.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010"><ce:label>1</ce:label><ce:section-title id="st0020">Motivation</ce:section-title><ce:para id="pr0010">Among all the proposed mechanisms toward deeply understanding the true origin of tiny masses of the three known neutrinos <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>), whose flavor eigenstates are commonly denoted as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:mi>α</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mi>μ</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi></mml:math>), the canonical seesaw mechanism <ce:cross-refs refid="br0010 br0020 br0030 br0040 br0050" id="crs0010">[1–5]</ce:cross-refs> stands out as being most economical and most natural. The simplicity of this mechanism lies in two aspects: (a) it just takes into account the right-handed neutrino fields <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math>, the chiral counterparts of the left-handed neutrino fields <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:mi>α</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mi>μ</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi></mml:math>), which were originally ignored from the particle content of the standard model (SM) <ce:cross-ref refid="br0060" id="crf0040">[6]</ce:cross-ref>; (b) it simply allows for lepton number violation or the Majorana nature of massive neutrinos <ce:cross-ref refid="br0070" id="crf0050">[7]</ce:cross-ref>, which is completely harmless to the theoretical framework of the SM itself. The naturalness of this mechanism is reflected in its attributing the small masses of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> to the existence of three heavy Majorana neutrinos <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>), whose masses are expected to be far above the fulcrum of the seesaw — presumably the electroweak symmetry breaking scale of the SM characterized by the vacuum expectation value of the Higgs field. On the other hand, the seesaw mechanism offers a big bonus to cosmology: the CP-violating and out-of-equilibrium decays of heavy Majorana neutrinos may give rise to a net lepton-antilepton asymmetry in the early Universe, and such a <ce:italic>leptogenesis</ce:italic> mechanism <ce:cross-ref refid="br0080" id="crf0060">[8]</ce:cross-ref> can finally lead to <ce:italic>baryogenesis</ce:italic> as a natural interpretation of the observed baryon-antibaryon asymmetry in today's Universe <ce:cross-ref refid="br0090" id="crf0070">[9]</ce:cross-ref>. In this sense the seesaw mechanism is the very <ce:italic>stone</ce:italic> that can kill two fundamental <ce:italic>birds</ce:italic> in particle physics and cosmology.</ce:para><ce:para id="pr0020">Note that the seesaw mechanism is expected to take effect at a superhigh energy scale Λ which is essentially of the order of the heavy Majorana neutrino masses. But the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:mi mathvariant="normal">SU</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mi mathvariant="normal">U</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Y</mml:mi></mml:mrow></mml:msub></mml:math> electroweak gauge symmetry has been unbroken until the Higgs field develops a nonzero vacuum expectation value <ce:italic>v</ce:italic> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mspace width="0.25em"/><mml:mtext>GeV</mml:mtext></mml:math>. In this situation the three active neutrinos are actually impossible to acquire their <ce:italic>true</ce:italic> masses of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> at the seesaw scale Λ due to the absence of a <ce:italic>real</ce:italic> fulcrum of the seesaw. On the other hand, thermal leptogenesis can be realized via the lepton-number-violating decays of heavy Majorana neutrinos into the leptonic and Higgs doublets at Λ. So we are well motivated to ask a conceptually important question: how can the seesaw mechanism <ce:italic>formally</ce:italic> survive with the unbroken electroweak gauge symmetry and work together with the leptogenesis mechanism? If the answer to this question is affirmative, we wonder whether the mass eigenstates of heavy Majorana neutrinos associated with thermal leptogenesis are exactly the same as those associated with the seesaw mechanism itself.<ce:cross-ref refid="fn0010" id="crf0080"><ce:sup>1</ce:sup></ce:cross-ref><ce:footnote id="fn0010"><ce:label>1</ce:label><ce:note-para id="np0010">A mismatch of this kind has been observed and discussed in the seesaw framework <ce:italic>after</ce:italic> spontaneous electroweak symmetry breaking and in an <ce:italic>approximate</ce:italic> way (see, e.g., Refs. <ce:cross-refs refid="br0100 br0110 br0120 br0130 br0140" id="crs0020">[10–14]</ce:cross-refs>). Here we shall take a new look at it <ce:italic>before</ce:italic> electroweak symmetry breaking and in an <ce:italic>exact</ce:italic> way at the tree level.</ce:note-para></ce:footnote> In case that there exists a mismatch between these two sets of mass bases, then the question becomes how small this mismatch is likely to be.</ce:para><ce:para id="pr0030">To answer the above questions and clarify some conceptual ambiguities that have never been taken seriously, we are going to study how to make the seesaw mechanism formally work before spontaneous electroweak symmetry breaking. We show that an exact seesaw relation between the light and heavy Majorana neutrinos can be established far above the electroweak scale, and it becomes the realistic seesaw relation after the Higgs field develops its vacuum expectation value. In this way it is straightforward to elucidate the reason why there is a mismatch between the mass eigenstates of heavy Majorana neutrinos associated with thermal leptogenesis and those associated with the seesaw mechanism. With the help of a full Euler-like parametrization of the flavor structure in the seesaw framework, we illuminate such a mismatch in a more specific way. The exact and explicit relations between the <ce:italic>original</ce:italic> and <ce:italic>derivational</ce:italic> parameters of massive Majorana neutrinos are obtained as a by-product, and they are expected to be useful in determining or constraining some of the original seesaw parameters from the low-energy neutrino experiments.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">A formal seesaw mechanism?</ce:section-title><ce:section id="se0030"><ce:label>2.1</ce:label><ce:section-title id="st0040">The leptonic Yukawa interactions</ce:section-title><ce:para id="pr0040">Let us begin with the gauge-invariant leptonic Yukawa interactions and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mi mathvariant="normal">SU</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math>-singlet Majorana neutrino mass term of the canonical seesaw mechanism at Λ<ce:cross-ref refid="fn0020" id="crf0090"><ce:sup>2</ce:sup></ce:cross-ref><ce:footnote id="fn0020"><ce:label>2</ce:label><ce:note-para id="np0020">Throughout this paper, our discussions are subject to the minimal extension of the SM with three right-handed neutrino fields and lepton number violation at <ce:italic>zero</ce:italic> temperature, so as to make our key point clear and avoid possible complications (e.g., thermal corrections to the masses of heavy Majorana neutrinos <ce:cross-ref refid="br0140" id="crf0100">[14]</ce:cross-ref>).</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> denotes the leptonic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mi mathvariant="normal">SU</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> doublet of the SM with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> standing respectively for the column vectors of the left-handed neutrino and charged lepton fields, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover><mml:mo>≡</mml:mo><mml:mi mathvariant="normal">i</mml:mi><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:mi>H</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mtd><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> being the Higgs doublet of the SM and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> being the second Pauli matrix, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> stand respectively for the column vectors of the right-handed charged lepton and neutrino fields which are the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mi mathvariant="normal">SU</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> singlets, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo>≡</mml:mo><mml:mi mathvariant="script">C</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:mi mathvariant="script">C</mml:mi></mml:math> being the charge-conjugation matrix and satisfying <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msup><mml:mrow><mml:mi mathvariant="script">C</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="script">C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="script">C</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="script">C</mml:mi></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> represent the respective Yukawa coupling matrices of charged leptons and neutrinos, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math> is the symmetric right-handed neutrino mass matrix. In Eq. <ce:cross-ref refid="fm0010" id="crf0110">(1)</ce:cross-ref> the hypercharges of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math>, <ce:italic>H</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover></mml:math> are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math>, −1, 0, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math>, respectively. Since <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math> is a Lorentz scalar and can be transformed into<ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msubsup><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo>≡</mml:mo><mml:mi mathvariant="script">C</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> is the charge-conjugated counterpart of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math>, one may easily rewrite Eq. <ce:cross-ref refid="fm0010" id="crf0120">(1)</ce:cross-ref> as<ce:display><ce:formula id="fm0030"><ce:label>(3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true" id="mmlbr0001"><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">]</mml:mo></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="before">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> This expression is highly nontrivial in the sense that it clearly shows a direct correlation between the left- and right-handed neutrino fields via their Yukawa couplings to the neutral component of the Higgs doublet even though the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:mi mathvariant="normal">SU</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mi mathvariant="normal">U</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Y</mml:mi></mml:mrow></mml:msub></mml:math> gauge symmetry is perfect at the seesaw scale Λ. In this situation the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math> Yukawa coupling matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> can be regarded as a “virtual” fulcrum of the seesaw before spontaneous electroweak symmetry breaking.</ce:para><ce:para id="pr0050">Note that both the scalar field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> and its charge-conjugated counterpart <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> have the mass dimension and act like two complex numbers in Eq. <ce:cross-ref refid="fm0030" id="crf0130">(3)</ce:cross-ref>. But of course they possess the respective hypercharges <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msup></mml:math> do. After spontaneous symmetry breaking <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> will acquire the same vacuum expectation value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mi>v</mml:mi><mml:mo>≃</mml:mo><mml:mn>246</mml:mn><mml:mspace width="0.25em"/><mml:mtext>GeV</mml:mtext></mml:math>, together with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, as in the SM. Then the formal seesaw will acquire a real fulcrum which allows one to naturally attribute the smallness of three active Majorana neutrino masses to the existence of three heavy Majorana neutrinos, as can be seen later on.</ce:para></ce:section><ce:section id="se0040"><ce:label>2.2</ce:label><ce:section-title id="st0050">The leptogenesis-associated basis</ce:section-title><ce:para id="pr0060">Now that all the SM particles are exactly massless in the early Universe when the temperature is far above the electroweak scale, a realization of thermal leptogenesis at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>≫</mml:mo><mml:mi>v</mml:mi></mml:math> only needs to calculate the lepton-number-violating decays of heavy Majorana neutrinos into the leptonic doublet and the Higgs doublet at the one-loop level by simply starting from Eq. <ce:cross-ref refid="fm0010" id="crf0140">(1)</ce:cross-ref> instead of Eq. <ce:cross-ref refid="fm0030" id="crf0150">(3)</ce:cross-ref> (see, e.g., Refs. <ce:cross-refs refid="br0080 br0150 br0160 br0170 br0180" id="crs0030">[8,15–18]</ce:cross-refs>). In this case the column vector of the mass eigenstates of three heavy Majorana neutrinos, denoted as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:msup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math>, can easily be obtained by making the Autonne-Takagi transformation <ce:cross-refs refid="br0190 br0200" id="crs0040">[19,20]</ce:cross-refs> as follows:<ce:display><ce:formula id="fm0040"><ce:label>(4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> is a unitary matrix, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mrow><mml:mi mathvariant="normal">Diag</mml:mi></mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">}</mml:mo></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> being the masses of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>). As a result, the Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0010" id="crf0160">(1)</ce:cross-ref> becomes<ce:display><ce:formula id="fm0050"><ce:label>(5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:math> is defined for the sake of simplicity. The rates of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> decaying into <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> and <ce:italic>H</ce:italic> or their CP-conjugated states are therefore determined by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>, so are the corresponding CP-violating asymmetries associated closely with thermal leptogenesis <ce:cross-refs refid="br0150 br0160 br0170 br0180" id="crs0050">[15–18]</ce:cross-refs>.<ce:cross-ref refid="fn0030" id="crf0170"><ce:sup>3</ce:sup></ce:cross-ref><ce:footnote id="fn0030"><ce:label>3</ce:label><ce:note-para id="np0030">Here we have used some <ce:italic>calligraphic</ce:italic> characters to denote the relevant physical quantities in the basis where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math> is diagonalized by the unitary transformation made in Eq. <ce:cross-ref refid="fm0040" id="crf0180">(4)</ce:cross-ref>. This basis is associated with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> decays and thermal leptogenesis, and it is conceptually different from the basis taken for the seesaw mechanism as can be seen below.</ce:note-para></ce:footnote> To be more specific, the flavor-dependent CP-violating asymmetries of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> decays are given by<ce:display><ce:formula id="fm0060"><ce:label>(6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mrow><mml:mtable align="axis -1" displaystyle="true" columnspacing="0.2em"><mml:mtr><mml:mtd columnalign="right"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo></mml:mtd><mml:mtd columnalign="left"><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mo>=</mml:mo></mml:mtd><mml:mtd columnalign="left"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>≠</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mi mathvariant="normal">Im</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi mathvariant="script">Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mi>ζ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></ce:formula></ce:display> where the Latin and Greek subscripts run respectively over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mi>μ</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> are defined, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:mi>ξ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi mathvariant="normal">ln</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:mi>ζ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:math> are the loop functions. A net lepton-antilepton asymmetry can therefore result from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:math> in the early Universe, and later on it can be partly converted into a net baryon-antibaryon asymmetry via the sphaleron interactions (see Ref. <ce:cross-ref refid="br0210" id="crf0190">[21]</ce:cross-ref> for a recent review).</ce:para><ce:para id="pr0070">At this point it is worth remarking that the right-handed neutrino fields <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math> have zero weak isospin and hypercharge, and hence they have no coupling with the charged and neutral gauge bosons of the SM. As a consequence, the mass eigenstates of heavy Majorana neutrinos obtained from Eq. <ce:cross-ref refid="fm0040" id="crf0200">(4)</ce:cross-ref> do not participate in the weak charged-current interactions of the SM,<ce:display><ce:formula id="fm0070"><ce:label>(7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cc</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>g</ce:italic> denotes the weak gauge coupling constant, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> represent the first and second Pauli matrices, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msubsup></mml:math> are two of the original <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mi mathvariant="normal">SU</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> gauge fields, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msubsup><mml:mo>≡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>∓</mml:mo><mml:mi mathvariant="normal">i</mml:mi><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math> stand for the fields of the physical charged gauge bosons <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msup></mml:math>. But in the seesaw framework we shall see that the expression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cc</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0070" id="crf0210">(7)</ce:cross-ref> will get modified, and the corresponding mass eigenstates of three heavy Majorana neutrinos can definitely take part in the weak charged-current interactions.</ce:para></ce:section><ce:section id="se0050"><ce:label>2.3</ce:label><ce:section-title id="st0060">The seesaw-associated basis</ce:section-title><ce:para id="pr0080">We proceed to show that the canonical seesaw mechanism can “formally” work before spontaneous electroweak symmetry breaking but the corresponding mass eigenstates of three heavy Majorana neutrinos are not exactly the same as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>) obtained above for the neutrino decays and thermal leptogenesis. To clarify this important point, let us diagonalize the symmetric <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> matrix in Eq. <ce:cross-ref refid="fm0030" id="crf0220">(3)</ce:cross-ref> in the following Autonne-Takagi way:<ce:display><ce:formula id="fm0080"><ce:label>(8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">U</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mi mathvariant="double-struck">U</mml:mi></mml:math> is a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> unitary matrix, and the diagonal and real matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> are defined as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mrow><mml:mi mathvariant="normal">Diag</mml:mi></mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">{</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">}</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mrow><mml:mi mathvariant="normal">Diag</mml:mi></mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">{</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">}</mml:mo></mml:math>. Meanwhile, the column vectors of left- and right-handed neutrino fields <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">]</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">]</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> undergo the transformations<ce:display><ce:formula id="fm0090"><ce:label>(9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:mrow><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo stretchy="false">⟶</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo stretchy="false">⟶</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">U</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> such that the Lagrangian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0030" id="crf0230">(3)</ce:cross-ref> keeps unchanged and thus its <ce:italic>overall</ce:italic> gauge symmetry is unbroken. Now that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> is dimensionless and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> has the same mass dimension as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math>, one may argue that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si79.svg"><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> should be the “working” or “virtual” mass parameters of three light Majorana neutrinos as the electroweak gauge symmetry is unbroken at the seesaw scale Λ. In comparison, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> are essentially the true masses of three heavy Majorana neutrinos in the existence of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si81.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mo>⁎</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>-mediated neutrino Yukawa interactions. Along this line of thought, we find that it is useful to decompose <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mi mathvariant="double-struck">U</mml:mi></mml:math> into the product of three matrices,<ce:display><ce:formula id="fm0100"><ce:label>(10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si82.svg"><mml:mrow><mml:mi mathvariant="double-struck">U</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>I</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>A</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>R</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mi>S</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>B</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:mi>I</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math> unitary matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> has been defined in Eq. <ce:cross-ref refid="fm0040" id="crf0240">(4)</ce:cross-ref> to primarily describe flavor mixing in the sterile (heavy) neutrino sector, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> denotes the other <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math> unitary matrix that is mainly responsible for flavor mixing in the active (light) neutrino sector, while the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math> matrices <ce:italic>A</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>R</ce:italic> and <ce:italic>S</ce:italic> signify the interplay between these two sectors <ce:cross-refs refid="br0220 br0230 br0240" id="crs0060">[22–24]</ce:cross-refs>. The unitarity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mi mathvariant="double-struck">U</mml:mi></mml:math> assures<ce:display><ce:formula id="fm0110"><ce:label>(11)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:mi id="mmlbr0002">A</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>B</mml:mi><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi><mml:mspace width="0.25em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="after">,</mml:mo><mml:mi>A</mml:mi><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mi>B</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn mathvariant="bold">0</mml:mn><mml:mspace width="0.25em"/><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="after">,</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mi>A</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mi>B</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> On the other hand, the arbitrary charged-lepton Yukawa coupling matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0030" id="crf0250">(3)</ce:cross-ref> can be diagonalized by a bi-unitary transformation:<ce:display><ce:formula id="fm0120"><ce:label>(12)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si85.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si86.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:math> are unitary, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mrow><mml:mi mathvariant="normal">Diag</mml:mi></mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">{</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">}</mml:mo></mml:math> stands for the “working” or “virtual” masses of three charged leptons <ce:italic>before</ce:italic> spontaneous electroweak symmetry breaking,<ce:cross-ref refid="fn0040" id="crf0260"><ce:sup>4</ce:sup></ce:cross-ref><ce:footnote id="fn0040"><ce:label>4</ce:label><ce:note-para id="np0040">Note that the scalar field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> in Eq. <ce:cross-ref refid="fm0120" id="crf0270">(12)</ce:cross-ref> carries a hypercharge, and hence <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si89.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:math> cannot be simply understood as a diagonal “mass” matrix. The physical meaning of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si89.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:math> is actually vague in our calculations which are mathematically exact and clear, so is the physical meaning of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0080" id="crf0280">(8)</ce:cross-ref>. But this vagueness will automatically disappear after spontaneous electroweak symmetry breaking, as can be subsequently seen.</ce:note-para></ce:footnote> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si90.svg"><mml:msup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>e</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>μ</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>τ</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> is defined as the column vector of the mass eigenstates of three charged leptons versus the column vector of their flavor eigenstates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:mi>l</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math>. Substituting Eqs. <ce:cross-ref refid="fm0080" id="crf0290">(8)</ce:cross-ref>—<ce:cross-ref refid="fm0100" id="crf0300">(10)</ce:cross-ref> and <ce:cross-ref refid="fm0120" id="crf0310">(12)</ce:cross-ref> into Eq. <ce:cross-ref refid="fm0030" id="crf0320">(3)</ce:cross-ref>, we immediately arrive at<ce:display><ce:formula id="fm0130"><ce:label>(13)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si92.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si93.svg"><mml:msup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> denotes the column vector of the <ce:italic>working</ce:italic> mass eigenstates of three light Majorana neutrinos far above the electroweak scale, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si94.svg"><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> stands for the column vectors of the mass eigenstates of three heavy Majorana neutrinos relevant to the seesaw mechanism at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>≫</mml:mo><mml:mi>v</mml:mi></mml:math>. In this case the flavor eigenstates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math> can be expressed in terms of the mass eigenstates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> or their charge-conjugated states as follows:<ce:display><ce:formula id="fm0140"><ce:label>(14)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si97.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>U</mml:mi><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:mi>U</mml:mi><mml:mo>≡</mml:mo><mml:mi>A</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si99.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>≡</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mi>B</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si100.svg"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>≡</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mi>S</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> are defined. Taking account of the Majorana property of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg"><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> <ce:cross-ref refid="br0070" id="crf0330">[7]</ce:cross-ref> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>), one simply obtains <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si103.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si104.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math>. One may then substitute the expression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.svg"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0120" id="crf0340">(12)</ce:cross-ref> and that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0140" id="crf0350">(14)</ce:cross-ref> into the standard form of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cc</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0070" id="crf0360">(7)</ce:cross-ref> and get at<ce:display><ce:formula id="fm0150"><ce:label>(15)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si106.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cc</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>e</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>μ</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>τ</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si107.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:mi>U</mml:mi></mml:math> is just the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix <ce:cross-refs refid="br0250 br0260 br0270" id="crs0070">[25–27]</ce:cross-refs> used to describe the flavor oscillations of three active neutrinos, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si108.svg"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msubsup><mml:mi>R</mml:mi></mml:math> is an analogue of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si109.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub></mml:math> in the seesaw mechanism which characterizes the strengths of weak charged-current interactions for three heavy Majorana neutrinos.</ce:para><ce:para id="pr0090">Without loss of generality, one may choose a convenient flavor basis in which the mass eigenstates of three charged leptons are identified with their corresponding flavor eigenstates (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si110.svg"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math>, or equivalently <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si111.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi></mml:math>). In this case we are simply left with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si112.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>U</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si113.svg"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PMNS</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>R</mml:mi></mml:math>, namely the effects of lepton flavor mixing originate purely from the active and sterile Majorana neutrino sectors and from the interplay between these two sectors. We shall take advantage of this flavor basis in the following discussions unless otherwise specified.</ce:para></ce:section><ce:section id="se0060"><ce:label>2.4</ce:label><ce:section-title id="st0070">Mismatch between the two bases</ce:section-title><ce:para id="pr0100">Before discussing a mismatch between the mass eigenstates of heavy Majorana neutrinos associated with thermal leptogenesis and those associated with the seesaw mechanism, let us take a look at the flavor structures of active and sterile neutrinos in the case that the electroweak gauge symmetry is unbroken at Λ. First of all, a combination of Eqs. <ce:cross-ref refid="fm0080" id="crf0370">(8)</ce:cross-ref> and <ce:cross-ref refid="fm0100" id="crf0380">(10)</ce:cross-ref> allows us to immediately derive the exact seesaw relation between the working masses of three light Majorana neutrinos and the real masses of three heavy Majorana neutrinos:<ce:display><ce:formula id="fm0160"><ce:label>(16)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si114.svg"><mml:mrow><mml:mi>U</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn mathvariant="bold">0</mml:mn><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> in which <ce:italic>U</ce:italic> and <ce:italic>R</ce:italic> are also correlated with each other via the unitarity condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si115.svg"><mml:mi>U</mml:mi><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi></mml:math>. Note that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si116.svg"><mml:mi>U</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>A</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> holds, where the unitary matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is primarily responsible for flavor mixing of the three active neutrinos. So we find it useful to rewrite Eq. <ce:cross-ref refid="fm0160" id="crf0390">(16)</ce:cross-ref> as<ce:display><ce:formula id="fm0170"><ce:label>(17)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si117.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi mathvariant="normal">i</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi mathvariant="normal">i</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> whose left- and right-hand sides are composed of the <ce:italic>derivational</ce:italic> and <ce:italic>original</ce:italic> seesaw parameters, respectively. This point will become more obvious when a complete Euler-like parametrization of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> unitary matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mi mathvariant="double-struck">U</mml:mi></mml:math> in Eq. <ce:cross-ref refid="fm0100" id="crf0400">(10)</ce:cross-ref> is adopted, as can be seen in section <ce:cross-ref refid="se0080" id="crf0410">3</ce:cross-ref>. Needless to say, the active-sterile flavor mixing matrix <ce:italic>R</ce:italic> essentially plays the role of the neutrino Yukawa coupling matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> in the canonical seesaw framework,<ce:display><ce:formula id="fm0180"><ce:label>(18)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si118.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi>I</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>;</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> and the right-handed Majorana neutrino mass matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:math> can be reconstructed into the form<ce:display><ce:formula id="fm0190"><ce:label>(19)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si119.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Note that all the quantities in Eqs. <ce:cross-ref refid="fm0180" id="crf0420">(18)</ce:cross-ref> and <ce:cross-ref refid="fm0190" id="crf0430">(19)</ce:cross-ref> belong to the <ce:italic>original</ce:italic> seesaw parameters in the sense that they have nothing to do with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> — the working masses and the primary flavor mixing matrix of three light Majorana neutrinos which are <ce:italic>derived</ce:italic> from the seesaw mechanism.</ce:para><ce:para id="pr0110">Now we turn to an unavoidable mismatch between the mass eigenstates of three heavy Majorana neutrinos associated with the seesaw and leptogenesis mechanisms. Eq. <ce:cross-ref refid="fm0140" id="crf0440">(14)</ce:cross-ref> tells us that the mass eigenstates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> in the seesaw basis can be expressed as<ce:display><ce:formula id="fm0200"><ce:label>(20)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where Eq. <ce:cross-ref refid="fm0040" id="crf0450">(4)</ce:cross-ref> has been used to link <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math>. To be more explicit, Eq. <ce:cross-ref refid="fm0200" id="crf0460">(20)</ce:cross-ref> means<ce:display><ce:formula id="fm0210"><ce:label>(21)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si122.svg"><mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> from which the differences between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> in the seesaw basis and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>) in the thermal leptogenesis basis can be clearly seen. Similarly, a combination of Eqs. <ce:cross-ref refid="fm0040" id="crf0470">(4)</ce:cross-ref> and <ce:cross-ref refid="fm0190" id="crf0480">(19)</ce:cross-ref> leads us to<ce:display><ce:formula id="fm0220"><ce:label>(22)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si123.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>B</mml:mi><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> from which one may easily see the difference between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si124.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>. Although <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub></mml:math>) would exactly coincide with each other if the Yukawa coupling matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> (or equivalently, <ce:italic>R</ce:italic> or <ce:italic>S</ce:italic>) were switched off, such a coincidence would make no sense because both the seesaw and leptogenesis mechanisms would fail in this special case. In the presence of the neutrino Yukawa interactions, thermal leptogenesis may take effect via the CP-violating and out-of-equilibrium decays of heavy Majorana neutrinos into the leptonic and Higgs doublets, while the seesaw mechanism can “formally” work with the help of an interplay between the active and sterile neutrino fields coupled only to the neutral component of the Higgs doublet. That is the key reason why there is an inevitable mismatch between the seesaw- and leptogenesis-associated bases for heavy Majorana neutrinos before spontaneous electroweak symmetry breaking.</ce:para></ce:section><ce:section id="se0070"><ce:label>2.5</ce:label><ce:section-title id="st0080">After gauge symmetry breaking</ce:section-title><ce:para id="pr0120">So far we have made some proper transformations of the charged lepton and neutrino fields in the flavor space to obtain their respective working or true mass eigenstates. All such unitary flavor basis transformations are completely reversible, and hence they do not affect the gauge invariance of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:msub></mml:math> at the seesaw scale. As already shown in Eqs. <ce:cross-ref refid="fm0200" id="crf0490">(20)</ce:cross-ref> and <ce:cross-ref refid="fm0220" id="crf0500">(22)</ce:cross-ref>, a seeable mismatch between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> or between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si124.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> results from the fact that the working seesaw mechanism itself is only associated with the neutral component of the Higgs doublet while the heavy Majorana neutrino decays and thermal leptogenesis at the seesaw scale Λ are associated with the whole Higgs doublet. This unavoidable mismatch deserves to be conceptually clarified as we have done, because it is an intrinsic issue of the seesaw and leptogenesis mechanisms.</ce:para><ce:para id="pr0130">It is now straightforward to prove that the <ce:italic>formal</ce:italic> seesaw mechanism far above the electroweak scale will become <ce:italic>real</ce:italic> after the Higgs potential of the SM is minimized at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si126.svg"><mml:mo stretchy="false">〈</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo>≡</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math> with a special direction characterized by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si127.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si128.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math>, by which the electroweak gauge symmetry is spontaneously broken and thus all the particles coupled to the Higgs field acquire their nonzero masses. In this case the Lagrangian in Eq. <ce:cross-ref refid="fm0030" id="crf0510">(3)</ce:cross-ref> can be simplified to a more popular form,<ce:display><ce:formula id="fm0230"><ce:label>(23)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si129.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">]</mml:mo></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mn mathvariant="bold">0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mi>v</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math> denotes the charged lepton mass matrix, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>v</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math> is usually referred to as the Dirac neutrino mass matrix. The expression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub></mml:math> in terms of the seesaw parameters can be directly read off from Eq. <ce:cross-ref refid="fm0180" id="crf0520">(18)</ce:cross-ref>, namely<ce:display><ce:formula id="fm0240"><ce:label>(24)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si133.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi>I</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> We find that the exact seesaw formula obtained in Eq. <ce:cross-ref refid="fm0160" id="crf0530">(16)</ce:cross-ref> and the analytical results obtained in Eqs. <ce:cross-ref refid="fm0190" id="crf0540">(19)</ce:cross-ref>—<ce:cross-ref refid="fm0220" id="crf0550">(22)</ce:cross-ref> formally keep unchanged after spontaneous gauge symmetry breaking, but they are now subject to the electroweak scale. In other words, the electroweak symmetry breaking itself does not really affect the flavor structures of the seesaw mechanism. This observation implies that it is possible to determine or constrain some of the original seesaw-associated flavor parameters in some low-energy neutrino experiments, after the radiative corrections to such parameters are properly taken into account with the help of the relevant renormaliztion-group equations (RGEs) between a superhigh seesaw scale and the electroweak scale <ce:cross-ref refid="br0280" id="crf0560">[28]</ce:cross-ref>.</ce:para><ce:para id="pr0140">Note that the exact seesaw formula obtained in Eq. <ce:cross-ref refid="fm0160" id="crf0570">(16)</ce:cross-ref> can be simplified to the more popular form in the leading-order approximations of Eqs. <ce:cross-ref refid="fm0190" id="crf0580">(19)</ce:cross-ref> and <ce:cross-ref refid="fm0240" id="crf0590">(24)</ce:cross-ref>. That is, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si134.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si135.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math>, so the <ce:italic>effective</ce:italic> mass matrix for three active Majorana neutrinos is given by<ce:display><ce:formula id="fm0250"><ce:label>(25)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si136.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:mo>≃</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"><mml:mi>A</mml:mi><mml:mo>≃</mml:mo><mml:mi>B</mml:mi><mml:mo>≃</mml:mo><mml:mi>I</mml:mi></mml:math> has been assumed (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si138.svg"><mml:mi>U</mml:mi><mml:mo>≃</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> holds in the neglect of the non-unitary effects characterized by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si139.svg"><mml:mi>A</mml:mi><mml:mo>≠</mml:mo><mml:mi>I</mml:mi></mml:math>). In this approximation the effective Majorana mass term for three active neutrinos at low energies turns out to be<ce:display><ce:formula id="fm0260"><ce:label>(26)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mo>‾</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mi mathvariant="normal">h</mml:mi><mml:mo>.</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where the column vector of the light neutrino mass eigenstates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> has already been defined below Eq. <ce:cross-ref refid="fm0130" id="crf0600">(13)</ce:cross-ref>, and the physical meaning of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> as the diagonal Majorana neutrino mass matrix becomes definite and obvious.</ce:para></ce:section></ce:section><ce:section id="se0080"><ce:label>3</ce:label><ce:section-title id="st0090">How small is the mismatch?</ce:section-title><ce:section id="se0090"><ce:label>3.1</ce:label><ce:section-title id="st0100">An Euler-like parametrization</ce:section-title><ce:para id="pr0150">To clearly see how small the difference between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub></mml:math>) is expected to be, let us follow Refs. <ce:cross-refs refid="br0220 br0230 br0240" id="crs0080">[22–24]</ce:cross-refs> to make an Euler-like parametrization of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> unitary matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mi mathvariant="double-struck">U</mml:mi></mml:math> in Eq. <ce:cross-ref refid="fm0100" id="crf0610">(10)</ce:cross-ref>. First of all we introduce fifteen <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> Euler-like unitary matrices of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si142.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mn>6</mml:mn></mml:math>): its <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si143.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si144.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> entries are both identical to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si145.svg"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mi mathvariant="normal">cos</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> being a flavor mixing angle and lying in the first quadrant, its other four diagonal elements are all equal to one, its <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si147.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si148.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> entries are given respectively by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si149.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">i</mml:mi><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mi mathvariant="normal">sin</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">i</mml:mi><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mi mathvariant="normal">sin</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si151.svg"><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> being a CP-violating phase, and its other off-diagonal elements are all equal to zero. These matrices are then grouped in the following way to respectively describe the <ce:italic>active</ce:italic> flavor sector, the <ce:italic>sterile</ce:italic> flavor sector and the <ce:italic>interplay</ce:italic> between these two sectors:<ce:display><ce:formula id="fm0270"><ce:label>(27)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si152.svg"><mml:mrow><mml:mo stretchy="true" id="mmlbr0003">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:mi>I</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="true" linebreak="newline" indentalign="id" indenttarget="mmlbr0003" linebreakstyle="before">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>I</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>56</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>46</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>45</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="true" linebreak="newline" indentalign="id" indenttarget="mmlbr0003" linebreakstyle="before">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>A</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>R</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mi>S</mml:mi></mml:mtd><mml:mtd columnalign="center"><mml:mi>B</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where the pattern of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is quite similar to the standard parametrization of a unitary PMNS matrix as advocated by the Particle Data Group <ce:cross-ref refid="br0090" id="crf0620">[9]</ce:cross-ref>,<ce:cross-ref refid="fn0050" id="crf0630"><ce:sup>5</ce:sup></ce:cross-ref><ce:footnote id="fn0050"><ce:label>5</ce:label><ce:note-para id="np0050">When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is applied to the phenomenology of neutrino physics in the basis of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si111.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi></mml:math>, it is the phase parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si153.svg"><mml:mi>δ</mml:mi><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub></mml:math> that characterizes the strength of CP violation in neutrino oscillations.</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0280"><ce:label>(28)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si154.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> and the expression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> can be directly read off from that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> with the subscript replacements <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si155.svg"><mml:mn>12</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>45</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si156.svg"><mml:mn>13</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>46</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si157.svg"><mml:mn>23</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>56</mml:mn></mml:math> for the three rotation angles and three CP-violating phases. The explicit expressions of <ce:italic>A</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>R</ce:italic> and <ce:italic>S</ce:italic> in terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si159.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>j</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:math>) are rather lengthy, and hence they are listed in Eqs. <ce:cross-ref refid="fm0360" id="crf0640">(A.1)</ce:cross-ref> and <ce:cross-ref refid="fm0370" id="crf0650">(A.2)</ce:cross-ref> in Appendix <ce:cross-ref refid="se0130" id="crf0660">A</ce:cross-ref> for the same of simplicity. Among the four active-sterile flavor mixing matrices, only <ce:italic>A</ce:italic> and <ce:italic>R</ce:italic> affect the physical processes in which the light and heavy Majorana neutrinos take part, as can be seen from Eq. <ce:cross-ref refid="fm0150" id="crf0670">(15)</ce:cross-ref>. As both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si116.svg"><mml:mi>U</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>A</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> and <ce:italic>R</ce:italic> appear in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cc</mml:mi></mml:mrow></mml:msub></mml:math> in the chosen flavor basis (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si111.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi></mml:math>), three of the nice CP-violating phases (or their combinations) of <ce:italic>A</ce:italic> and <ce:italic>R</ce:italic> can always be rotated away by properly redefining the phases of three charged lepton fields <ce:cross-refs refid="br0290 br0300" id="crs0090">[29,30]</ce:cross-refs>.</ce:para><ce:para id="pr0160">The PMNS matrix <ce:italic>U</ce:italic> is obviously non-unitary because of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si161.svg"><mml:mi>U</mml:mi><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>A</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>I</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup><mml:mo>≠</mml:mo><mml:mi>I</mml:mi></mml:math>, but its deviation from exact unitarity (i.e., from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math>) is found to be very small. A detailed and careful analysis of currently available electroweak precision measurements and neutrino oscillation data has put a stringent constraint on the non-unitarity of <ce:italic>U</ce:italic> — the latter is below or far below <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si162.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> <ce:cross-refs refid="br0310 br0320 br0330 br0340 br0350" id="crs0100">[31–35]</ce:cross-refs>. This result implies that the deviation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si163.svg"><mml:mi>A</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">†</mml:mi></mml:mrow></mml:msup></mml:math> from <ce:italic>I</ce:italic> ought to be smaller than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si162.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>, and thus the nine active-sterile flavor mixing angles in <ce:italic>R</ce:italic> should be smaller than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si164.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>. The advantage of such a phenomenological observation is that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si138.svg"><mml:mi>U</mml:mi><mml:mo>≃</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> can be a quite reliable approximation in most cases, but its disadvantage is that an experimental exploration of the seesaw-induced non-unitary effects of <ce:italic>U</ce:italic> at low energies will be rather challenging.</ce:para></ce:section><ce:section id="se0100"><ce:label>3.2</ce:label><ce:section-title id="st0110">Smallness of the mismatch</ce:section-title><ce:para id="pr0170">Eq. <ce:cross-ref refid="fm0200" id="crf0680">(20)</ce:cross-ref> tells us that a difference between the mass eigenstates of three heavy Majorana neutrinos associated with the seesaw mechanism (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math>) and those associated with thermal leptogenesis (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math>) is mainly characterized by the deviation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si165.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> from the identity matrix <ce:italic>I</ce:italic>. With the help of Eq. <ce:cross-ref refid="fm0370" id="crf0690">(A.2)</ce:cross-ref>, we arrive at<ce:display><ce:formula id="fm0290"><ce:label>(29)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si166.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="newline" indentalign="left" linebreakstyle="before" id="mmlbr0004">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0004" linebreakstyle="before">≃</mml:mo><mml:mi>I</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mspace width="0.25em"/><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/></mml:mtd><mml:mtd columnalign="center"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si167.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">i</mml:mi><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mi mathvariant="normal">tan</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> is defined, and all the terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si168.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math> or smaller have been omitted from the second equation as an excellent approximation due to the smallness of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>j</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:math>). We see that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si165.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> is also a lower triangular matrix like <ce:italic>B</ce:italic> itself. On the other hand, the factor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si169.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> appearing in Eq. <ce:cross-ref refid="fm0200" id="crf0700">(20)</ce:cross-ref> can be explicitly expressed as follows:<ce:display><ce:formula id="fm0300"><ce:label>(30)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si170.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where Eq. <ce:cross-ref refid="fm0370" id="crf0710">(A.2)</ce:cross-ref> has been used, and the terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si171.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math> or smaller have been omitted from the second equation as a very good approximation. Now we conclude that the heavy Majorana neutrino mass basis <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> is identical to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> up to the accuracy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math>, but it contains a small contribution of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si173.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math> from the light Majorana neutrino mass basis <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si174.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:math> in the seesaw framework. Since the magnitudes of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>j</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:math>) are highly suppressed in a realistic seesaw model with little fine-tuning, the mismatch between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> is expected to be negligible in most cases.</ce:para><ce:para id="pr0180">Let us proceed to examine how small the difference between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub></mml:math> in Eq. <ce:cross-ref refid="fm0210" id="crf0720">(21)</ce:cross-ref> can be. First of all, Eq. <ce:cross-ref refid="fm0370" id="crf0730">(A.2)</ce:cross-ref> allows us to make the approximation<ce:display><ce:formula id="fm0310"><ce:label>(31)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si175.svg"><mml:mrow><mml:mi>B</mml:mi><mml:mo>≃</mml:mo><mml:mi>I</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mspace width="0.25em"/><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.25em"/></mml:mtd><mml:mtd columnalign="center"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where the terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si168.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math> or smaller have been omitted. Secondly, we obtain<ce:display><ce:formula id="fm0320"><ce:label>(32)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si176.svg"><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo linebreak="newline" indentalign="left" linebreakstyle="before" id="mmlbr0005">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0005" linebreakstyle="before">≃</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> from Eq. <ce:cross-ref refid="fm0360" id="crf0740">(A.1)</ce:cross-ref>, where the terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si171.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math> or smaller have been neglected in the second equation as a reasonably good approximation. The exact expression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si177.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi></mml:math> can be directly read off from that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si178.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> with the help of Eq. <ce:cross-ref refid="fm0320" id="crf0750">(32)</ce:cross-ref> by making the subscript replacements <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si179.svg"><mml:mn>15</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>24</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si180.svg"><mml:mn>16</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>34</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si181.svg"><mml:mn>26</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>35</mml:mn></mml:math>, so can its approximate expression. As a consequence,<ce:display><ce:formula id="fm0330"><ce:label>(33)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si182.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo>≃</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:mspace width="0.25em"/><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mspace width="0.25em"/><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.25em"/></mml:mtd><mml:mtd columnalign="center"><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:math></ce:formula></ce:display> holds in the same approximation as made above. This result implies that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> are identical to each other up to the accuracy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math>, simply because on the right-hand side of Eq. <ce:cross-ref refid="fm0220" id="crf0760">(22)</ce:cross-ref> the second term is suppressed in magnitude to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si168.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math> as compared with the first term.</ce:para><ce:para id="pr0190">It is worth remarking that our above analytical approximations are more or less subject to the canonical seesaw mechanism at an energy scale far above the electroweak scale, and thus the mismatch between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup></mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:msub></mml:math>) is very small. This situation will change when the low-scale seesaw and leptogenesis scenarios, in which a mismatch between the two sets of mass bases for heavy Majorana neutrinos is crucial, are taken into account (see, e.g., Refs. <ce:cross-refs refid="br0110 br0120" id="crs0110">[11,12]</ce:cross-refs>).</ce:para></ce:section><ce:section id="se0110"><ce:label>3.3</ce:label><ce:section-title id="st0120">Determination of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></ce:section-title><ce:para id="pr0200">As already shown in Eq. <ce:cross-ref refid="fm0170" id="crf0770">(17)</ce:cross-ref>, the nine effective flavor parameters of three light Majorana neutrinos in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> (i.e., three effective masses, three flavor mixing angles and three CP-violating phases) can be expressed in terms of the eighteen seesaw parameters hidden in <ce:italic>A</ce:italic>, <ce:italic>R</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> (i.e., three heavy Majorana neutrino masses, nine active-sterile flavor mixing angles and six CP-violating phases). It is obvious that all the derivational seesaw parameters on the left-hand side of Eq. <ce:cross-ref refid="fm0170" id="crf0780">(17)</ce:cross-ref> would vanish if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si183.svg"><mml:mi>R</mml:mi><mml:mo>∝</mml:mo><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> were switched off. So this equation provides an unambiguous way to determine the light degrees of freedom from the heavy degrees of freedom in the seesaw framework.</ce:para><ce:para id="pr0210">To be more specific, the six independent elements of the effective Majorana neutrino mass matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si184.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup></mml:math> are given as follows:<ce:display><ce:formula id="fm0340"><ce:label>(34)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si185.svg"><mml:msub id="mmlbr0006"><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub id="mmlbr0007"><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0006" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0007" linebreakstyle="before">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0006" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0007" linebreakstyle="before">−</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0006" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0006" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0007" linebreakstyle="before">−</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0006" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> On the other hand, Eq. <ce:cross-ref refid="fm0170" id="crf0790">(17)</ce:cross-ref> tells us that these six matrix elements can originally be determined by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si186.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> thanks to the exact seesaw relation bridging the big gap between the light and heavy Majorana neutrinos. With the help of the explicit expression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si187.svg"><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>R</mml:mi></mml:math> given in Eq. <ce:cross-ref refid="fm0320" id="crf0800">(32)</ce:cross-ref>, it is straightforward to obtain the expressions for the elements of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si188.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> in terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si151.svg"><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>j</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:math>). Instead of presenting the exact analytical results, which are rather lengthy and hence less instructive, here we make the leading-order approximations for the expressions of <ce:italic>A</ce:italic> and <ce:italic>R</ce:italic> given in Eq. <ce:cross-ref refid="fm0360" id="crf0810">(A.1)</ce:cross-ref> and then arrive at<ce:display><ce:formula id="fm0350"><ce:label>(35)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si189.svg"><mml:msub id="mmlbr0008"><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex" linebreak="newline" indentalign="id" indenttarget="mmlbr0008" linebreakstyle="before">(</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo>≃</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Let us emphasize that there appear nine CP-violating phases in Eq. <ce:cross-ref refid="fm0350" id="crf0820">(35)</ce:cross-ref>, but three of them (or their combinations) are redundant and can always be removed by rephasing the charged lepton fields in a proper way.<ce:cross-ref refid="fn0060" id="crf0830"><ce:sup>6</ce:sup></ce:cross-ref><ce:footnote id="fn0060"><ce:label>6</ce:label><ce:note-para id="np0060">A straightforward way to remove the three redundant phase parameters of <ce:italic>A</ce:italic> and <ce:italic>R</ce:italic> is just to switch off three of the nine phases in the nine active-sterile flavor mixing matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>j</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:math>) in Eq. <ce:cross-ref refid="fm0270" id="crf0840">(27)</ce:cross-ref> from the very beginning. As there are many options in doing so, we do not go into details here.</ce:note-para></ce:footnote> A combination of Eqs. <ce:cross-ref refid="fm0340" id="crf0850">(34)</ce:cross-ref> and <ce:cross-ref refid="fm0350" id="crf0860">(35)</ce:cross-ref> allows us to establish the direct relations between the nine derivational and eighteen original seesaw parameters. So the former can in principle be determined from the latter for a given seesaw model (a top-down approach), and the latter may be partly probed or constrained from the former with the help of some low-energy neutrino experiments (a bottom-up approach). A careful and detailed analysis of the parameter space along this line of thought will be made elsewhere.</ce:para></ce:section></ce:section><ce:section id="se0120"><ce:label>4</ce:label><ce:section-title id="st0130">Summary</ce:section-title><ce:para id="pr0220">We have reformulated the canonical seesaw mechanism by considering the fact that the electroweak gauge symmetry is unbroken at the seesaw scale characterized by the masses of heavy Majorana neutrinos, and shown that it can <ce:italic>formally</ce:italic> work and allow us to derive an exact seesaw relation between the active (light) and sterile (heavy) Majorana neutrinos. In this way we have elucidated the reason why there is an unavoidable mismatch between the mass eigenstates of heavy Majorana neutrinos associated with the seesaw and thermal leptogenesis mechanisms. The smallness of this mismatch has been discussed with the help of a complete Euler-like parametrization of the flavor structure in the seesaw framework, and the exact and explicit relations between the <ce:italic>original</ce:italic> and <ce:italic>derivational</ce:italic> seesaw parameters have been established as a by-product.</ce:para><ce:para id="pr0230">We hope that this work may help clarify some conceptual ambiguities associated with the validity of the seesaw mechanism before and after spontaneous electroweak symmetry breaking, because such ambiguities have never been taken serious in the literature. It should also be helpful to clarify the ambiguities associated with the RGE evolution between the “virtual” flavor parameters of Majorana neutrinos at the seesaw scale and those “real” ones at the electroweak scale, which is crucial to bridge the gap between a well-motivated UV-complete flavor theory including the seesaw mechanism and all the possible low-energy flavor experiments.</ce:para> </ce:section><ce:section id="se0140"><ce:section-title id="st0180">CRediT authorship contribution statement</ce:section-title><ce:para id="pr0270"><ce:bold>Zhi-zhong Xing:</ce:bold> Conceptualization, Investigation, Methodology, Writing – original draft, Writing – review & editing.</ce:para></ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0190">Declaration of Competing Interest</ce:section-title><ce:para id="pr0280">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0140">Acknowledgements</ce:section-title><ce:para id="pr0240">I am greatly indebted to Di Zhang and Shun Zhou for numerous helpful discussions and comments about this paper, a work dedicated to the 50th birthday of my home institute, the Institute of High Energy Physics, which was founded on 1 February 1973. I would also like to thank Marco Drewes for very useful discussions and comments. My research is supported in part by the <ce:grant-sponsor id="gsp0010" sponsor-id="https://doi.org/10.13039/501100001809">National Natural Science Foundation of China</ce:grant-sponsor> under grant No. <ce:grant-number refid="gsp0010">12075254</ce:grant-number> and grant No. <ce:grant-number refid="gsp0010">11835013</ce:grant-number>.</ce:para></ce:acknowledgment><ce:appendices><ce:section id="se0130"><ce:label>Appendix A</ce:label><ce:section-title id="st0150">The expressions of <ce:italic>A</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>R</ce:italic> and <ce:italic>S</ce:italic></ce:section-title><ce:para id="pr0250">Given the Euler-like parametrization of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:mn>6</mml:mn></mml:math> unitary flavor mixing matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mi mathvariant="double-struck">U</mml:mi></mml:math> decomposed in Eq. <ce:cross-ref refid="fm0270" id="crf0870">(27)</ce:cross-ref>, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math> active-sterile flavor mixing matrices <ce:italic>A</ce:italic>, <ce:italic>B</ce:italic>, <ce:italic>R</ce:italic> and <ce:italic>S</ce:italic> depend on the same nine rotation angles <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> and nine phase angles <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si151.svg"><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>j</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:math>). To be explicit <ce:cross-refs refid="br0220 br0230" id="crs0120">[22,23]</ce:cross-refs>,<ce:display><ce:formula id="fm0360"><ce:label>(A.1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si190.svg"><mml:mi id="mmlbr0009">A</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd 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stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>26</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mtd><mml:mtd columnalign="center"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>35</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>36</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> We see that both <ce:italic>A</ce:italic> and <ce:italic>B</ce:italic> are the lower triangular matrices, and the expression of <ce:italic>B</ce:italic> can be read off from that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si192.svg"><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> with the subscript replacements <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si179.svg"><mml:mn>15</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>24</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si180.svg"><mml:mn>16</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>34</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si181.svg"><mml:mn>26</mml:mn><mml:mo stretchy="false">↔</mml:mo><mml:mn>35</mml:mn></mml:math>. The expression of <ce:italic>S</ce:italic> can be similarly obtained from that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si193.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> with the same subscript replacements <ce:cross-ref refid="br0240" id="crf0880">[24]</ce:cross-ref>. Note, however, that <ce:italic>B</ce:italic> and <ce:italic>S</ce:italic> do not affect any physical processes in the seesaw mechanism.</ce:para></ce:section></ce:appendices></body><tail><ce:bibliography id="bl0010"><ce:section-title id="st0160">References</ce:section-title><ce:bibliography-sec id="bs0010"><ce:bib-reference id="br0010"><ce:label>[1]</ce:label><sb:reference id="bibADD2B090BFBF49417BB3AB7A01295D54s1"><sb:contribution><sb:authors><sb:author><ce:given-name>P.</ce:given-name><ce:surname>Minkowski</ce:surname></sb:author></sb:authors><sb:title><sb:maintitle><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si194.svg"><mml:mi>μ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>e</mml:mi><mml:mi>γ</mml:mi></mml:math> at a rate of one out of 10<ce:sup>9</ce:sup> muon decays?</sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Phys. Lett. B</sb:maintitle></sb:title><sb:volume-nr>67</sb:volume-nr></sb:series><sb:date>1977</sb:date></sb:issue><sb:pages><sb:first-page>421</sb:first-page></sb:pages></sb:host></sb:reference><ce:source-text id="srct0015">P. Minkowski, “μ→eγ at a rate of one out of 109 muon decays?” Phys. Lett. 67B (1977) 421.</ce:source-text></ce:bib-reference><ce:bib-reference id="br0020"><ce:label>[2]</ce:label><sb:reference id="bib5BF9E2DBBCA42A64CDAE66A10E170751s1"><sb:contribution><sb:authors><sb:author><ce:given-name>T.</ce:given-name><ce:surname>Yanagida</ce:surname></sb:author></sb:authors><sb:title><sb:maintitle>Horizontal gauge symmetry and masses of neutrinos</sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Conf. Proc. C</sb:maintitle></sb:title><sb:volume-nr>7902131</sb:volume-nr></sb:series><sb:date>1979</sb:date></sb:issue><sb:pages><sb:first-page>95</sb:first-page></sb:pages></sb:host></sb:reference><ce:source-text id="srct0020">T. Yanagida, “Horizontal gauge symmetry and masses of neutrinos,” Conf. Proc. C 7902131 (1979) 95.</ce:source-text></ce:bib-reference><ce:bib-reference id="br0030"><ce:label>[3]</ce:label><sb:reference id="bib882BADE8E896CBC1DEA7386BD15AA05Es1"><sb:contribution><sb:authors><sb:author><ce:given-name>M.</ce:given-name><ce:surname>Gell-Mann</ce:surname></sb:author><sb:author><ce:given-name>P.</ce:given-name><ce:surname>Ramond</ce:surname></sb:author><sb:author><ce:given-name>R.</ce:given-name><ce:surname>Slansky</ce:surname></sb:author></sb:authors><sb:title><sb:maintitle>Complex spinors and unified theories</sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Conf. Proc. 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The square nodes represent the fusion of the original tensor indices into the combined fat indices <ce:cross-ref refid="fm0030" id="crf0020">(3)</ce:cross-ref> of <ce:italic>M</ce:italic>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0020">Fig. 2</ce:alt-text><ce:link locator="gr002" xlink:type="simple" xlink:href="pii:S0550321323000366/gr002" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0020"/></ce:figure><ce:figure id="fg0030"><ce:label>Fig. 3</ce:label><ce:caption id="cp0030"><ce:simple-para id="sp0030">Two tensors <ce:italic>T</ce:italic> are contracted over their shared vertical link, producing a tensor <ce:italic>M</ce:italic> with fat horizontal modes (green box). The fat modes of <ce:italic>M</ce:italic> are projected onto subspaces with the projectors <ce:italic>P</ce:italic><ce:inf><ce:italic>U</ce:italic></ce:inf> = <ce:italic>UU</ce:italic><ce:sup><ce:italic>T</ce:italic></ce:sup> and <ce:italic>P</ce:italic><ce:inf><ce:italic>V</ce:italic></ce:inf> = <ce:italic>VV</ce:italic><ce:sup><ce:italic>T</ce:italic></ce:sup>, respectively, to form the lower rank approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> of <ce:cross-ref refid="fm0200" id="crf0030">(20)</ce:cross-ref> (blue box). As part of the construction one recognizes the core tensor <ce:italic>S</ce:italic> of <ce:cross-ref refid="fm0190" id="crf0040">(19)</ce:cross-ref> (red box). Note that the matrices <ce:italic>U</ce:italic>, <ce:italic>U</ce:italic><ce:sup><ce:italic>T</ce:italic></ce:sup>, <ce:italic>V</ce:italic> and <ce:italic>V</ce:italic><ce:sup><ce:italic>T</ce:italic></ce:sup>, described by diamonds in the figure, are applied from the inside to the outside, in correspondence with <ce:cross-ref refid="fm0190" id="crf0050">(19)</ce:cross-ref> and <ce:cross-ref refid="fm0200" id="crf0060">(20)</ce:cross-ref>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0030">Fig. 3</ce:alt-text><ce:link locator="gr003" xlink:type="simple" xlink:href="pii:S0550321323000366/gr003" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0030"/></ce:figure><ce:figure id="fg0040"><ce:label>Fig. 4</ce:label><ce:caption id="cp0040"><ce:simple-para id="sp0040">Two approximations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math>, constructed in <ce:cross-ref refid="fg0030" id="crf0070">Fig. 3</ce:cross-ref>, are contracted in the horizontal direction. This illustrates how the projections performed in the first contraction are concatenated when making this second contraction, leading to a product <ce:italic>P</ce:italic><ce:inf><ce:italic>U</ce:italic></ce:inf><ce:italic>P</ce:italic><ce:inf><ce:italic>V</ce:italic></ce:inf>. Here a new building block <ce:italic>G</ce:italic> = <ce:italic>U</ce:italic><ce:sup><ce:italic>T</ce:italic></ce:sup><ce:italic>V</ce:italic> arises, which we call a <ce:italic>merger</ce:italic> between two core tensors <ce:italic>S</ce:italic>. Note that, for consistency, the matrices operate in chronological order (from the inside to the outside with respect to <ce:italic>M</ce:italic> of <ce:cross-ref refid="fg0030" id="crf0080">Fig. 3</ce:cross-ref>) and not from left to right. The half-mergers on the left and right will connect to their counter parts in further contractions.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0040">Fig. 4</ce:alt-text><ce:link locator="gr004" xlink:type="simple" xlink:href="pii:S0550321323000366/gr004" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0040"/></ce:figure><ce:figure id="fg0050"><ce:label>Fig. 5</ce:label><ce:caption id="cp0050"><ce:simple-para id="sp0050">The construction in <ce:cross-ref refid="fg0040" id="crf0090">Fig. 4</ce:cross-ref> drastically simplifies when choosing <ce:italic>U</ce:italic> = <ce:italic>V</ce:italic>, as the merger <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi>G</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mn mathvariant="double-struck">1</mml:mn></mml:mrow><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:math> in this case. The frames are solely needed to construct the core tensor <ce:italic>S</ce:italic>. This property will spread through the entire blocking procedure (including the final trace), such that the calculation can be performed using the core tensor only.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0050">Fig. 5</ce:alt-text><ce:link locator="gr005" xlink:type="simple" xlink:href="pii:S0550321323000366/gr005" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0050"/></ce:figure><ce:figure id="fg0060"><ce:label>Fig. 6</ce:label><ce:caption id="cp0060"><ce:simple-para id="sp0060">Comparison of (<ce:italic>ϵ</ce:italic> − <ce:italic>ϵ</ce:italic><ce:inf>hooi</ce:inf>)/<ce:italic>ϵ</ce:italic><ce:inf>hooi</ce:inf> for approximations of 100 independent random tensors of dimension 10 × 10 × 100 × 100 truncated to dimension 10 × 10 × 10 × 10 using HOSVD, the Xie method, SuperQ and ISQ. The tensor elements are chosen randomly from a uniform distribution over [0,1] (left plot) and from a Gaussian distribution <ce:italic>N</ce:italic>(0;1) (right plot). The horizontal axis represents different random tensors.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0060">Fig. 6</ce:alt-text><ce:link locator="gr006" xlink:type="simple" xlink:href="pii:S0550321323000366/gr006" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0060"/></ce:figure><ce:figure id="fg0070"><ce:label>Fig. 7</ce:label><ce:caption id="cp0070"><ce:simple-para id="sp0070">Comparison of (<ce:italic>ϵ</ce:italic> − <ce:italic>ϵ</ce:italic><ce:inf>hooi</ce:inf>)/<ce:italic>ϵ</ce:italic><ce:inf>hooi</ce:inf> for random tensors of dimension 30 × 30 × 30 × 30 truncated to dimension 10 × 10 × 10 × 10, using the same approximation methods and the same probability distributions for the tensor elements as in <ce:cross-ref refid="fg0060" id="crf0100">Fig. 6</ce:cross-ref>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0070">Fig. 7</ce:alt-text><ce:link locator="gr007" xlink:type="simple" xlink:href="pii:S0550321323000366/gr007" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0070"/></ce:figure></ce:floats><head><ce:title id="ti0010">Improved local truncation schemes for the higher-order tensor renormalization group method</ce:title><ce:author-group id="ag0010"><ce:author orcid="0000-0002-8443-4804" id="au0010" author-id="S0550321323000366-90624b61ad93431930b6562552c19df0"><ce:given-name>Jacques</ce:given-name><ce:surname>Bloch</ce:surname><ce:cross-ref refid="aff0010" id="crf0110"><ce:sup>a</ce:sup></ce:cross-ref><ce:e-address type="email" xlink:href="mailto:jacques.bloch@ur.de" id="ea0010">jacques.bloch@ur.de</ce:e-address></ce:author><ce:author id="au0020" author-id="S0550321323000366-fe5401542e27ee7a6c5d41c731520f95"><ce:given-name>Robert</ce:given-name><ce:surname>Lohmayer</ce:surname><ce:cross-ref refid="aff0010" id="crf0120"><ce:sup>a</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0020" id="crf0130"><ce:sup>b</ce:sup></ce:cross-ref><ce:e-address type="email" xlink:href="mailto:robert.lohmayer@ur.de" id="ea0020">robert.lohmayer@ur.de</ce:e-address></ce:author><ce:author id="au0030" author-id="S0550321323000366-1b6acdaa5f29277a4d5e446535dfa32c"><ce:given-name>Maximilian</ce:given-name><ce:surname>Meister</ce:surname><ce:cross-ref refid="aff0010" id="crf0140"><ce:sup>a</ce:sup></ce:cross-ref></ce:author><ce:author id="au0040" author-id="S0550321323000366-ec2957601dd286a31c9014e12efc3769"><ce:given-name>Michael</ce:given-name><ce:surname>Nunhofer</ce:surname><ce:cross-ref refid="aff0010" id="crf0150"><ce:sup>a</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="aff0010" affiliation-id="S0550321323000366-e3a23c135363bac74bbab5d41f20f508"><ce:label>a</ce:label><ce:textfn>Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany</ce:textfn><sa:affiliation><sa:organization>Institute for Theoretical Physics</sa:organization><sa:organization>University of Regensburg</sa:organization><sa:city>Regensburg</sa:city><sa:postal-code>93040</sa:postal-code><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0005">Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0020" affiliation-id="S0550321323000366-3425d587d16c5d27239eb2362996a3b9"><ce:label>b</ce:label><ce:textfn>Leibniz Institute for Immunotherapy (LIT), 93053 Regensburg, Germany</ce:textfn><sa:affiliation><sa:organization>Leibniz Institute for Immunotherapy (LIT)</sa:organization><sa:city>Regensburg</sa:city><sa:postal-code>93053</sa:postal-code><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0010">Leibniz Institute for Immunotherapy (LIT), 93053 Regensburg, Germany</ce:source-text></ce:affiliation></ce:author-group><ce:date-received day="5" month="10" year="2022"/><ce:date-revised day="12" month="1" year="2023"/><ce:date-accepted day="1" month="2" year="2023"/><ce:miscellaneous id="ms0010">Editor: Hubert Saleur</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0080">The higher-order tensor renormalization group is a tensor-network method providing estimates for the partition function and thermodynamical observables of classical and quantum systems in thermal equilibrium. At every step of the iterative blocking procedure, the coarse-grid tensor is truncated to keep the tensor dimension under control. For a consistent tensor blocking procedure, it is crucial that the forward and backward tensor modes are projected on the same lower dimensional subspaces. In this paper we present two methods, the SuperQ and the iterative SuperQ method, to construct tensor truncations that reduce or even minimize the local approximation errors, while satisfying this constraint.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0100">Data availability</ce:section-title><ce:para id="pr0590">Data will be made available on request.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">Physical systems in thermal equilibrium are described by their partition function, whose complexity grows exponentially in the volume. The standard method to simulate such statistical systems is the Markov chain Monte Carlo method (MC), which efficiently samples the relevant states of the system to produce reliable estimates of observables. A fundamental prerequisite for the MC method is the positivity of the sampling weights. Models which do not satisfy this condition cause the infamous sign problem and require alternative simulation methods. Quantum systems with complex actions are typical examples of systems with a sign problem. An important topical application in high energy physics is the simulation of quantum chromodynamics (QCD) at nonzero quark chemical potential, which allows for the investigation of the QCD phase diagram as a function of temperature and baryon density.</ce:para><ce:para id="pr0020">There exist numerous methods to circumvent the sign problem, and some even solve it for particular systems <ce:cross-refs refid="br0010 br0020 br0030" id="crs0010">[1–3]</ce:cross-refs>. Very mild sign problems can be circumvented by reweighting, which uses the Monte Carlo method on an auxiliary ensemble with positive weights, and reweights the observables to the target ensemble. The main issue with this method is that the statistical error increases exponentially with the volume such that it is hardly usable in any realistic situation, except for the validation of other methods in regions where the sign problem is small. Other methods which have shown their merit on some models, but are known to have fundamental problems for other ones, are the complex Langevin method, the thimbles, the density of states method and the method of dual variables, where the simulations are usually performed with the worm algorithm. Common to those methods is the stochastic sampling of the partition function.</ce:para><ce:para id="pr0030">An alternative approach that has recently drawn a lot of interest is that of tensor networks, see <ce:cross-ref refid="br0040" id="crf0160">[4]</ce:cross-ref> for a review. In these methods the partition function is first rewritten as a full contraction of a tensor network covering the entire lattice. The exact computation of the partition function and observables in this formulation would have an exponential complexity. The tensor renormalization group (TRG) <ce:cross-ref refid="br0050" id="crf0170">[5]</ce:cross-ref> and higher order tensor renormalization group (HOTRG) <ce:cross-ref refid="br0060" id="crf0180">[6]</ce:cross-ref> methods avoid this exponential cost by blocking the lattice iteratively and truncating the inflated dimensions of the coarse grid tensor at each blocking step using truncated higher order singular value decompositions (HOSVD) <ce:cross-ref refid="br0070" id="crf0190">[7]</ce:cross-ref>, which are based on the matrix singular value decomposition (SVD).</ce:para><ce:para id="pr0040">We consider the partition function of a <ce:italic>d</ce:italic>-dimensional classical or quantum system in thermal equilibrium, written as a fully contracted tensor network <ce:cross-ref refid="br0080" id="crf0200">[8]</ce:cross-ref>,<ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mrow><mml:mi>Z</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="normal">tTr</mml:mi><mml:mspace width="0.2em"/><mml:munderover><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>…</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> with a tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> at each site <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>x</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>V</mml:mi></mml:math>. In general the local tensor is the same on all sites, i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>T</mml:mi></mml:math> for all <ce:italic>x</ce:italic>. For each lattice direction <ce:italic>ν</ce:italic>, the tensor has one mode for the forward and one mode for the backward orientation, corresponding to the indices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo>,</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>, respectively, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> is a unit step in the <ce:italic>ν</ce:italic> direction. We will often refer to these modes as backward and forward modes of the physical tensor. The trace in the partition function stands for a full contraction over all tensor indices, where two adjacent tensors share exactly one index.</ce:para><ce:para id="pr0050">Thermodynamical observables, which are defined as derivatives of the partition function with respect to one of its parameters, can be computed using either a finite-difference approximation or an impurity tensor formulation involving the analytical derivative of <ce:italic>T</ce:italic> <ce:cross-ref refid="br0090" id="crf0210">[9]</ce:cross-ref>.</ce:para><ce:para id="pr0060">In the following we will restrict our discussion to HOTRG, because it can be applied to any number of dimensions, whereas TRG is limited to two-dimensional systems. The HOTRG method uses an iterative blocking procedure that reduces the size of the lattice by a factor of two during each blocking step by contracting pairs of adjacent tensors. The procedure is illustrated for a two-dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:mn>4</mml:mn><mml:mo>×</mml:mo><mml:mn>4</mml:mn></mml:math> lattice in <ce:cross-ref refid="fg0010" id="crf0220">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/>. Its extension to higher dimensions is obvious, and below we will further discuss the HOTRG method for the three-dimensional case.</ce:para><ce:para id="pr0070">When contracting two adjacent tensors <ce:italic>T</ce:italic> over their shared link, a tensor <ce:italic>M</ce:italic> of higher order is produced. Such a contraction in the 1-direction is illustrated for the three-dimensional case in <ce:cross-ref refid="fg0020" id="crf0230">Fig. 2</ce:cross-ref><ce:float-anchor refid="fg0020"/> and can be written as<ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:mi>y</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>x</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> and therefore <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>, by definition, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mi>X</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> labels sites on the coarse grid and<ce:display><ce:formula id="fm0030"><ce:label>(3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mrow><mml:mtable displaystyle="true" columnspacing="0.2em"><mml:mtr><mml:mtd columnalign="right"/><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mspace width="2em"/><mml:mspace width="0.2em"/><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"/><mml:mtd columnalign="left"><mml:mtable displaystyle="true" columnspacing="0.2em"><mml:mtr><mml:mtd columnalign="right"/><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mtd><mml:mtd columnalign="right"/><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"/><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mtd><mml:mtd columnalign="right"/><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mspace width="1em"/><mml:mo stretchy="true" maxsize="6.6ex" minsize="6.6ex">}</mml:mo><mml:mspace width="1em"/><mml:mrow><mml:mtext mathvariant="bold">fat</mml:mtext><mml:mspace width="0.25em"/><mml:mtext>indices</mml:mtext></mml:mrow><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></ce:formula></ce:display> For any direction perpendicular to the direction of contraction, the tensor <ce:italic>M</ce:italic> has modes originating from both contracted tensors. To keep the order of the tensor unchanged, we gather every such pair of modes in a new fat mode corresponding to its direct product space. Assuming that the modes of the local tensor have dimension <ce:italic>D</ce:italic>, then the fat modes will have dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. In HOTRG these fat modes are truncated back to dimension <ce:italic>D</ce:italic> using a modified version of the HOSVD approximation method, such that the dimension of the coarse grid tensor remains the same as that of the original local tensor throughout the entire blocking procedure.</ce:para><ce:para id="pr0080">In general, step <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>k</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn></mml:math> of the HOTRG procedure can be summarized as<ce:display><ce:formula id="fm0040"><ce:label>(4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>:</mml:mo><mml:mi>M</mml:mi><mml:mover accent="true"><mml:mrow><mml:mo stretchy="false">⟶</mml:mo></mml:mrow><mml:mrow><mml:mtext>truncate</mml:mtext></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>-operation symbolically represents a forward-backward contraction in direction <ce:italic>ν</ce:italic>. The precise construction of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup></mml:math> will be discussed in Sec. <ce:cross-ref refid="se0030" id="crf0240">3</ce:cross-ref>.</ce:para><ce:para id="pr0090">In the standard approximation procedure using HOSVD <ce:cross-ref refid="br0070" id="crf0250">[7]</ce:cross-ref>, referred to as <ce:italic>HOSVD approximation</ce:italic> in the following, the dimension of each tensor mode gets reduced by projecting it on a lower dimensional subspace, which is generically different for each mode. This HOSVD approximation is modified when used as part of the iterative blocking procedure in the standard HOTRG algorithm, as it is essential for the accuracy and effectiveness of the method that the backward and forward modes for every direction get projected on the same subspace. Each of these subspaces will be characterized by a frame, which is a set of orthonormal basis vectors spanning the subspace. Constructing appropriate frames will be the major subject of this paper.</ce:para><ce:para id="pr0100">The standard HOTRG procedure for the construction of frames <ce:cross-ref refid="br0060" id="crf0260">[6]</ce:cross-ref> is not optimal, in particular when the local tensor is not symmetric in its backward and forward modes. In this paper we present two improved methods for the construction of common subspaces for pairs of backward and forward modes: the <ce:italic>SuperQ</ce:italic> and the <ce:italic>iterative SuperQ</ce:italic> method (ISQ), which is an iterative improvement of the former in search of the optimal subspaces. Note that the discussion in this paper solely focuses on the optimization of the rank reduction of the local tensors at every blocking step, but does not take into account global effects on the full contraction of the tensor network.</ce:para><ce:para id="pr0110">Here is a brief outline of the paper. In Sec. <ce:cross-ref refid="se0020" id="crf0270">2</ce:cross-ref> we review the standard HOSVD method to construct a reduced rank approximation for an arbitrary tensor. In Sec. <ce:cross-ref refid="se0030" id="crf0280">3</ce:cross-ref> we explain why the HOTRG method uses a modification of this rank reduction procedure such that the backward and forward modes are projected on the same subspace. We then propose two methods to improve the standard HOTRG truncation: In Sec. <ce:cross-ref refid="se0040" id="crf0290">4</ce:cross-ref> we present the SuperQ method, and in Sec. <ce:cross-ref refid="se0050" id="crf0300">5</ce:cross-ref> we derive the more sophisticated ISQ method. Finally, we summarize and conclude in Sec. <ce:cross-ref refid="se0070" id="crf0310">7</ce:cross-ref>.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">Rank reduction and HOSVD approximation</ce:section-title><ce:para id="pr0120">Below we first review the general idea of rank reduction for an arbitrary tensor, before describing the HOSVD procedure <ce:cross-ref refid="br0070" id="crf0320">[7]</ce:cross-ref> which can be used to generate a quasi-optimal rank-reduced approximation in an efficient way.</ce:para><ce:para id="pr0130">For a real tensor <ce:italic>M</ce:italic> of order <ce:italic>n</ce:italic> with dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo>⋯</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>, the left <ce:italic>matrix-tensor multiplication</ce:italic> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mi>A</mml:mi><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mi>M</mml:mi></mml:math> is defined as a contraction of the second index of the matrix <ce:italic>A</ce:italic> with the <ce:italic>r</ce:italic>-th index of the tensor <ce:italic>M</ce:italic>,<ce:display><ce:formula id="fm0050"><ce:label>(5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>A</mml:mi><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0140">A lower-rank approximation of <ce:italic>M</ce:italic> can be constructed as<ce:display><ce:formula id="fm0060"><ce:label>(6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> projectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:math>. In this approximation, the <ce:italic>r</ce:italic>-th tensor mode of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> is projected onto a subspace of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math>, embedded in the original space. Typically the Frobenius norm <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:math> is used as a measure for the quality of the low-rank approximation and the aim is to determine optimal projectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> with fixed ranks <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math>.</ce:para><ce:para id="pr0150">The projectors can be represented as<ce:display><ce:formula id="fm0070"><ce:label>(7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow></mml:math></ce:formula></ce:display> with semi-orthogonal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, which we will call <ce:italic>frames</ce:italic> in the following. Semi-orthogonal means that the columns of the frames are orthonormal, but not their rows (unless <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math>). The columns of each frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> provide an orthonormal basis of the corresponding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math>-dimensional subspace,<ce:display><ce:formula id="fm0080"><ce:label>(8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mn mathvariant="double-struck">1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0160">The approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> can then be rewritten as<ce:display><ce:formula id="fm0090"><ce:label>(9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mi>S</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo>…</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> dimensional <ce:italic>core tensor</ce:italic><ce:display><ce:formula id="fm0100"><ce:label>(10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mrow><mml:mi>S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> The core tensor represents <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> in the bases of the subspaces, which are spanned by the columns of the frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. Note that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> has the same dimension as <ce:italic>M</ce:italic>, but is generically of lower rank as the <ce:italic>r</ce:italic>-th mode is projected from a space of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> on a subspace of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:math>. In practice, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> is usually not constructed explicitly, as many operations involving <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> can be performed at much lower cost using only the core tensor <ce:italic>S</ce:italic> and the frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, which contain the same information as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> but condensed in lower-dimensional objects. A typical example of such an operation is the contraction of two tensors, as will be discussed in the next section.</ce:para><ce:para id="pr0170">The squared Frobenius norm of <ce:italic>M</ce:italic> is given by<ce:display><ce:formula id="fm0110"><ce:label>(11)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where for brevity we introduce the notation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> for the summation indices, and the inner product between two real tensors of equal dimension is defined as<ce:display><ce:formula id="fm0120"><ce:label>(12)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:mrow><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0180">Since the reduced-rank tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> is a projection of <ce:italic>M</ce:italic>, we have<ce:display><ce:formula id="fm0130"><ce:label>(13)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mrow><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Therefore, the squared approximation error is given by<ce:display><ce:formula id="fm0140"><ce:label>(14)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> which does not require the explicit computation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math>.</ce:para><ce:para id="pr0190">In the HOSVD approximation procedure <ce:cross-ref refid="br0070" id="crf0330">[7]</ce:cross-ref> the semi-orthogonal frames used to approximate <ce:italic>M</ce:italic> are constructed using properties of matrix SVDs. We first introduce the <ce:italic>r</ce:italic>-unfolding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, which is a matrix containing the same entries as the tensor <ce:italic>M</ce:italic>, but reordered such that its rows correspond to the <ce:italic>r</ce:italic>-th mode of <ce:italic>M</ce:italic> and its columns correspond to a combination of all other tensor modes. The entries of the <ce:italic>r</ce:italic>-unfolding of <ce:italic>M</ce:italic> are given by<ce:display><ce:formula id="fm0150"><ce:label>(15)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where the column index <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math> is a linear index of coordinates in a space of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:msub><mml:mrow><mml:mo>∏</mml:mo></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>≠</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math>. The multi-rank of a tensor is defined by the <ce:italic>n</ce:italic>-tuple of the ranks of the individual unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:math>. Therefore <ce:italic>M</ce:italic> has at most multi-rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math> and the approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> of <ce:cross-ref refid="fm0060" id="crf0340">(6)</ce:cross-ref> at most multi-rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>. Note that the squared Frobenius norm <ce:cross-ref refid="fm0110" id="crf0350">(11)</ce:cross-ref> of a tensor is identical to that of any of its unfoldings, as it is just a sum over all squared components,<ce:display><ce:formula id="fm0160"><ce:label>(16)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow></mml:mrow></mml:math></ce:formula></ce:display> for any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:math>. To construct the HOSVD approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math>, we first consider the singular value decomposition for each <ce:italic>r</ce:italic>-unfolding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of <ce:italic>M</ce:italic> (assuming real tensors for simplicity),<ce:display><ce:formula id="fm0170"><ce:label>(17)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">Σ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where the columns of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> orthogonal matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> are the left singular vectors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. The columns of the orthogonal matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> contain the corresponding right singular vectors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. The diagonal entries of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">Σ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> are the singular values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, which are always real and non-negative, while all other entries are zero.</ce:para><ce:para id="pr0200">It is well-known in linear algebra that retaining the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> largest singular values in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">Σ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, while setting all others to zero, yields the best-possible matrix approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> (best-possible referring to a minimization of the Frobenius norm <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mo stretchy="false">‖</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">‖</mml:mo></mml:math>). The relative truncation error is given by<ce:display><ce:formula id="fm0180"><ce:label>(18)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">‖</mml:mo></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> are the eigenvalues of the Gramian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:math>, i.e., the squared singular values of the unfolding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, ordered such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>≥</mml:mo><mml:mo>…</mml:mo><mml:mo>≥</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math>.</ce:para><ce:para id="pr0210">This matrix property is used in HOSVD by separately performing the matrix SVDs of all individual unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of <ce:italic>M</ce:italic> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:math> and constructing the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> with the singular vectors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> corresponding to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> largest singular values of the unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. These frames are then used to construct the core tensor <ce:cross-ref refid="fm0100" id="crf0360">(10)</ce:cross-ref> and the matrix approximation <ce:cross-ref refid="fm0090" id="crf0370">(9)</ce:cross-ref> of HOSVD. Unlike for the matrix case, the HOSVD tensor approximation is in general not the best-possible approximation of a given multi-rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>, even though it is usually quite close to it <ce:cross-ref refid="br0070" id="crf0380">[7]</ce:cross-ref>.</ce:para><ce:para id="pr0220">In a variant of the HOSVD approximation, called interlaced HOSVD approximation, the rank reduction procedure is carried out in the following way: Starting with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:mi>S</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>M</mml:mi></mml:math>, the frames are computed on successive unfoldings of the core tensor, which gets updated every time a new truncation frame is constructed until the core tensor is of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo>…</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>. For the interlaced HOSVD approximation, the result depends on the order of the updates, but is usually close to that of the ordinary HOSVD approximation.</ce:para><ce:para id="pr0230">The best-possible approximation of multi-rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>, which minimizes the Frobenius norm <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:math>, can be constructed numerically using the Higher Order Orthogonal Iteration (HOOI) <ce:cross-ref refid="br0100" id="crf0390">[10]</ce:cross-ref>. Nevertheless, the HOSVD approximation is especially appealing because of its relative simplicity to produce an almost optimal approximation.</ce:para></ce:section><ce:section id="se0030"><ce:label>3</ce:label><ce:section-title id="st0040">Backward-forward symmetric truncation in HOTRG</ce:section-title><ce:para id="pr0240">We now discuss how the HOSVD formalism is used in HOTRG to avoid the exponential blow up of the tensor dimension during the blocking procedure, and why the standard HOSVD truncation is modified to avoid drawbacks related to accuracy and efficiency. To make our point we will use the two-dimensional case as it can be easiest illustrated and contains all the ingredients necessary for the discussion. Extending it to higher dimensions is straightforward.</ce:para><ce:para id="pr0250">We consider the contraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mi>M</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>T</mml:mi></mml:math> of two local tensors <ce:italic>T</ce:italic> along the 2-direction. According to the discussion in the introduction, <ce:italic>M</ce:italic> will have thin backward and forward modes of dimension <ce:italic>D</ce:italic> in the contracted 2-direction, and fat backward and forward modes of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> in the perpendicular 1-direction, which we want to reduce to lower rank by projecting on a <ce:italic>D</ce:italic>-dimensional subspace using <ce:cross-ref refid="fm0060" id="crf0400">(6)</ce:cross-ref>. This procedure of contraction and truncation, which we detail below, is illustrated in <ce:cross-ref refid="fg0030" id="crf0410">Fig. 3</ce:cross-ref><ce:float-anchor refid="fg0030"/>. To reduce the dimension of the fat modes back from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> to <ce:italic>D</ce:italic>, while minimizing the loss of information, we apply the HOSVD approximation procedure, explained in Sec. <ce:cross-ref refid="se0020" id="crf0420">2</ce:cross-ref>, where we only truncate the fat modes. The SVDs are computed for the unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> for the backward and forward modes in the 1-direction, respectively, and the frames <ce:italic>U</ce:italic> and <ce:italic>V</ce:italic> of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:mi>D</mml:mi></mml:math> are constructed with the singular vectors corresponding to their <ce:italic>D</ce:italic> largest singular values. For the modes in the contracted 2-direction no truncation is required. With these frames we construct a core tensor <ce:italic>S</ce:italic> of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:mi>D</mml:mi><mml:mo>×</mml:mo><mml:mi>D</mml:mi><mml:mo>×</mml:mo><mml:mi>D</mml:mi><mml:mo>×</mml:mo><mml:mi>D</mml:mi></mml:math>, according to <ce:cross-ref refid="fm0100" id="crf0430">(10)</ce:cross-ref>,<ce:display><ce:formula id="fm0190"><ce:label>(19)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mrow><mml:mi>S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> The corresponding approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math>, defined in <ce:cross-ref refid="fm0090" id="crf0440">(9)</ce:cross-ref>, with the same dimension as <ce:italic>M</ce:italic>, but typically much lower rank, is given by<ce:display><ce:formula id="fm0200"><ce:label>(20)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>U</mml:mi><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>V</mml:mi><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>S</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>U</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> projectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>U</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>U</mml:mi><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>V</mml:mi><mml:msup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math>, with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mi>U</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mi>V</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mn mathvariant="double-struck">1</mml:mn></mml:mrow><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:math>. As mentioned in Sec. <ce:cross-ref refid="se0020" id="crf0450">2</ce:cross-ref>, operations involving <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> can typically be performed at much lower cost by using only the core tensor <ce:italic>S</ce:italic> and the frames <ce:italic>U</ce:italic> and <ce:italic>V</ce:italic>.</ce:para><ce:para id="pr0260">Assume that in the next blocking step two <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> tensors are contracted in the 1-direction, as is illustrated in <ce:cross-ref refid="fg0040" id="crf0460">Fig. 4</ce:cross-ref><ce:float-anchor refid="fg0040"/>. When using the standard HOSVD approximation <ce:cross-ref refid="fm0200" id="crf0470">(20)</ce:cross-ref> the backward and forward modes in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> will have been projected on different subspaces using the projectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>U</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:msub></mml:math>, respectively. In a contraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> the two projectors will be multiplied, as can be seen in the center of the figure. The decomposition of the approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> in <ce:cross-ref refid="fm0200" id="crf0480">(20)</ce:cross-ref> can be used to reduce the computational effort, as the original contraction can be replaced by contractions of two core tensors <ce:italic>S</ce:italic> with a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:mi>D</mml:mi><mml:mo>×</mml:mo><mml:mi>D</mml:mi></mml:math> dimensional merger <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si79.svg"><mml:mi>G</mml:mi><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mi>V</mml:mi></mml:math> in between, as can be seen in the figure.<ce:cross-ref refid="fn0010" id="crf0490"><ce:sup>1</ce:sup></ce:cross-ref><ce:footnote id="fn0010"><ce:label>1</ce:label><ce:note-para id="np0010">In fact this produces an amputated version of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math>, which together with the mergers is all we need in the full contraction of the tensor network, see also <ce:cross-ref refid="fg0040" id="crf0500">Fig. 4</ce:cross-ref>.</ce:note-para></ce:footnote> The entries of <ce:italic>G</ce:italic> are scalar products of the basis vectors in <ce:italic>U</ce:italic> and <ce:italic>V</ce:italic>.</ce:para><ce:para id="pr0270">At this point it is important to discuss a crucial modification introduced by the HOTRG method to the HOSVD truncation procedure presented above, which is rarely discussed in the literature. Although the HOSVD approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> gives a close-to-best lower-rank approximation to <ce:italic>M</ce:italic>, it is in general not such a good and useful truncation when viewed as part of the iterative blocking procedure. Indeed, the product of projectors corresponds to a projection of a projection, which will unavoidably loose additional information if the projectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>U</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:msub></mml:math> are different. In this case, the contraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> will no longer necessarily be a good approximation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:mi>M</mml:mi><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>M</mml:mi></mml:math>, even if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> itself was close to the best-possible approximation of <ce:italic>M</ce:italic>.</ce:para><ce:para id="pr0280">We now observe that, due to the idempotence of projectors, there would be no additional loss if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si81.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>U</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:msub></mml:math>, i.e., if the backward and forward modes of <ce:italic>M</ce:italic> in the 1-direction were projected on the same subspace. Note that in this case the merger <ce:italic>G</ce:italic> is an orthogonal matrix. Moreover, we can also get a serious gain in algorithmic simplicity, on top of this accuracy improvement, if we choose the same basis for both modes in the common subspace, i.e., we choose frames satisfying <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si82.svg"><mml:mi>U</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>V</mml:mi></mml:math>, for which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi>G</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mn mathvariant="double-struck">1</mml:mn></mml:mrow><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:math>. When looking back at <ce:cross-ref refid="fg0040" id="crf0510">Fig. 4</ce:cross-ref> we see that, in this case, the central merger just drops out, and the contraction can be replaced by a contraction of two core tensors, as is illustrated in <ce:cross-ref refid="fg0050" id="crf0520">Fig. 5</ce:cross-ref><ce:float-anchor refid="fg0050"/> (the frames on the left and right of <ce:cross-ref refid="fg0040" id="crf0530">Fig. 4</ce:cross-ref> will connect to their counter parts in further contractions, to form another merger <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi>G</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mn mathvariant="double-struck">1</mml:mn></mml:mrow><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:math>, which will also drop out).</ce:para><ce:para id="pr0290">For this reason, in HOTRG the backward and forward modes of each direction are truncated using a common frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si166.svg"><mml:mi>ν</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:math>, even when the HOSVD frames are different, which is typically the case for systems at nonzero chemical potential. With this backward-forward symmetric truncation, the core tensor <ce:italic>S</ce:italic> can be used as new coarse grid tensor after each blocking step, where at the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si86.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>-th blocking step two tensors of step <ce:italic>k</ce:italic> are contracted to form a new coarse grid tensor,<ce:display><ce:formula id="fm0210"><ce:label>(21)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>:</mml:mo><mml:mi>M</mml:mi><mml:mover accent="true"><mml:mrow><mml:mo stretchy="false">⟶</mml:mo></mml:mrow><mml:mrow><mml:mtext>BF</mml:mtext></mml:mrow></mml:mover><mml:mi>S</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>:</mml:mo><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> The abbreviation BF on the arrow means that we apply a backward-forward symmetric truncation to construct the core tensor, which then becomes the new local tensor on the coarse grid. The frames are only needed to construct the core tensor with <ce:cross-ref refid="fm0190" id="crf0540">(19)</ce:cross-ref>, and can then be discarded.</ce:para><ce:para id="pr0300">The same reasoning also holds for a contraction in the other direction, where the directions of thin and fat modes are interchanged. Moreover, the procedure naturally generalizes to <ce:italic>d</ce:italic> dimensions, where we have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:mi>d</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:math> directions <ce:italic>ν</ce:italic> with backward and forward fat modes: If the backward and forward frames are chosen to be identical for each direction, the core tensor <ce:cross-ref refid="fm0100" id="crf0550">(10)</ce:cross-ref> can be used as the new coarse grid tensor in the HOTRG blocking procedure.</ce:para><ce:para id="pr0310">This strategy of choosing the same frame to truncate the backward and forward mode for each individual direction in HOTRG, makes it fundamentally different from the HOSVD approximation, as it can no longer directly rely on the optimal low-rank properties of matrix SVD. Geometrically, truncating the backward and forward modes with the same frame means that these modes get projected on the same subspace and are described in the same basis. This explains why the full contraction of the tensor network into a scalar can be rewritten in terms of the core tensors only.</ce:para><ce:para id="pr0320">Note that if one would use the standard HOSVD approximation procedure and work with different backward and forward frames, we would have to use both the core tensors and the mergers <ce:italic>G</ce:italic> defined above when performing the iterative contractions, to ensure that the different subspaces are matched onto one another, see <ce:cross-ref refid="fg0040" id="crf0560">Fig. 4</ce:cross-ref>. Although this is no conceptual problem, it would complicate the algorithm, require additional computational work, and most of all the product of projectors would deteriorate the results further.</ce:para><ce:para id="pr0330">The construction of the shared backward-forward frames in HOTRG has not been given a lot of attention in the literature until now. There is a brief discussion of this issue in the original HOTRG paper <ce:cross-ref refid="br0060" id="crf0570">[6]</ce:cross-ref>, where either the backward or forward frame is chosen and applied to both modes, depending on which one gives the smallest SVD truncation error. The error introduced by this choice on the other mode is however not taken into account. We observed that for tensors lacking a backward-forward symmetry, this choice of frame is not optimal and can be improved upon.</ce:para><ce:para id="pr0340">Below we present two new methods to improve the construction of shared frames for the backward and forward modes. The first one, called SuperQ method and presented in Sec. <ce:cross-ref refid="se0040" id="crf0580">4</ce:cross-ref>, minimizes a combined error on the backward and forward unfoldings for each individual direction. The second method, which we call iterative SuperQ (ISQ) method is presented in Sec. <ce:cross-ref refid="se0050" id="crf0590">5</ce:cross-ref>. This iterative method aims at determining the best-possible approximation to <ce:italic>M</ce:italic> for a given multi-rank, satisfying the requirement that the backward and forward frames for each direction are identical. The ISQ method leans on ideas developed for the higher order orthogonal iteration (HOOI) method <ce:cross-ref refid="br0100" id="crf0600">[10]</ce:cross-ref>, which constructs the best-possible approximation of a given multi-rank with independent frames for all modes. We will see that the SuperQ solution can be used as a natural starting point for the ISQ procedure. Note that the SuperQ and ISQ methods are specifically conceived for tensors which are part of a physical tensor network on a space-time lattice and have modes corresponding to backward and forward orientations.</ce:para></ce:section><ce:section id="se0040"><ce:label>4</ce:label><ce:section-title id="st0050">The SuperQ method</ce:section-title><ce:para id="pr0350">To discuss the construction of truncations satisfying the requirement that the frames for the backward and forward modes are identical, we consider a tensor <ce:italic>M</ce:italic> with <ce:italic>d</ce:italic> pairs of backward and forward modes. The tensor is thus of order 2<ce:italic>d</ce:italic> with dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si89.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo>⋯</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:math>, which will be truncated to dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si90.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo>⋯</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:math> using <ce:italic>d</ce:italic> semi-orthogonal frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>. With these frames the core tensor is constructed using<ce:display><ce:formula id="fm0220"><ce:label>(22)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si92.svg"><mml:mrow><mml:mi>S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>d</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Consider the positive semi-definite Gram matrices<ce:display><ce:formula id="fm0230"><ce:label>(23)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si93.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si104.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> are the unfoldings of <ce:italic>M</ce:italic> with respect to the backward and forward modes for direction <ce:italic>ν</ce:italic>, i.e.,<ce:display><ce:formula id="fm0240"><ce:label>(24)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> is the <ce:italic>r</ce:italic>-unfolding of the tensor <ce:italic>M</ce:italic> defined in <ce:cross-ref refid="fm0150" id="crf0610">(15)</ce:cross-ref>. We denote the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> frames constructed with the eigenvectors corresponding to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> largest eigenvalues of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si99.svg"><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si100.svg"><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math>, respectively. As the backward and forward Gramians <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si99.svg"><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si100.svg"><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> are in general not identical, the corresponding subspaces spanned by the vectors of the frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> will be different too.<ce:cross-ref refid="fn0020" id="crf0620"><ce:sup>2</ce:sup></ce:cross-ref><ce:footnote id="fn0020"><ce:label>2</ce:label><ce:note-para id="np0020">This can even be the case if the eigenvalues of both Gramians coincide, as we have observed for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si103.svg"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> model with chemical potential.</ce:note-para></ce:footnote> In the standard HOTRG procedure <ce:cross-ref refid="br0060" id="crf0630">[6]</ce:cross-ref> it is suggested to choose either <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> for the unique <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, depending which of both gives the smallest SVD truncation error <ce:cross-ref refid="fm0180" id="crf0640">(18)</ce:cross-ref>. Even though this choice of frame optimizes the truncation error for one mode, it does not take into account its effect on the mode corresponding to the opposite orientation. Therefore, it is clear that, generically, better choices of frames should exist, and our aim is to construct frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> that reduce the combined truncation error when applied simultaneously to the backward and forward modes for the <ce:italic>ν</ce:italic> direction.</ce:para><ce:para id="pr0360">Let us now consider a single truncation frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> which we use to reduce the rank of the unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si104.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math>. Using <ce:cross-ref refid="fm0140" id="crf0650">(14)</ce:cross-ref> and <ce:cross-ref refid="fm0160" id="crf0660">(16)</ce:cross-ref>, the relative truncation errors on the backward and forward unfoldings are<ce:display><ce:formula id="fm0250"><ce:label>(25)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si106.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">‖</mml:mo></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0260"><ce:label>(26)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si107.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">‖</mml:mo></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si108.svg"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si109.svg"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> are the rank-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> approximations to the unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si104.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math>, respectively, obtained with the same frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>.</ce:para><ce:para id="pr0370">To improve upon using either <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math>, we determine the common <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> by minimizing the combination of both errors in<ce:display><ce:formula id="fm0270"><ce:label>(27)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si112.svg"><mml:mrow><mml:msup><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>S</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where we also used <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si113.svg"><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. We define the SuperQ matrix for direction <ce:italic>ν</ce:italic> as<ce:display><ce:formula id="fm0280"><ce:label>(28)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si114.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>S</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> which is symmetric and positive semi-definite as it is a sum of two symmetric positive semi-definite matrices. Therefore, if we diagonalize the SuperQ matrix and truncate the eigenvector matrix, retaining the eigenvectors corresponding to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> largest eigenvalues, then this semi-orthogonal frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> minimizes the truncation error <ce:cross-ref refid="fm0270" id="crf0670">(27)</ce:cross-ref> on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si115.svg"><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>S</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math>. This SuperQ procedure is repeated on all <ce:italic>d</ce:italic> directions to determine all frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, which can then be used to approximate <ce:italic>M</ce:italic> and to construct the corresponding core tensor <ce:italic>S</ce:italic>, see <ce:cross-ref refid="fm0220" id="crf0680">(22)</ce:cross-ref>.</ce:para><ce:para id="pr0380">The SuperQ method is computationally efficient since it only requires a single eigenvalue decomposition for each pair of backward and forward fat modes, while the standard HOTRG procedure <ce:cross-ref refid="br0060" id="crf0690">[6]</ce:cross-ref> performs separate decompositions on these modes.</ce:para><ce:para id="pr0390">When applying the SuperQ method to HOTRG, where <ce:italic>M</ce:italic> is a contraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si116.svg"><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>T</mml:mi></mml:math> along one of the directions, only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si117.svg"><mml:mn>2</mml:mn><mml:mi>d</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:math> modes will actually be truncated, as the backward and forward modes for the contracted direction need not be truncated.</ce:para><ce:para id="pr0400">In analogy to the interlaced HOSVD approximation, see Sec. <ce:cross-ref refid="se0020" id="crf0700">2</ce:cross-ref>, we can also define an interlaced version of the SuperQ method where we determine the frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> by applying the method to an intermediate core tensor, which gets updated direction-by-direction (starting from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:mi>S</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>M</mml:mi></mml:math>) by truncating the respective backward and forward mode each time a frame has been computed. This interlaced SuperQ method is also of interest in the light of the iterative procedure derived in the next section.</ce:para></ce:section><ce:section id="se0050"><ce:label>5</ce:label><ce:section-title id="st0060">Optimized frames with iterative SuperQ</ce:section-title><ce:para id="pr0410">Although the HOSVD method, see Sec. <ce:cross-ref refid="se0020" id="crf0710">2</ce:cross-ref>, typically yields a good tensor approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> <ce:cross-ref refid="br0070" id="crf0720">[7]</ce:cross-ref>, the best-possible one, which minimizes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si118.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, can be constructed with an iterative procedure called higher order orthogonal iteration (HOOI) method <ce:cross-ref refid="br0100" id="crf0730">[10]</ce:cross-ref>.</ce:para><ce:para id="pr0420">According to the discussion of the backward-forward symmetric truncation in Sec. <ce:cross-ref refid="se0030" id="crf0740">3</ce:cross-ref>, it is clear that HOOI is itself not applicable in a tensor network approach to statistical physics, because the backward and forward modes should be truncated with the same semi-orthogonal frame for each direction, while HOOI very generically generates different frames for all modes. Below we present the iterative SuperQ (ISQ) method, which is inspired by the original HOOI procedure but imposes the requirement that the same frame has to be used to truncate the backward and forward modes of each direction.</ce:para><ce:para id="pr0430">As in Sec. <ce:cross-ref refid="se0040" id="crf0750">4</ce:cross-ref>, we consider a tensor <ce:italic>M</ce:italic> with <ce:italic>d</ce:italic> pairs of backward and forward modes, i.e., the tensor is of order 2<ce:italic>d</ce:italic> with dimensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si119.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:math>, which will be truncated to dimensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si90.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo>⋯</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:math>, see <ce:cross-ref refid="fm0220" id="crf0760">(22)</ce:cross-ref>. Our aim is to minimize the squared Frobenius norm <ce:cross-ref refid="fm0140" id="crf0770">(14)</ce:cross-ref><ce:display><ce:formula id="fm0290"><ce:label>(29)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> over all semi-orthogonal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:mi>ν</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>…</mml:mo><mml:mi>d</mml:mi></mml:math>, for fixed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>, with the additional condition that the backward and forward modes for each direction <ce:italic>ν</ce:italic> are truncated with the same frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>.</ce:para><ce:para id="pr0440">The semi-orthogonality of the frames is imposed explicitly by orthonormality conditions for the column vectors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>,<ce:display><ce:formula id="fm0300"><ce:label>(30)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si122.svg"><mml:mrow><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mtext>for </mml:mtext><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mtext> and </mml:mtext><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>ν</mml:mi><mml:mo>≤</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> in the constrained minimization of <ce:cross-ref refid="fm0290" id="crf0780">(29)</ce:cross-ref>. This leads to the cost function<ce:display><ce:formula id="fm0310"><ce:label>(31)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si123.svg"><mml:mi>g</mml:mi><mml:mo id="mmlbr0001" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>C</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>d</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mspace linebreak="newline"/><mml:mspace width="1em"/><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:munderover><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn mathvariant="double-struck">1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> with matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si124.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> containing the Lagrange multipliers. The orthonormalization conditions are symmetric under the exchange <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:mi>a</mml:mi><mml:mo stretchy="false">↔</mml:mo><mml:mi>b</mml:mi></mml:math>, and so <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si124.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> will be symmetric too. If we diagonalize <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si126.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mtext>diag</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, redefine <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si127.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si128.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mtext>diag</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, and use the orthogonality of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si129.svg"><mml:msup><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, then Eq. <ce:cross-ref refid="fm0310" id="crf0790">(31)</ce:cross-ref> remains unaltered albeit now with diagonal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mtext>diag</mml:mtext><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:math>, and this without loss of generality. Written out in components this is<ce:display><ce:formula id="fm0320"><ce:label>(32)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:mi>g</mml:mi><mml:mo id="mmlbr0002" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:munderover><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>c</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="before">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munderover><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:munderover><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>c</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> For a constrained maximum of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, the partial derivative of <ce:italic>g</ce:italic> with respect to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si133.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-entry of the <ce:italic>ν</ce:italic>-th orthogonal frame has to satisfy<ce:display><ce:formula id="fm0330"><ce:label>(33)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si134.svg"><mml:mrow><mml:mn>0</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munderover><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si135.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>k</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si136.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>c</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>ν</mml:mi><mml:mo>≤</mml:mo><mml:mi>d</mml:mi></mml:math>. Note that the same frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> appears twice in <ce:italic>S</ce:italic>, as it is used to truncate the modes in the backward and forward <ce:italic>ν</ce:italic> direction. We therefore obtain<ce:display><ce:formula id="fm0340"><ce:label>(34)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si138.svg"><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true" id="mmlbr0003">(</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munder><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0003" linebreakstyle="before">+</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munder><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> After eliminating the Kronecker deltas we get<ce:display><ce:formula id="fm0350"><ce:label>(35)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si139.svg"><mml:munder id="mmlbr0004"><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:mi>c</mml:mi><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munder><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:mi>k</mml:mi><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0004" linebreakstyle="before">+</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>c</mml:mi><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munder><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>k</mml:mi><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Let us define the unfolding matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> with dimensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si142.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>⋯</mml:mo><mml:msubsup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>⋯</mml:mo><mml:msubsup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math>, where all directions of <ce:italic>M</ce:italic> are truncated, except for the backward-<ce:italic>ν</ce:italic> mode for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, and the forward-<ce:italic>ν</ce:italic> mode for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, and the unfolding is performed with respect to the untruncated mode,<ce:display><ce:formula id="fm0360"><ce:label>(36)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si143.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munder><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0370"><ce:label>(37)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si144.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munder><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where we used the notation introduced in <ce:cross-ref refid="fm0150" id="crf0800">(15)</ce:cross-ref> for the matrix indices. The core tensor can also be written in terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> by truncating the remaining untruncated index:<ce:display><ce:formula id="fm0380"><ce:label>(38)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si145.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> After substituting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> in <ce:cross-ref refid="fm0350" id="crf0810">(35)</ce:cross-ref> we obtain<ce:display><ce:formula id="fm0390"><ce:label>(39)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:munder id="mmlbr0005"><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mspace linebreak="newline"/><mml:mspace width="1em"/><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> or<ce:display><ce:formula id="fm0400"><ce:label>(40)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si147.svg"><mml:mrow><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> If we introduce the positive semi-definite matrices<ce:display><ce:formula id="fm0410"><ce:label>(41)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si148.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mspace width="2em"/><mml:mi>ν</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> we can identify <ce:cross-ref refid="fm0400" id="crf0820">(40)</ce:cross-ref> as a coupled nonlinear eigenvalue problem (which is nonlinear in the eigenvectors)<ce:display><ce:formula id="fm0420"><ce:label>(42)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si149.svg"><mml:mrow><mml:mphantom><mml:mspace width="2em"/><mml:mspace width="2em"/><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>…</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo></mml:mphantom><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mspace width="2em"/><mml:mi>ν</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> for the semi-orthogonal frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>-dimensional diagonal matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si124.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. Note that all frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si151.svg"><mml:mi>μ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>…</mml:mo><mml:mi>d</mml:mi></mml:math>, appear in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si152.svg"><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> (as projections <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si153.svg"><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:math>). This is even true for the direction <ce:italic>ν</ce:italic> itself, as its projector is applied to the forward mode in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si154.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:math> and to the backward mode in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si155.svg"><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:math>. Therefore the <ce:italic>ν</ce:italic>-th equation is cubic in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and quartic in all other <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si156.svg"><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:math>. It is crucial to keep in mind that the self-consistent solutions to <ce:cross-ref refid="fm0420" id="crf0830">(42)</ce:cross-ref>, which we are looking for, are required to be semi-orthogonal matrices with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si157.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> columns, in order to satisfy the constraints <ce:cross-ref refid="fm0300" id="crf0840">(30)</ce:cross-ref>.</ce:para><ce:para id="pr0450">It is useful to note that if we replace the matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si152.svg"><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> in <ce:cross-ref refid="fm0420" id="crf0850">(42)</ce:cross-ref> by fixed matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, the <ce:italic>d</ce:italic> matrix equations decouple and each one of them is a linear eigenvalue equation for the frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. The solutions of these linearized equations are however in general no solution of the original nonlinear equations <ce:cross-ref refid="fm0420" id="crf0860">(42)</ce:cross-ref>. On the other hand, if <ce:cross-ref refid="fm0420" id="crf0870">(42)</ce:cross-ref> is satisfied, then <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> is a solution of the linear eigenvalue problem for the specific matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si159.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>.</ce:para><ce:para id="pr0460">Therefore, we propose to solve the coupled system of equations using an iterative procedure, where at each iteration step, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> is computed with <ce:cross-ref refid="fm0410" id="crf0880">(41)</ce:cross-ref> using the current frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>μ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:math>, and the eigenvalue problem<ce:display><ce:formula id="fm0430"><ce:label>(43)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si161.svg"><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> is solved for the eigenvalues <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si162.svg"><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and eigenvectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si163.svg"><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of the positive semi-definite matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. We then take the normalized eigenvectors corresponding to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> largest eigenvalues to form a new frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. Note that the eigenvectors of the symmetric matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> are orthogonal and therefore the constraint <ce:cross-ref refid="fm0300" id="crf0890">(30)</ce:cross-ref> is automatically satisfied for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>.</ce:para><ce:para id="pr0470">This strategy can be motivated in the following way. We are looking for the solution of <ce:cross-ref refid="fm0420" id="crf0900">(42)</ce:cross-ref> which maximizes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. Using the expression <ce:cross-ref refid="fm0380" id="crf0910">(38)</ce:cross-ref> for the core tensor <ce:italic>S</ce:italic> in terms of the unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, the squared norm in the cost function <ce:cross-ref refid="fm0310" id="crf0920">(31)</ce:cross-ref> can be written as<ce:display><ce:formula id="fm0440"><ce:label>(44)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si164.svg"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi mathvariant="normal">Tr</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">]</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="normal">Tr</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">]</mml:mo><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">Tr</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">]</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> which explicitly contains the matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si165.svg"><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> defined in <ce:cross-ref refid="fm0410" id="crf0930">(41)</ce:cross-ref> (note that this equation yields the same <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> for each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si166.svg"><mml:mi>ν</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:math>). It is straightforward to show that, at each iteration step, the solution obtained from the linearization <ce:cross-ref refid="fm0430" id="crf0940">(43)</ce:cross-ref> corresponds to the constrained optimization of the approximation<ce:display><ce:formula id="fm0450"><ce:label>(45)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si167.svg"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>≈</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">Tr</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> to <ce:cross-ref refid="fm0440" id="crf0950">(44)</ce:cross-ref>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> is fixed and computed with the most recent frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>μ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:math>. As this approximation is quadratic in the new frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, its maximum will be given by the eigenvectors corresponding to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> largest eigenvalues of the symmetric, positive semi-definite matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>.</ce:para><ce:para id="pr0480">The iterative procedure can be interpreted as an iterative SuperQ method, where at each step all modes of <ce:italic>M</ce:italic> are truncated using the last known frames, except for a backward mode in <ce:italic>B</ce:italic> and the corresponding forward mode in <ce:italic>F</ce:italic>.</ce:para><ce:para id="pr0490">During the iterative procedure we cycle through the <ce:italic>d</ce:italic> dimensions <ce:italic>ν</ce:italic> and determine a new frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> at each step using <ce:cross-ref refid="fm0430" id="crf0960">(43)</ce:cross-ref>. Then, we repeat these <ce:italic>d</ce:italic> iteration steps until all frames have converged. In practice we observed that the first iteration for each direction is the most important one, and further iterations of the same direction only give small corrections.</ce:para><ce:para id="pr0500">An alternative procedure would be to iterate <ce:cross-ref refid="fm0430" id="crf0970">(43)</ce:cross-ref> for a single <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> (keeping all other <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> fixed) until convergence has been reached (reevaluating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> with the most recent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> at every step), and then go on to the next frame. Once all frames have been iterated, this whole procedure is repeated until all frames converge together. Note that the inner iterations are computationally cheap, as all frames but one are kept fixed and all matrix-tensor multiplications in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si168.svg"><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> involving these fixed frames have to be computed only once. However, this procedure does not seem to give an overall faster convergence.</ce:para><ce:para id="pr0510">A natural choice for the starting frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:mi>ν</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>…</mml:mo><mml:mi>d</mml:mi></mml:math>, in the iterative procedure are the frames obtained from the interlaced SuperQ method, see Sec. <ce:cross-ref refid="se0040" id="crf0980">4</ce:cross-ref>.</ce:para><ce:para id="pr0520">Note that the iterative procedure is not guaranteed to converge, and even when it does, the solution is not necessarily the global maximum. This can be improved upon by tuning the starting frames or by applying an under-relaxation procedure to the intermediate <ce:italic>Q</ce:italic> matrix. In this procedure, we replace <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> in <ce:cross-ref refid="fm0430" id="crf0990">(43)</ce:cross-ref> by<ce:display><ce:formula id="fm0460"><ce:label>(46)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si169.svg"><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mo>‾</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>ω</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>ω</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mtext>prev</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si170.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mtext>prev</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> was used to obtain the previous <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> in the iterative procedure. The relaxation procedure can be used to optimize <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> by tuning the local parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si171.svg"><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>ω</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:math>. We observed that a coarse tuning of <ce:italic>ω</ce:italic> is sufficient to improve the overall convergence of the iterative procedure.</ce:para><ce:para id="pr0530">When applying the ISQ method to HOTRG, where <ce:italic>M</ce:italic> is a contraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si116.svg"><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>T</mml:mi></mml:math> along one of the directions, only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si117.svg"><mml:mn>2</mml:mn><mml:mi>d</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:math> of 2<ce:italic>d</ce:italic> modes will actually be truncated, as the backward and forward modes for the contracted direction are left unchanged. For the two-dimensional case, where only one frame has to be determined after each contraction, an alternative method to optimize the truncation using a linearization was proposed in the <ce:italic>projective truncation</ce:italic> of Ref. <ce:cross-ref refid="br0110" id="crf1000">[11]</ce:cross-ref>.</ce:para></ce:section><ce:section id="se0060"><ce:label>6</ce:label><ce:section-title id="st0070">Examples</ce:section-title><ce:para id="pr0540">In the following, we illustrate the effect of the SuperQ and ISQ methods. For various random tensors <ce:italic>A</ce:italic>, we compute core tensors <ce:italic>S</ce:italic> using the standard HOSVD approximation <ce:cross-ref refid="br0070" id="crf1010">[7]</ce:cross-ref> and the best possible approximation of a given multi-rank with the HOOI method <ce:cross-ref refid="br0100" id="crf1020">[10]</ce:cross-ref>. These results are compared with the following backward-forward symmetric approximations: the method proposed by Xie et al. (used in standard HOTRG <ce:cross-ref refid="br0060" id="crf1030">[6]</ce:cross-ref>), the SuperQ approximation of Sec. <ce:cross-ref refid="se0040" id="crf1040">4</ce:cross-ref>, and the ISQ approximation of Sec. <ce:cross-ref refid="se0050" id="crf1050">5</ce:cross-ref>. The relative error for each of these approximations is given by<ce:display><ce:formula id="fm0470"><ce:label>(47)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.svg"><mml:mrow><mml:mi>ϵ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> The comparison of the different methods will be illustrated by plotting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si173.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>ϵ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>hooi</mml:mtext></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>hooi</mml:mtext></mml:mrow></mml:msub></mml:math> in the figures below. For the ISQ method the iterations are initialized with the frames obtained using the interlaced SuperQ method. The ISQ method is iterated until the relative error <ce:italic>ϵ</ce:italic> has a relative change of less than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si174.svg"><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msup></mml:math> between two major iterations, where a major iteration corresponds to one update of all frames. For each iteration we tune the relaxation parameter <ce:italic>ω</ce:italic> to reduce the error. The number of major iterations varies between 20 and 100 with an average of around 50.</ce:para><ce:para id="pr0550">In a first example we consider random tensors <ce:italic>A</ce:italic> of order 4 with dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si175.svg"><mml:mn>10</mml:mn><mml:mo>×</mml:mo><mml:mn>10</mml:mn><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mo>×</mml:mo><mml:mn>100</mml:mn></mml:math>, whose rank is reduced by truncating the last two indices to dimension 10. The results shown in <ce:cross-ref refid="fg0060" id="crf1060">Fig. 6</ce:cross-ref><ce:float-anchor refid="fg0060"/> were computed for initial tensors filled with uniformly distributed elements in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si176.svg"><mml:mo stretchy="false">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math> (left panel) and normally distributed elements with mean <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si177.svg"><mml:mi>μ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> and standard deviation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si178.svg"><mml:mi>σ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:math> (right panel). In another example, shown in <ce:cross-ref refid="fg0070" id="crf1070">Fig. 7</ce:cross-ref><ce:float-anchor refid="fg0070"/>, all modes of random <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si179.svg"><mml:mn>30</mml:mn><mml:mo>×</mml:mo><mml:mn>30</mml:mn><mml:mo>×</mml:mo><mml:mn>30</mml:mn><mml:mo>×</mml:mo><mml:mn>30</mml:mn></mml:math> tensors are reduced to dimension 10. Again the random tensors are filled with elements from a uniform distribution (left) and a normal distribution (right).</ce:para><ce:para id="pr0560">In all examples, the hierarchy between the approximations is the same. In decreasing order of accuracy we find: HOOI, ISQ, HOSVD, SuperQ, and finally the Xie-method. We notice that, as we suggested in the derivation of Sec. <ce:cross-ref refid="se0040" id="crf1080">4</ce:cross-ref>, the SuperQ method performs better than the Xie-method. Both of them are superseded by HOSVD, which is logical as the latter does not have to satisfy the additional backward-forward symmetry constraint. However, a somewhat unexpected result is that in all examples, the ISQ method performs better than the standard HOSVD approximation, even though the former does satisfy the additional backward-forward symmetry constraint. In all cases HOOI performs best, as it is the best possible approximation of the given multi-rank.</ce:para></ce:section><ce:section id="se0070" role="conclusion"><ce:label>7</ce:label><ce:section-title id="st0080">Conclusions</ce:section-title><ce:para id="pr0570">In this paper we consider the reduction of the local truncation error in a single blocking step of the HOTRG procedure. We have discussed in detail the constraints imposed on the semi-orthogonal truncation frames in the HOTRG algorithm, where the backward and forward modes for each direction have to be projected on the same lower-dimensional subspace at each blocking step. We first introduced the SuperQ method, which minimizes a combined error on the backward and forward unfoldings for each individual direction. The method is computationally more efficient and generically yields a reduced local truncation error when compared to the original HOTRG truncation.</ce:para><ce:para id="pr0580">As a further improvement, we presented the iterative SuperQ method, where we formulate a constrained minimization problem, which leads to equations that have to be satisfied by the semi-orthogonal truncation frames in order to minimize the error on the lower-rank tensor approximation, while satisfying the backward-forward symmetry constraints. The method is inspired by the HOOI method, with the additional requirement that the same frames are used on the backward and forward modes of each direction. The equations form a coupled nonlinear eigenvalue problem, which we propose to solve using an iterative procedure, where decoupled linear eigenvalue problems are solved at each iteration step. Computing the optimal backward-forward symmetric truncation frames with the ISQ method is more expensive than the truncation applied in the original HOTRG method, as each iteration step requires new eigenvalue decompositions. 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B95 (2017) 045117. arXiv:1509.07484, doi:10.1103/PhysRevB.95.045117.</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> \ No newline at end of file +<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd" [<!ENTITY gr001 SYSTEM "gr001" NDATA IMAGE><!ENTITY gr002 SYSTEM "gr002" NDATA IMAGE><!ENTITY gr003 SYSTEM "gr003" NDATA IMAGE><!ENTITY gr004 SYSTEM "gr004" NDATA IMAGE><!ENTITY gr005 SYSTEM "gr005" NDATA IMAGE><!ENTITY gr006 SYSTEM "gr006" NDATA IMAGE><!ENTITY gr007 SYSTEM "gr007" NDATA IMAGE>]><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="fla" xml:lang="en"><item-info><jid>NUPHB</jid><aid>116107</aid><ce:article-number>116107</ce:article-number><ce:pii>S0550-3213(23)00036-6</ce:pii><ce:doi>10.1016/j.nuclphysb.2023.116107</ce:doi><ce:copyright year="2023" type="other">The Author(s)</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>Quantum Field Theory and Statistical Systems</ce:text></ce:doctopic></ce:doctopics></item-info><ce:floats><ce:figure id="fg0010"><ce:label>Fig. 1</ce:label><ce:caption id="cp0010"><ce:simple-para id="sp0010">Blocking procedure to reduce a two-dimensional 4 × 4 lattice to a single tensor using alternating contractions in the horizontal and vertical directions.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0010">Fig. 1</ce:alt-text><ce:link locator="gr001" xlink:type="simple" xlink:href="pii:S0550321323000366/gr001" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0010"/></ce:figure><ce:figure id="fg0020"><ce:label>Fig. 2</ce:label><ce:caption id="cp0020"><ce:simple-para id="sp0020">Illustration of the contraction <ce:italic>T</ce:italic>⋆<ce:inf>1</ce:inf><ce:italic>T</ce:italic> = <ce:italic>M</ce:italic> along the 1-direction in a three-dimensional system, as in <ce:cross-ref refid="fm0020" id="crf0010">(2)</ce:cross-ref>. The square nodes represent the fusion of the original tensor indices into the combined fat indices <ce:cross-ref refid="fm0030" id="crf0020">(3)</ce:cross-ref> of <ce:italic>M</ce:italic>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0020">Fig. 2</ce:alt-text><ce:link locator="gr002" xlink:type="simple" xlink:href="pii:S0550321323000366/gr002" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0020"/></ce:figure><ce:figure id="fg0030"><ce:label>Fig. 3</ce:label><ce:caption id="cp0030"><ce:simple-para id="sp0030">Two tensors <ce:italic>T</ce:italic> are contracted over their shared vertical link, producing a tensor <ce:italic>M</ce:italic> with fat horizontal modes (green box). The fat modes of <ce:italic>M</ce:italic> are projected onto subspaces with the projectors <ce:italic>P</ce:italic><ce:inf><ce:italic>U</ce:italic></ce:inf> = <ce:italic>UU</ce:italic><ce:sup><ce:italic>T</ce:italic></ce:sup> and <ce:italic>P</ce:italic><ce:inf><ce:italic>V</ce:italic></ce:inf> = <ce:italic>VV</ce:italic><ce:sup><ce:italic>T</ce:italic></ce:sup>, respectively, to form the lower rank approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> of <ce:cross-ref refid="fm0200" id="crf0030">(20)</ce:cross-ref> (blue box). As part of the construction one recognizes the core tensor <ce:italic>S</ce:italic> of <ce:cross-ref refid="fm0190" id="crf0040">(19)</ce:cross-ref> (red box). Note that the matrices <ce:italic>U</ce:italic>, <ce:italic>U</ce:italic><ce:sup><ce:italic>T</ce:italic></ce:sup>, <ce:italic>V</ce:italic> and <ce:italic>V</ce:italic><ce:sup><ce:italic>T</ce:italic></ce:sup>, described by diamonds in the figure, are applied from the inside to the outside, in correspondence with <ce:cross-ref refid="fm0190" id="crf0050">(19)</ce:cross-ref> and <ce:cross-ref refid="fm0200" id="crf0060">(20)</ce:cross-ref>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0030">Fig. 3</ce:alt-text><ce:link locator="gr003" xlink:type="simple" xlink:href="pii:S0550321323000366/gr003" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0030"/></ce:figure><ce:figure id="fg0040"><ce:label>Fig. 4</ce:label><ce:caption id="cp0040"><ce:simple-para id="sp0040">Two approximations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math>, constructed in <ce:cross-ref refid="fg0030" id="crf0070">Fig. 3</ce:cross-ref>, are contracted in the horizontal direction. This illustrates how the projections performed in the first contraction are concatenated when making this second contraction, leading to a product <ce:italic>P</ce:italic><ce:inf><ce:italic>U</ce:italic></ce:inf><ce:italic>P</ce:italic><ce:inf><ce:italic>V</ce:italic></ce:inf>. Here a new building block <ce:italic>G</ce:italic> = <ce:italic>U</ce:italic><ce:sup><ce:italic>T</ce:italic></ce:sup><ce:italic>V</ce:italic> arises, which we call a <ce:italic>merger</ce:italic> between two core tensors <ce:italic>S</ce:italic>. Note that, for consistency, the matrices operate in chronological order (from the inside to the outside with respect to <ce:italic>M</ce:italic> of <ce:cross-ref refid="fg0030" id="crf0080">Fig. 3</ce:cross-ref>) and not from left to right. The half-mergers on the left and right will connect to their counter parts in further contractions.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0040">Fig. 4</ce:alt-text><ce:link locator="gr004" xlink:type="simple" xlink:href="pii:S0550321323000366/gr004" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0040"/></ce:figure><ce:figure id="fg0050"><ce:label>Fig. 5</ce:label><ce:caption id="cp0050"><ce:simple-para id="sp0050">The construction in <ce:cross-ref refid="fg0040" id="crf0090">Fig. 4</ce:cross-ref> drastically simplifies when choosing <ce:italic>U</ce:italic> = <ce:italic>V</ce:italic>, as the merger <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi>G</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mn mathvariant="double-struck">1</mml:mn></mml:mrow><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:math> in this case. The frames are solely needed to construct the core tensor <ce:italic>S</ce:italic>. This property will spread through the entire blocking procedure (including the final trace), such that the calculation can be performed using the core tensor only.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0050">Fig. 5</ce:alt-text><ce:link locator="gr005" xlink:type="simple" xlink:href="pii:S0550321323000366/gr005" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0050"/></ce:figure><ce:figure id="fg0060"><ce:label>Fig. 6</ce:label><ce:caption id="cp0060"><ce:simple-para id="sp0060">Comparison of (<ce:italic>ϵ</ce:italic> − <ce:italic>ϵ</ce:italic><ce:inf>hooi</ce:inf>)/<ce:italic>ϵ</ce:italic><ce:inf>hooi</ce:inf> for approximations of 100 independent random tensors of dimension 10 × 10 × 100 × 100 truncated to dimension 10 × 10 × 10 × 10 using HOSVD, the Xie method, SuperQ and ISQ. The tensor elements are chosen randomly from a uniform distribution over [0,1] (left plot) and from a Gaussian distribution <ce:italic>N</ce:italic>(0;1) (right plot). The horizontal axis represents different random tensors.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0060">Fig. 6</ce:alt-text><ce:link locator="gr006" xlink:type="simple" xlink:href="pii:S0550321323000366/gr006" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0060"/></ce:figure><ce:figure id="fg0070"><ce:label>Fig. 7</ce:label><ce:caption id="cp0070"><ce:simple-para id="sp0070">Comparison of (<ce:italic>ϵ</ce:italic> − <ce:italic>ϵ</ce:italic><ce:inf>hooi</ce:inf>)/<ce:italic>ϵ</ce:italic><ce:inf>hooi</ce:inf> for random tensors of dimension 30 × 30 × 30 × 30 truncated to dimension 10 × 10 × 10 × 10, using the same approximation methods and the same probability distributions for the tensor elements as in <ce:cross-ref refid="fg0060" id="crf0100">Fig. 6</ce:cross-ref>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0070">Fig. 7</ce:alt-text><ce:link locator="gr007" xlink:type="simple" xlink:href="pii:S0550321323000366/gr007" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0070"/></ce:figure></ce:floats><head><ce:title id="ti0010">Improved local truncation schemes for the higher-order tensor renormalization group method</ce:title><ce:author-group id="ag0010"><ce:author orcid="0000-0002-8443-4804" id="au0010" author-id="S0550321323000366-90624b61ad93431930b6562552c19df0"><ce:given-name>Jacques</ce:given-name><ce:surname>Bloch</ce:surname><ce:cross-ref refid="aff0010" id="crf0110"><ce:sup>a</ce:sup></ce:cross-ref><ce:e-address type="email" xlink:href="mailto:jacques.bloch@ur.de" id="ea0010">jacques.bloch@ur.de</ce:e-address></ce:author><ce:author id="au0020" author-id="S0550321323000366-fe5401542e27ee7a6c5d41c731520f95"><ce:given-name>Robert</ce:given-name><ce:surname>Lohmayer</ce:surname><ce:cross-ref refid="aff0010" id="crf0120"><ce:sup>a</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0020" id="crf0130"><ce:sup>b</ce:sup></ce:cross-ref><ce:e-address type="email" xlink:href="mailto:robert.lohmayer@ur.de" id="ea0020">robert.lohmayer@ur.de</ce:e-address></ce:author><ce:author id="au0030" author-id="S0550321323000366-1b6acdaa5f29277a4d5e446535dfa32c"><ce:given-name>Maximilian</ce:given-name><ce:surname>Meister</ce:surname><ce:cross-ref refid="aff0010" id="crf0140"><ce:sup>a</ce:sup></ce:cross-ref></ce:author><ce:author id="au0040" author-id="S0550321323000366-ec2957601dd286a31c9014e12efc3769"><ce:given-name>Michael</ce:given-name><ce:surname>Nunhofer</ce:surname><ce:cross-ref refid="aff0010" id="crf0150"><ce:sup>a</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="aff0010" affiliation-id="S0550321323000366-e3a23c135363bac74bbab5d41f20f508"><ce:label>a</ce:label><ce:textfn>Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany</ce:textfn><sa:affiliation><sa:organization>Institute for Theoretical Physics</sa:organization><sa:organization>University of Regensburg</sa:organization><sa:city>Regensburg</sa:city><sa:postal-code>93040</sa:postal-code><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0005">Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0020" affiliation-id="S0550321323000366-3425d587d16c5d27239eb2362996a3b9"><ce:label>b</ce:label><ce:textfn>Leibniz Institute for Immunotherapy (LIT), 93053 Regensburg, Germany</ce:textfn><sa:affiliation><sa:organization>Leibniz Institute for Immunotherapy (LIT)</sa:organization><sa:city>Regensburg</sa:city><sa:postal-code>93053</sa:postal-code><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0010">Leibniz Institute for Immunotherapy (LIT), 93053 Regensburg, Germany</ce:source-text></ce:affiliation></ce:author-group><ce:date-received day="5" month="10" year="2022"/><ce:date-revised day="12" month="1" year="2023"/><ce:date-accepted day="1" month="2" year="2023"/><ce:miscellaneous id="ms0010">Editor: Hubert Saleur</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0080">The higher-order tensor renormalization group is a tensor-network method providing estimates for the partition function and thermodynamical observables of classical and quantum systems in thermal equilibrium. At every step of the iterative blocking procedure, the coarse-grid tensor is truncated to keep the tensor dimension under control. For a consistent tensor blocking procedure, it is crucial that the forward and backward tensor modes are projected on the same lower dimensional subspaces. In this paper we present two methods, the SuperQ and the iterative SuperQ method, to construct tensor truncations that reduce or even minimize the local approximation errors, while satisfying this constraint.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0100">Data availability</ce:section-title><ce:para id="pr0590">Data will be made available on request.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">Physical systems in thermal equilibrium are described by their partition function, whose complexity grows exponentially in the volume. The standard method to simulate such statistical systems is the Markov chain Monte Carlo method (MC), which efficiently samples the relevant states of the system to produce reliable estimates of observables. A fundamental prerequisite for the MC method is the positivity of the sampling weights. Models which do not satisfy this condition cause the infamous sign problem and require alternative simulation methods. Quantum systems with complex actions are typical examples of systems with a sign problem. An important topical application in high energy physics is the simulation of quantum chromodynamics (QCD) at nonzero quark chemical potential, which allows for the investigation of the QCD phase diagram as a function of temperature and baryon density.</ce:para><ce:para id="pr0020">There exist numerous methods to circumvent the sign problem, and some even solve it for particular systems <ce:cross-refs refid="br0010 br0020 br0030" id="crs0010">[1–3]</ce:cross-refs>. Very mild sign problems can be circumvented by reweighting, which uses the Monte Carlo method on an auxiliary ensemble with positive weights, and reweights the observables to the target ensemble. The main issue with this method is that the statistical error increases exponentially with the volume such that it is hardly usable in any realistic situation, except for the validation of other methods in regions where the sign problem is small. Other methods which have shown their merit on some models, but are known to have fundamental problems for other ones, are the complex Langevin method, the thimbles, the density of states method and the method of dual variables, where the simulations are usually performed with the worm algorithm. Common to those methods is the stochastic sampling of the partition function.</ce:para><ce:para id="pr0030">An alternative approach that has recently drawn a lot of interest is that of tensor networks, see <ce:cross-ref refid="br0040" id="crf0160">[4]</ce:cross-ref> for a review. In these methods the partition function is first rewritten as a full contraction of a tensor network covering the entire lattice. The exact computation of the partition function and observables in this formulation would have an exponential complexity. The tensor renormalization group (TRG) <ce:cross-ref refid="br0050" id="crf0170">[5]</ce:cross-ref> and higher order tensor renormalization group (HOTRG) <ce:cross-ref refid="br0060" id="crf0180">[6]</ce:cross-ref> methods avoid this exponential cost by blocking the lattice iteratively and truncating the inflated dimensions of the coarse grid tensor at each blocking step using truncated higher order singular value decompositions (HOSVD) <ce:cross-ref refid="br0070" id="crf0190">[7]</ce:cross-ref>, which are based on the matrix singular value decomposition (SVD).</ce:para><ce:para id="pr0040">We consider the partition function of a <ce:italic>d</ce:italic>-dimensional classical or quantum system in thermal equilibrium, written as a fully contracted tensor network <ce:cross-ref refid="br0080" id="crf0200">[8]</ce:cross-ref>,<ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mrow><mml:mi>Z</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="normal">tTr</mml:mi><mml:mspace width="0.2em"/><mml:munderover><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>…</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> with a tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> at each site <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mi>x</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>V</mml:mi></mml:math>. In general the local tensor is the same on all sites, i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>T</mml:mi></mml:math> for all <ce:italic>x</ce:italic>. For each lattice direction <ce:italic>ν</ce:italic>, the tensor has one mode for the forward and one mode for the backward orientation, corresponding to the indices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo>,</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>, respectively, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> is a unit step in the <ce:italic>ν</ce:italic> direction. We will often refer to these modes as backward and forward modes of the physical tensor. The trace in the partition function stands for a full contraction over all tensor indices, where two adjacent tensors share exactly one index.</ce:para><ce:para id="pr0050">Thermodynamical observables, which are defined as derivatives of the partition function with respect to one of its parameters, can be computed using either a finite-difference approximation or an impurity tensor formulation involving the analytical derivative of <ce:italic>T</ce:italic> <ce:cross-ref refid="br0090" id="crf0210">[9]</ce:cross-ref>.</ce:para><ce:para id="pr0060">In the following we will restrict our discussion to HOTRG, because it can be applied to any number of dimensions, whereas TRG is limited to two-dimensional systems. The HOTRG method uses an iterative blocking procedure that reduces the size of the lattice by a factor of two during each blocking step by contracting pairs of adjacent tensors. The procedure is illustrated for a two-dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:mn>4</mml:mn><mml:mo>×</mml:mo><mml:mn>4</mml:mn></mml:math> lattice in <ce:cross-ref refid="fg0010" id="crf0220">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/>. Its extension to higher dimensions is obvious, and below we will further discuss the HOTRG method for the three-dimensional case.</ce:para><ce:para id="pr0070">When contracting two adjacent tensors <ce:italic>T</ce:italic> over their shared link, a tensor <ce:italic>M</ce:italic> of higher order is produced. Such a contraction in the 1-direction is illustrated for the three-dimensional case in <ce:cross-ref refid="fg0020" id="crf0230">Fig. 2</ce:cross-ref><ce:float-anchor refid="fg0020"/> and can be written as<ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:mi>y</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>x</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> and therefore <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>, by definition, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mi>X</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> labels sites on the coarse grid and<ce:display><ce:formula id="fm0030"><ce:label>(3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mrow><mml:mtable displaystyle="true" columnspacing="0.2em"><mml:mtr><mml:mtd columnalign="right"/><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mspace width="2em"/><mml:mspace width="0.2em"/><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"/><mml:mtd columnalign="left"><mml:mtable displaystyle="true" columnspacing="0.2em"><mml:mtr><mml:mtd columnalign="right"/><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mtd><mml:mtd columnalign="right"/><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"/><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mtd><mml:mtd columnalign="right"/><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mspace width="1em"/><mml:mo stretchy="true" maxsize="6.6ex" minsize="6.6ex">}</mml:mo><mml:mspace width="1em"/><mml:mrow><mml:mtext mathvariant="bold">fat</mml:mtext><mml:mspace width="0.25em"/><mml:mtext>indices</mml:mtext></mml:mrow><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></ce:formula></ce:display> For any direction perpendicular to the direction of contraction, the tensor <ce:italic>M</ce:italic> has modes originating from both contracted tensors. To keep the order of the tensor unchanged, we gather every such pair of modes in a new fat mode corresponding to its direct product space. Assuming that the modes of the local tensor have dimension <ce:italic>D</ce:italic>, then the fat modes will have dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. In HOTRG these fat modes are truncated back to dimension <ce:italic>D</ce:italic> using a modified version of the HOSVD approximation method, such that the dimension of the coarse grid tensor remains the same as that of the original local tensor throughout the entire blocking procedure.</ce:para><ce:para id="pr0080">In general, step <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mi>k</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn></mml:math> of the HOTRG procedure can be summarized as<ce:display><ce:formula id="fm0040"><ce:label>(4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>:</mml:mo><mml:mi>M</mml:mi><mml:mover accent="true"><mml:mrow><mml:mo stretchy="false">⟶</mml:mo></mml:mrow><mml:mrow><mml:mtext>truncate</mml:mtext></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>-operation symbolically represents a forward-backward contraction in direction <ce:italic>ν</ce:italic>. The precise construction of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup></mml:math> will be discussed in Sec. <ce:cross-ref refid="se0030" id="crf0240">3</ce:cross-ref>.</ce:para><ce:para id="pr0090">In the standard approximation procedure using HOSVD <ce:cross-ref refid="br0070" id="crf0250">[7]</ce:cross-ref>, referred to as <ce:italic>HOSVD approximation</ce:italic> in the following, the dimension of each tensor mode gets reduced by projecting it on a lower dimensional subspace, which is generically different for each mode. This HOSVD approximation is modified when used as part of the iterative blocking procedure in the standard HOTRG algorithm, as it is essential for the accuracy and effectiveness of the method that the backward and forward modes for every direction get projected on the same subspace. Each of these subspaces will be characterized by a frame, which is a set of orthonormal basis vectors spanning the subspace. Constructing appropriate frames will be the major subject of this paper.</ce:para><ce:para id="pr0100">The standard HOTRG procedure for the construction of frames <ce:cross-ref refid="br0060" id="crf0260">[6]</ce:cross-ref> is not optimal, in particular when the local tensor is not symmetric in its backward and forward modes. In this paper we present two improved methods for the construction of common subspaces for pairs of backward and forward modes: the <ce:italic>SuperQ</ce:italic> and the <ce:italic>iterative SuperQ</ce:italic> method (ISQ), which is an iterative improvement of the former in search of the optimal subspaces. Note that the discussion in this paper solely focuses on the optimization of the rank reduction of the local tensors at every blocking step, but does not take into account global effects on the full contraction of the tensor network.</ce:para><ce:para id="pr0110">Here is a brief outline of the paper. In Sec. <ce:cross-ref refid="se0020" id="crf0270">2</ce:cross-ref> we review the standard HOSVD method to construct a reduced rank approximation for an arbitrary tensor. In Sec. <ce:cross-ref refid="se0030" id="crf0280">3</ce:cross-ref> we explain why the HOTRG method uses a modification of this rank reduction procedure such that the backward and forward modes are projected on the same subspace. We then propose two methods to improve the standard HOTRG truncation: In Sec. <ce:cross-ref refid="se0040" id="crf0290">4</ce:cross-ref> we present the SuperQ method, and in Sec. <ce:cross-ref refid="se0050" id="crf0300">5</ce:cross-ref> we derive the more sophisticated ISQ method. Finally, we summarize and conclude in Sec. <ce:cross-ref refid="se0070" id="crf0310">7</ce:cross-ref>.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">Rank reduction and HOSVD approximation</ce:section-title><ce:para id="pr0120">Below we first review the general idea of rank reduction for an arbitrary tensor, before describing the HOSVD procedure <ce:cross-ref refid="br0070" id="crf0320">[7]</ce:cross-ref> which can be used to generate a quasi-optimal rank-reduced approximation in an efficient way.</ce:para><ce:para id="pr0130">For a real tensor <ce:italic>M</ce:italic> of order <ce:italic>n</ce:italic> with dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo>⋯</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>, the left <ce:italic>matrix-tensor multiplication</ce:italic> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mi>A</mml:mi><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mi>M</mml:mi></mml:math> is defined as a contraction of the second index of the matrix <ce:italic>A</ce:italic> with the <ce:italic>r</ce:italic>-th index of the tensor <ce:italic>M</ce:italic>,<ce:display><ce:formula id="fm0050"><ce:label>(5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>A</mml:mi><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0140">A lower-rank approximation of <ce:italic>M</ce:italic> can be constructed as<ce:display><ce:formula id="fm0060"><ce:label>(6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> projectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:math>. In this approximation, the <ce:italic>r</ce:italic>-th tensor mode of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> is projected onto a subspace of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math>, embedded in the original space. Typically the Frobenius norm <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:math> is used as a measure for the quality of the low-rank approximation and the aim is to determine optimal projectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> with fixed ranks <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math>.</ce:para><ce:para id="pr0150">The projectors can be represented as<ce:display><ce:formula id="fm0070"><ce:label>(7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow></mml:math></ce:formula></ce:display> with semi-orthogonal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, which we will call <ce:italic>frames</ce:italic> in the following. Semi-orthogonal means that the columns of the frames are orthonormal, but not their rows (unless <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math>). The columns of each frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> provide an orthonormal basis of the corresponding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math>-dimensional subspace,<ce:display><ce:formula id="fm0080"><ce:label>(8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mn mathvariant="double-struck">1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0160">The approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> can then be rewritten as<ce:display><ce:formula id="fm0090"><ce:label>(9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mi>S</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo>…</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> dimensional <ce:italic>core tensor</ce:italic><ce:display><ce:formula id="fm0100"><ce:label>(10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mrow><mml:mi>S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> The core tensor represents <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> in the bases of the subspaces, which are spanned by the columns of the frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. Note that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> has the same dimension as <ce:italic>M</ce:italic>, but is generically of lower rank as the <ce:italic>r</ce:italic>-th mode is projected from a space of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> on a subspace of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:math>. In practice, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> is usually not constructed explicitly, as many operations involving <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> can be performed at much lower cost using only the core tensor <ce:italic>S</ce:italic> and the frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, which contain the same information as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> but condensed in lower-dimensional objects. A typical example of such an operation is the contraction of two tensors, as will be discussed in the next section.</ce:para><ce:para id="pr0170">The squared Frobenius norm of <ce:italic>M</ce:italic> is given by<ce:display><ce:formula id="fm0110"><ce:label>(11)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where for brevity we introduce the notation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> for the summation indices, and the inner product between two real tensors of equal dimension is defined as<ce:display><ce:formula id="fm0120"><ce:label>(12)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:mrow><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0180">Since the reduced-rank tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> is a projection of <ce:italic>M</ce:italic>, we have<ce:display><ce:formula id="fm0130"><ce:label>(13)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mrow><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Therefore, the squared approximation error is given by<ce:display><ce:formula id="fm0140"><ce:label>(14)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> which does not require the explicit computation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math>.</ce:para><ce:para id="pr0190">In the HOSVD approximation procedure <ce:cross-ref refid="br0070" id="crf0330">[7]</ce:cross-ref> the semi-orthogonal frames used to approximate <ce:italic>M</ce:italic> are constructed using properties of matrix SVDs. We first introduce the <ce:italic>r</ce:italic>-unfolding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, which is a matrix containing the same entries as the tensor <ce:italic>M</ce:italic>, but reordered such that its rows correspond to the <ce:italic>r</ce:italic>-th mode of <ce:italic>M</ce:italic> and its columns correspond to a combination of all other tensor modes. The entries of the <ce:italic>r</ce:italic>-unfolding of <ce:italic>M</ce:italic> are given by<ce:display><ce:formula id="fm0150"><ce:label>(15)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where the column index <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math> is a linear index of coordinates in a space of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:msub><mml:mrow><mml:mo>∏</mml:mo></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>≠</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math>. The multi-rank of a tensor is defined by the <ce:italic>n</ce:italic>-tuple of the ranks of the individual unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:math>. Therefore <ce:italic>M</ce:italic> has at most multi-rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math> and the approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> of <ce:cross-ref refid="fm0060" id="crf0340">(6)</ce:cross-ref> at most multi-rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>. Note that the squared Frobenius norm <ce:cross-ref refid="fm0110" id="crf0350">(11)</ce:cross-ref> of a tensor is identical to that of any of its unfoldings, as it is just a sum over all squared components,<ce:display><ce:formula id="fm0160"><ce:label>(16)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow></mml:mrow></mml:math></ce:formula></ce:display> for any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:math>. To construct the HOSVD approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math>, we first consider the singular value decomposition for each <ce:italic>r</ce:italic>-unfolding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of <ce:italic>M</ce:italic> (assuming real tensors for simplicity),<ce:display><ce:formula id="fm0170"><ce:label>(17)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">Σ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where the columns of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> orthogonal matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> are the left singular vectors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. The columns of the orthogonal matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> contain the corresponding right singular vectors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. The diagonal entries of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">Σ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> are the singular values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, which are always real and non-negative, while all other entries are zero.</ce:para><ce:para id="pr0200">It is well-known in linear algebra that retaining the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> largest singular values in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">Σ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, while setting all others to zero, yields the best-possible matrix approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> (best-possible referring to a minimization of the Frobenius norm <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mo stretchy="false">‖</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">‖</mml:mo></mml:math>). The relative truncation error is given by<ce:display><ce:formula id="fm0180"><ce:label>(18)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">‖</mml:mo></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> are the eigenvalues of the Gramian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:math>, i.e., the squared singular values of the unfolding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, ordered such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>≥</mml:mo><mml:mo>…</mml:mo><mml:mo>≥</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math>.</ce:para><ce:para id="pr0210">This matrix property is used in HOSVD by separately performing the matrix SVDs of all individual unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of <ce:italic>M</ce:italic> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:math> and constructing the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> with the singular vectors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> corresponding to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math> largest singular values of the unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. These frames are then used to construct the core tensor <ce:cross-ref refid="fm0100" id="crf0360">(10)</ce:cross-ref> and the matrix approximation <ce:cross-ref refid="fm0090" id="crf0370">(9)</ce:cross-ref> of HOSVD. Unlike for the matrix case, the HOSVD tensor approximation is in general not the best-possible approximation of a given multi-rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>, even though it is usually quite close to it <ce:cross-ref refid="br0070" id="crf0380">[7]</ce:cross-ref>.</ce:para><ce:para id="pr0220">In a variant of the HOSVD approximation, called interlaced HOSVD approximation, the rank reduction procedure is carried out in the following way: Starting with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:mi>S</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>M</mml:mi></mml:math>, the frames are computed on successive unfoldings of the core tensor, which gets updated every time a new truncation frame is constructed until the core tensor is of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo>…</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>. For the interlaced HOSVD approximation, the result depends on the order of the updates, but is usually close to that of the ordinary HOSVD approximation.</ce:para><ce:para id="pr0230">The best-possible approximation of multi-rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>, which minimizes the Frobenius norm <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:math>, can be constructed numerically using the Higher Order Orthogonal Iteration (HOOI) <ce:cross-ref refid="br0100" id="crf0390">[10]</ce:cross-ref>. Nevertheless, the HOSVD approximation is especially appealing because of its relative simplicity to produce an almost optimal approximation.</ce:para></ce:section><ce:section id="se0030"><ce:label>3</ce:label><ce:section-title id="st0040">Backward-forward symmetric truncation in HOTRG</ce:section-title><ce:para id="pr0240">We now discuss how the HOSVD formalism is used in HOTRG to avoid the exponential blow up of the tensor dimension during the blocking procedure, and why the standard HOSVD truncation is modified to avoid drawbacks related to accuracy and efficiency. To make our point we will use the two-dimensional case as it can be easiest illustrated and contains all the ingredients necessary for the discussion. Extending it to higher dimensions is straightforward.</ce:para><ce:para id="pr0250">We consider the contraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mi>M</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>T</mml:mi></mml:math> of two local tensors <ce:italic>T</ce:italic> along the 2-direction. According to the discussion in the introduction, <ce:italic>M</ce:italic> will have thin backward and forward modes of dimension <ce:italic>D</ce:italic> in the contracted 2-direction, and fat backward and forward modes of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> in the perpendicular 1-direction, which we want to reduce to lower rank by projecting on a <ce:italic>D</ce:italic>-dimensional subspace using <ce:cross-ref refid="fm0060" id="crf0400">(6)</ce:cross-ref>. This procedure of contraction and truncation, which we detail below, is illustrated in <ce:cross-ref refid="fg0030" id="crf0410">Fig. 3</ce:cross-ref><ce:float-anchor refid="fg0030"/>. To reduce the dimension of the fat modes back from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> to <ce:italic>D</ce:italic>, while minimizing the loss of information, we apply the HOSVD approximation procedure, explained in Sec. <ce:cross-ref refid="se0020" id="crf0420">2</ce:cross-ref>, where we only truncate the fat modes. The SVDs are computed for the unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> for the backward and forward modes in the 1-direction, respectively, and the frames <ce:italic>U</ce:italic> and <ce:italic>V</ce:italic> of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:mi>D</mml:mi></mml:math> are constructed with the singular vectors corresponding to their <ce:italic>D</ce:italic> largest singular values. For the modes in the contracted 2-direction no truncation is required. With these frames we construct a core tensor <ce:italic>S</ce:italic> of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:mi>D</mml:mi><mml:mo>×</mml:mo><mml:mi>D</mml:mi><mml:mo>×</mml:mo><mml:mi>D</mml:mi><mml:mo>×</mml:mo><mml:mi>D</mml:mi></mml:math>, according to <ce:cross-ref refid="fm0100" id="crf0430">(10)</ce:cross-ref>,<ce:display><ce:formula id="fm0190"><ce:label>(19)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mrow><mml:mi>S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> The corresponding approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math>, defined in <ce:cross-ref refid="fm0090" id="crf0440">(9)</ce:cross-ref>, with the same dimension as <ce:italic>M</ce:italic>, but typically much lower rank, is given by<ce:display><ce:formula id="fm0200"><ce:label>(20)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>U</mml:mi><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>V</mml:mi><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>S</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>U</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> projectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>U</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>U</mml:mi><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>V</mml:mi><mml:msup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math>, with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mi>U</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mi>V</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mn mathvariant="double-struck">1</mml:mn></mml:mrow><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:math>. As mentioned in Sec. <ce:cross-ref refid="se0020" id="crf0450">2</ce:cross-ref>, operations involving <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> can typically be performed at much lower cost by using only the core tensor <ce:italic>S</ce:italic> and the frames <ce:italic>U</ce:italic> and <ce:italic>V</ce:italic>.</ce:para><ce:para id="pr0260">Assume that in the next blocking step two <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> tensors are contracted in the 1-direction, as is illustrated in <ce:cross-ref refid="fg0040" id="crf0460">Fig. 4</ce:cross-ref><ce:float-anchor refid="fg0040"/>. When using the standard HOSVD approximation <ce:cross-ref refid="fm0200" id="crf0470">(20)</ce:cross-ref> the backward and forward modes in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> will have been projected on different subspaces using the projectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>U</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:msub></mml:math>, respectively. In a contraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> the two projectors will be multiplied, as can be seen in the center of the figure. The decomposition of the approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> in <ce:cross-ref refid="fm0200" id="crf0480">(20)</ce:cross-ref> can be used to reduce the computational effort, as the original contraction can be replaced by contractions of two core tensors <ce:italic>S</ce:italic> with a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:mi>D</mml:mi><mml:mo>×</mml:mo><mml:mi>D</mml:mi></mml:math> dimensional merger <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si79.svg"><mml:mi>G</mml:mi><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mi>V</mml:mi></mml:math> in between, as can be seen in the figure.<ce:cross-ref refid="fn0010" id="crf0490"><ce:sup>1</ce:sup></ce:cross-ref><ce:footnote id="fn0010"><ce:label>1</ce:label><ce:note-para id="np0010">In fact this produces an amputated version of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math>, which together with the mergers is all we need in the full contraction of the tensor network, see also <ce:cross-ref refid="fg0040" id="crf0500">Fig. 4</ce:cross-ref>.</ce:note-para></ce:footnote> The entries of <ce:italic>G</ce:italic> are scalar products of the basis vectors in <ce:italic>U</ce:italic> and <ce:italic>V</ce:italic>.</ce:para><ce:para id="pr0270">At this point it is important to discuss a crucial modification introduced by the HOTRG method to the HOSVD truncation procedure presented above, which is rarely discussed in the literature. Although the HOSVD approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> gives a close-to-best lower-rank approximation to <ce:italic>M</ce:italic>, it is in general not such a good and useful truncation when viewed as part of the iterative blocking procedure. Indeed, the product of projectors corresponds to a projection of a projection, which will unavoidably loose additional information if the projectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>U</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:msub></mml:math> are different. In this case, the contraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> will no longer necessarily be a good approximation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:mi>M</mml:mi><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>M</mml:mi></mml:math>, even if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> itself was close to the best-possible approximation of <ce:italic>M</ce:italic>.</ce:para><ce:para id="pr0280">We now observe that, due to the idempotence of projectors, there would be no additional loss if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si81.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>U</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:msub></mml:math>, i.e., if the backward and forward modes of <ce:italic>M</ce:italic> in the 1-direction were projected on the same subspace. Note that in this case the merger <ce:italic>G</ce:italic> is an orthogonal matrix. Moreover, we can also get a serious gain in algorithmic simplicity, on top of this accuracy improvement, if we choose the same basis for both modes in the common subspace, i.e., we choose frames satisfying <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si82.svg"><mml:mi>U</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>V</mml:mi></mml:math>, for which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi>G</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mn mathvariant="double-struck">1</mml:mn></mml:mrow><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:math>. When looking back at <ce:cross-ref refid="fg0040" id="crf0510">Fig. 4</ce:cross-ref> we see that, in this case, the central merger just drops out, and the contraction can be replaced by a contraction of two core tensors, as is illustrated in <ce:cross-ref refid="fg0050" id="crf0520">Fig. 5</ce:cross-ref><ce:float-anchor refid="fg0050"/> (the frames on the left and right of <ce:cross-ref refid="fg0040" id="crf0530">Fig. 4</ce:cross-ref> will connect to their counter parts in further contractions, to form another merger <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi>G</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mn mathvariant="double-struck">1</mml:mn></mml:mrow><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:math>, which will also drop out).</ce:para><ce:para id="pr0290">For this reason, in HOTRG the backward and forward modes of each direction are truncated using a common frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si166.svg"><mml:mi>ν</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:math>, even when the HOSVD frames are different, which is typically the case for systems at nonzero chemical potential. With this backward-forward symmetric truncation, the core tensor <ce:italic>S</ce:italic> can be used as new coarse grid tensor after each blocking step, where at the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si86.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>-th blocking step two tensors of step <ce:italic>k</ce:italic> are contracted to form a new coarse grid tensor,<ce:display><ce:formula id="fm0210"><ce:label>(21)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>:</mml:mo><mml:mi>M</mml:mi><mml:mover accent="true"><mml:mrow><mml:mo stretchy="false">⟶</mml:mo></mml:mrow><mml:mrow><mml:mtext>BF</mml:mtext></mml:mrow></mml:mover><mml:mi>S</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>:</mml:mo><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> The abbreviation BF on the arrow means that we apply a backward-forward symmetric truncation to construct the core tensor, which then becomes the new local tensor on the coarse grid. The frames are only needed to construct the core tensor with <ce:cross-ref refid="fm0190" id="crf0540">(19)</ce:cross-ref>, and can then be discarded.</ce:para><ce:para id="pr0300">The same reasoning also holds for a contraction in the other direction, where the directions of thin and fat modes are interchanged. Moreover, the procedure naturally generalizes to <ce:italic>d</ce:italic> dimensions, where we have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:mi>d</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:math> directions <ce:italic>ν</ce:italic> with backward and forward fat modes: If the backward and forward frames are chosen to be identical for each direction, the core tensor <ce:cross-ref refid="fm0100" id="crf0550">(10)</ce:cross-ref> can be used as the new coarse grid tensor in the HOTRG blocking procedure.</ce:para><ce:para id="pr0310">This strategy of choosing the same frame to truncate the backward and forward mode for each individual direction in HOTRG, makes it fundamentally different from the HOSVD approximation, as it can no longer directly rely on the optimal low-rank properties of matrix SVD. Geometrically, truncating the backward and forward modes with the same frame means that these modes get projected on the same subspace and are described in the same basis. This explains why the full contraction of the tensor network into a scalar can be rewritten in terms of the core tensors only.</ce:para><ce:para id="pr0320">Note that if one would use the standard HOSVD approximation procedure and work with different backward and forward frames, we would have to use both the core tensors and the mergers <ce:italic>G</ce:italic> defined above when performing the iterative contractions, to ensure that the different subspaces are matched onto one another, see <ce:cross-ref refid="fg0040" id="crf0560">Fig. 4</ce:cross-ref>. Although this is no conceptual problem, it would complicate the algorithm, require additional computational work, and most of all the product of projectors would deteriorate the results further.</ce:para><ce:para id="pr0330">The construction of the shared backward-forward frames in HOTRG has not been given a lot of attention in the literature until now. There is a brief discussion of this issue in the original HOTRG paper <ce:cross-ref refid="br0060" id="crf0570">[6]</ce:cross-ref>, where either the backward or forward frame is chosen and applied to both modes, depending on which one gives the smallest SVD truncation error. The error introduced by this choice on the other mode is however not taken into account. We observed that for tensors lacking a backward-forward symmetry, this choice of frame is not optimal and can be improved upon.</ce:para><ce:para id="pr0340">Below we present two new methods to improve the construction of shared frames for the backward and forward modes. The first one, called SuperQ method and presented in Sec. <ce:cross-ref refid="se0040" id="crf0580">4</ce:cross-ref>, minimizes a combined error on the backward and forward unfoldings for each individual direction. The second method, which we call iterative SuperQ (ISQ) method is presented in Sec. <ce:cross-ref refid="se0050" id="crf0590">5</ce:cross-ref>. This iterative method aims at determining the best-possible approximation to <ce:italic>M</ce:italic> for a given multi-rank, satisfying the requirement that the backward and forward frames for each direction are identical. The ISQ method leans on ideas developed for the higher order orthogonal iteration (HOOI) method <ce:cross-ref refid="br0100" id="crf0600">[10]</ce:cross-ref>, which constructs the best-possible approximation of a given multi-rank with independent frames for all modes. We will see that the SuperQ solution can be used as a natural starting point for the ISQ procedure. Note that the SuperQ and ISQ methods are specifically conceived for tensors which are part of a physical tensor network on a space-time lattice and have modes corresponding to backward and forward orientations.</ce:para></ce:section><ce:section id="se0040"><ce:label>4</ce:label><ce:section-title id="st0050">The SuperQ method</ce:section-title><ce:para id="pr0350">To discuss the construction of truncations satisfying the requirement that the frames for the backward and forward modes are identical, we consider a tensor <ce:italic>M</ce:italic> with <ce:italic>d</ce:italic> pairs of backward and forward modes. The tensor is thus of order 2<ce:italic>d</ce:italic> with dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si89.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo>⋯</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:math>, which will be truncated to dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si90.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo>⋯</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:math> using <ce:italic>d</ce:italic> semi-orthogonal frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>. With these frames the core tensor is constructed using<ce:display><ce:formula id="fm0220"><ce:label>(22)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si92.svg"><mml:mrow><mml:mi>S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>d</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Consider the positive semi-definite Gram matrices<ce:display><ce:formula id="fm0230"><ce:label>(23)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si93.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si104.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> are the unfoldings of <ce:italic>M</ce:italic> with respect to the backward and forward modes for direction <ce:italic>ν</ce:italic>, i.e.,<ce:display><ce:formula id="fm0240"><ce:label>(24)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> is the <ce:italic>r</ce:italic>-unfolding of the tensor <ce:italic>M</ce:italic> defined in <ce:cross-ref refid="fm0150" id="crf0610">(15)</ce:cross-ref>. We denote the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> frames constructed with the eigenvectors corresponding to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> largest eigenvalues of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si99.svg"><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si100.svg"><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math>, respectively. As the backward and forward Gramians <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si99.svg"><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si100.svg"><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> are in general not identical, the corresponding subspaces spanned by the vectors of the frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> will be different too.<ce:cross-ref refid="fn0020" id="crf0620"><ce:sup>2</ce:sup></ce:cross-ref><ce:footnote id="fn0020"><ce:label>2</ce:label><ce:note-para id="np0020">This can even be the case if the eigenvalues of both Gramians coincide, as we have observed for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si103.svg"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> model with chemical potential.</ce:note-para></ce:footnote> In the standard HOTRG procedure <ce:cross-ref refid="br0060" id="crf0630">[6]</ce:cross-ref> it is suggested to choose either <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> for the unique <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, depending which of both gives the smallest SVD truncation error <ce:cross-ref refid="fm0180" id="crf0640">(18)</ce:cross-ref>. Even though this choice of frame optimizes the truncation error for one mode, it does not take into account its effect on the mode corresponding to the opposite orientation. Therefore, it is clear that, generically, better choices of frames should exist, and our aim is to construct frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> that reduce the combined truncation error when applied simultaneously to the backward and forward modes for the <ce:italic>ν</ce:italic> direction.</ce:para><ce:para id="pr0360">Let us now consider a single truncation frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> which we use to reduce the rank of the unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si104.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math>. Using <ce:cross-ref refid="fm0140" id="crf0650">(14)</ce:cross-ref> and <ce:cross-ref refid="fm0160" id="crf0660">(16)</ce:cross-ref>, the relative truncation errors on the backward and forward unfoldings are<ce:display><ce:formula id="fm0250"><ce:label>(25)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si106.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">‖</mml:mo></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0260"><ce:label>(26)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si107.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">‖</mml:mo></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si108.svg"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si109.svg"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> are the rank-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> approximations to the unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si104.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.svg"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math>, respectively, obtained with the same frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>.</ce:para><ce:para id="pr0370">To improve upon using either <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math>, we determine the common <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> by minimizing the combination of both errors in<ce:display><ce:formula id="fm0270"><ce:label>(27)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si112.svg"><mml:mrow><mml:msup><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>S</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where we also used <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si113.svg"><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. We define the SuperQ matrix for direction <ce:italic>ν</ce:italic> as<ce:display><ce:formula id="fm0280"><ce:label>(28)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si114.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>S</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>b</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> which is symmetric and positive semi-definite as it is a sum of two symmetric positive semi-definite matrices. Therefore, if we diagonalize the SuperQ matrix and truncate the eigenvector matrix, retaining the eigenvectors corresponding to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> largest eigenvalues, then this semi-orthogonal frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> minimizes the truncation error <ce:cross-ref refid="fm0270" id="crf0670">(27)</ce:cross-ref> on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si115.svg"><mml:msubsup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mtext>S</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math>. This SuperQ procedure is repeated on all <ce:italic>d</ce:italic> directions to determine all frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, which can then be used to approximate <ce:italic>M</ce:italic> and to construct the corresponding core tensor <ce:italic>S</ce:italic>, see <ce:cross-ref refid="fm0220" id="crf0680">(22)</ce:cross-ref>.</ce:para><ce:para id="pr0380">The SuperQ method is computationally efficient since it only requires a single eigenvalue decomposition for each pair of backward and forward fat modes, while the standard HOTRG procedure <ce:cross-ref refid="br0060" id="crf0690">[6]</ce:cross-ref> performs separate decompositions on these modes.</ce:para><ce:para id="pr0390">When applying the SuperQ method to HOTRG, where <ce:italic>M</ce:italic> is a contraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si116.svg"><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>T</mml:mi></mml:math> along one of the directions, only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si117.svg"><mml:mn>2</mml:mn><mml:mi>d</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:math> modes will actually be truncated, as the backward and forward modes for the contracted direction need not be truncated.</ce:para><ce:para id="pr0400">In analogy to the interlaced HOSVD approximation, see Sec. <ce:cross-ref refid="se0020" id="crf0700">2</ce:cross-ref>, we can also define an interlaced version of the SuperQ method where we determine the frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> by applying the method to an intermediate core tensor, which gets updated direction-by-direction (starting from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:mi>S</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>M</mml:mi></mml:math>) by truncating the respective backward and forward mode each time a frame has been computed. This interlaced SuperQ method is also of interest in the light of the iterative procedure derived in the next section.</ce:para></ce:section><ce:section id="se0050"><ce:label>5</ce:label><ce:section-title id="st0060">Optimized frames with iterative SuperQ</ce:section-title><ce:para id="pr0410">Although the HOSVD method, see Sec. <ce:cross-ref refid="se0020" id="crf0710">2</ce:cross-ref>, typically yields a good tensor approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:math> <ce:cross-ref refid="br0070" id="crf0720">[7]</ce:cross-ref>, the best-possible one, which minimizes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si118.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, can be constructed with an iterative procedure called higher order orthogonal iteration (HOOI) method <ce:cross-ref refid="br0100" id="crf0730">[10]</ce:cross-ref>.</ce:para><ce:para id="pr0420">According to the discussion of the backward-forward symmetric truncation in Sec. <ce:cross-ref refid="se0030" id="crf0740">3</ce:cross-ref>, it is clear that HOOI is itself not applicable in a tensor network approach to statistical physics, because the backward and forward modes should be truncated with the same semi-orthogonal frame for each direction, while HOOI very generically generates different frames for all modes. Below we present the iterative SuperQ (ISQ) method, which is inspired by the original HOOI procedure but imposes the requirement that the same frame has to be used to truncate the backward and forward modes of each direction.</ce:para><ce:para id="pr0430">As in Sec. <ce:cross-ref refid="se0040" id="crf0750">4</ce:cross-ref>, we consider a tensor <ce:italic>M</ce:italic> with <ce:italic>d</ce:italic> pairs of backward and forward modes, i.e., the tensor is of order 2<ce:italic>d</ce:italic> with dimensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si119.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:math>, which will be truncated to dimensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si90.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo>⋯</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:math>, see <ce:cross-ref refid="fm0220" id="crf0760">(22)</ce:cross-ref>. Our aim is to minimize the squared Frobenius norm <ce:cross-ref refid="fm0140" id="crf0770">(14)</ce:cross-ref><ce:display><ce:formula id="fm0290"><ce:label>(29)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> over all semi-orthogonal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:mi>ν</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>…</mml:mo><mml:mi>d</mml:mi></mml:math>, for fixed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>, with the additional condition that the backward and forward modes for each direction <ce:italic>ν</ce:italic> are truncated with the same frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>.</ce:para><ce:para id="pr0440">The semi-orthogonality of the frames is imposed explicitly by orthonormality conditions for the column vectors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>,<ce:display><ce:formula id="fm0300"><ce:label>(30)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si122.svg"><mml:mrow><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mtext>for </mml:mtext><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mtext> and </mml:mtext><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>ν</mml:mi><mml:mo>≤</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> in the constrained minimization of <ce:cross-ref refid="fm0290" id="crf0780">(29)</ce:cross-ref>. This leads to the cost function<ce:display><ce:formula id="fm0310"><ce:label>(31)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si123.svg"><mml:mi>g</mml:mi><mml:mo id="mmlbr0001" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>C</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>d</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mi>M</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mspace linebreak="newline"/><mml:mspace width="1em"/><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:munderover><mml:mi mathvariant="normal">Tr</mml:mi><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn mathvariant="double-struck">1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> with matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si124.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> containing the Lagrange multipliers. The orthonormalization conditions are symmetric under the exchange <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.svg"><mml:mi>a</mml:mi><mml:mo stretchy="false">↔</mml:mo><mml:mi>b</mml:mi></mml:math>, and so <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si124.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> will be symmetric too. If we diagonalize <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si126.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mtext>diag</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, redefine <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si127.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si128.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mtext>diag</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, and use the orthogonality of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si129.svg"><mml:msup><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, then Eq. <ce:cross-ref refid="fm0310" id="crf0790">(31)</ce:cross-ref> remains unaltered albeit now with diagonal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mtext>diag</mml:mtext><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:math>, and this without loss of generality. Written out in components this is<ce:display><ce:formula id="fm0320"><ce:label>(32)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.svg"><mml:mi>g</mml:mi><mml:mo id="mmlbr0002" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:munderover><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>c</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="before">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munderover><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:munderover><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>c</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> For a constrained maximum of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, the partial derivative of <ce:italic>g</ce:italic> with respect to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si133.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-entry of the <ce:italic>ν</ce:italic>-th orthogonal frame has to satisfy<ce:display><ce:formula id="fm0330"><ce:label>(33)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si134.svg"><mml:mrow><mml:mn>0</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munderover><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si135.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>k</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si136.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>c</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>ν</mml:mi><mml:mo>≤</mml:mo><mml:mi>d</mml:mi></mml:math>. Note that the same frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> appears twice in <ce:italic>S</ce:italic>, as it is used to truncate the modes in the backward and forward <ce:italic>ν</ce:italic> direction. We therefore obtain<ce:display><ce:formula id="fm0340"><ce:label>(34)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si138.svg"><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true" id="mmlbr0003">(</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munder><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0003" linebreakstyle="before">+</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munder><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> After eliminating the Kronecker deltas we get<ce:display><ce:formula id="fm0350"><ce:label>(35)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si139.svg"><mml:munder id="mmlbr0004"><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:mi>c</mml:mi><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munder><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:mi>k</mml:mi><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0004" linebreakstyle="before">+</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>c</mml:mi><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munder><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>k</mml:mi><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Let us define the unfolding matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> with dimensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si142.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>⋯</mml:mo><mml:msubsup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>⋯</mml:mo><mml:msubsup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math>, where all directions of <ce:italic>M</ce:italic> are truncated, except for the backward-<ce:italic>ν</ce:italic> mode for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, and the forward-<ce:italic>ν</ce:italic> mode for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, and the unfolding is performed with respect to the untruncated mode,<ce:display><ce:formula id="fm0360"><ce:label>(36)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si143.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munder><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0370"><ce:label>(37)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si144.svg"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:munder><mml:mo movablelimits="false">∏</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where we used the notation introduced in <ce:cross-ref refid="fm0150" id="crf0800">(15)</ce:cross-ref> for the matrix indices. The core tensor can also be written in terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> by truncating the remaining untruncated index:<ce:display><ce:formula id="fm0380"><ce:label>(38)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si145.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> After substituting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> in <ce:cross-ref refid="fm0350" id="crf0810">(35)</ce:cross-ref> we obtain<ce:display><ce:formula id="fm0390"><ce:label>(39)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si146.svg"><mml:munder id="mmlbr0005"><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mspace linebreak="newline"/><mml:mspace width="1em"/><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> or<ce:display><ce:formula id="fm0400"><ce:label>(40)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si147.svg"><mml:mrow><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msubsup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> If we introduce the positive semi-definite matrices<ce:display><ce:formula id="fm0410"><ce:label>(41)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si148.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mspace width="2em"/><mml:mi>ν</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> we can identify <ce:cross-ref refid="fm0400" id="crf0820">(40)</ce:cross-ref> as a coupled nonlinear eigenvalue problem (which is nonlinear in the eigenvectors)<ce:display><ce:formula id="fm0420"><ce:label>(42)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si149.svg"><mml:mrow><mml:mphantom><mml:mspace width="2em"/><mml:mspace width="2em"/><mml:mi>ν</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>…</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo></mml:mphantom><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mspace width="2em"/><mml:mi>ν</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> for the semi-orthogonal frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>-dimensional diagonal matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si124.svg"><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. Note that all frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si151.svg"><mml:mi>μ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>…</mml:mo><mml:mi>d</mml:mi></mml:math>, appear in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si152.svg"><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> (as projections <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si153.svg"><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:math>). This is even true for the direction <ce:italic>ν</ce:italic> itself, as its projector is applied to the forward mode in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si154.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:math> and to the backward mode in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si155.svg"><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:math>. Therefore the <ce:italic>ν</ce:italic>-th equation is cubic in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and quartic in all other <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si156.svg"><mml:mi>μ</mml:mi><mml:mo>≠</mml:mo><mml:mi>ν</mml:mi></mml:math>. It is crucial to keep in mind that the self-consistent solutions to <ce:cross-ref refid="fm0420" id="crf0830">(42)</ce:cross-ref>, which we are looking for, are required to be semi-orthogonal matrices with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si157.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> columns, in order to satisfy the constraints <ce:cross-ref refid="fm0300" id="crf0840">(30)</ce:cross-ref>.</ce:para><ce:para id="pr0450">It is useful to note that if we replace the matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si152.svg"><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> in <ce:cross-ref refid="fm0420" id="crf0850">(42)</ce:cross-ref> by fixed matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, the <ce:italic>d</ce:italic> matrix equations decouple and each one of them is a linear eigenvalue equation for the frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. The solutions of these linearized equations are however in general no solution of the original nonlinear equations <ce:cross-ref refid="fm0420" id="crf0860">(42)</ce:cross-ref>. On the other hand, if <ce:cross-ref refid="fm0420" id="crf0870">(42)</ce:cross-ref> is satisfied, then <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> is a solution of the linear eigenvalue problem for the specific matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si159.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>.</ce:para><ce:para id="pr0460">Therefore, we propose to solve the coupled system of equations using an iterative procedure, where at each iteration step, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> is computed with <ce:cross-ref refid="fm0410" id="crf0880">(41)</ce:cross-ref> using the current frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>μ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:math>, and the eigenvalue problem<ce:display><ce:formula id="fm0430"><ce:label>(43)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si161.svg"><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> is solved for the eigenvalues <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si162.svg"><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and eigenvectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si163.svg"><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> of the positive semi-definite matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. We then take the normalized eigenvectors corresponding to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> largest eigenvalues to form a new frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. Note that the eigenvectors of the symmetric matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> are orthogonal and therefore the constraint <ce:cross-ref refid="fm0300" id="crf0890">(30)</ce:cross-ref> is automatically satisfied for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>.</ce:para><ce:para id="pr0470">This strategy can be motivated in the following way. We are looking for the solution of <ce:cross-ref refid="fm0420" id="crf0900">(42)</ce:cross-ref> which maximizes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. Using the expression <ce:cross-ref refid="fm0380" id="crf0910">(38)</ce:cross-ref> for the core tensor <ce:italic>S</ce:italic> in terms of the unfoldings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si141.svg"><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, the squared norm in the cost function <ce:cross-ref refid="fm0310" id="crf0920">(31)</ce:cross-ref> can be written as<ce:display><ce:formula id="fm0440"><ce:label>(44)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si164.svg"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi mathvariant="normal">Tr</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">]</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="normal">Tr</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">]</mml:mo><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">Tr</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">]</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> which explicitly contains the matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si165.svg"><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> defined in <ce:cross-ref refid="fm0410" id="crf0930">(41)</ce:cross-ref> (note that this equation yields the same <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> for each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si166.svg"><mml:mi>ν</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:math>). It is straightforward to show that, at each iteration step, the solution obtained from the linearization <ce:cross-ref refid="fm0430" id="crf0940">(43)</ce:cross-ref> corresponds to the constrained optimization of the approximation<ce:display><ce:formula id="fm0450"><ce:label>(45)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si167.svg"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>≈</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">Tr</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> to <ce:cross-ref refid="fm0440" id="crf0950">(44)</ce:cross-ref>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> is fixed and computed with the most recent frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si160.svg"><mml:mi>μ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:math>. As this approximation is quadratic in the new frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, its maximum will be given by the eigenvectors corresponding to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> largest eigenvalues of the symmetric, positive semi-definite matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>.</ce:para><ce:para id="pr0480">The iterative procedure can be interpreted as an iterative SuperQ method, where at each step all modes of <ce:italic>M</ce:italic> are truncated using the last known frames, except for a backward mode in <ce:italic>B</ce:italic> and the corresponding forward mode in <ce:italic>F</ce:italic>.</ce:para><ce:para id="pr0490">During the iterative procedure we cycle through the <ce:italic>d</ce:italic> dimensions <ce:italic>ν</ce:italic> and determine a new frame <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> at each step using <ce:cross-ref refid="fm0430" id="crf0960">(43)</ce:cross-ref>. Then, we repeat these <ce:italic>d</ce:italic> iteration steps until all frames have converged. In practice we observed that the first iteration for each direction is the most important one, and further iterations of the same direction only give small corrections.</ce:para><ce:para id="pr0500">An alternative procedure would be to iterate <ce:cross-ref refid="fm0430" id="crf0970">(43)</ce:cross-ref> for a single <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> (keeping all other <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> fixed) until convergence has been reached (reevaluating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> with the most recent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> at every step), and then go on to the next frame. Once all frames have been iterated, this whole procedure is repeated until all frames converge together. Note that the inner iterations are computationally cheap, as all frames but one are kept fixed and all matrix-tensor multiplications in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si168.svg"><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> involving these fixed frames have to be computed only once. However, this procedure does not seem to give an overall faster convergence.</ce:para><ce:para id="pr0510">A natural choice for the starting frames <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si121.svg"><mml:mi>ν</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo>…</mml:mo><mml:mi>d</mml:mi></mml:math>, in the iterative procedure are the frames obtained from the interlaced SuperQ method, see Sec. <ce:cross-ref refid="se0040" id="crf0980">4</ce:cross-ref>.</ce:para><ce:para id="pr0520">Note that the iterative procedure is not guaranteed to converge, and even when it does, the solution is not necessarily the global maximum. This can be improved upon by tuning the starting frames or by applying an under-relaxation procedure to the intermediate <ce:italic>Q</ce:italic> matrix. In this procedure, we replace <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> in <ce:cross-ref refid="fm0430" id="crf0990">(43)</ce:cross-ref> by<ce:display><ce:formula id="fm0460"><ce:label>(46)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si169.svg"><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mo>‾</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>ω</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>ω</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mtext>prev</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si170.svg"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mtext>prev</mml:mtext></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> was used to obtain the previous <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> in the iterative procedure. The relaxation procedure can be used to optimize <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> by tuning the local parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si171.svg"><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>ω</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:math>. We observed that a coarse tuning of <ce:italic>ω</ce:italic> is sufficient to improve the overall convergence of the iterative procedure.</ce:para><ce:para id="pr0530">When applying the ISQ method to HOTRG, where <ce:italic>M</ce:italic> is a contraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si116.svg"><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>T</mml:mi></mml:math> along one of the directions, only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si117.svg"><mml:mn>2</mml:mn><mml:mi>d</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:math> of 2<ce:italic>d</ce:italic> modes will actually be truncated, as the backward and forward modes for the contracted direction are left unchanged. For the two-dimensional case, where only one frame has to be determined after each contraction, an alternative method to optimize the truncation using a linearization was proposed in the <ce:italic>projective truncation</ce:italic> of Ref. <ce:cross-ref refid="br0110" id="crf1000">[11]</ce:cross-ref>.</ce:para></ce:section><ce:section id="se0060"><ce:label>6</ce:label><ce:section-title id="st0070">Examples</ce:section-title><ce:para id="pr0540">In the following, we illustrate the effect of the SuperQ and ISQ methods. For various random tensors <ce:italic>A</ce:italic>, we compute core tensors <ce:italic>S</ce:italic> using the standard HOSVD approximation <ce:cross-ref refid="br0070" id="crf1010">[7]</ce:cross-ref> and the best possible approximation of a given multi-rank with the HOOI method <ce:cross-ref refid="br0100" id="crf1020">[10]</ce:cross-ref>. These results are compared with the following backward-forward symmetric approximations: the method proposed by Xie et al. (used in standard HOTRG <ce:cross-ref refid="br0060" id="crf1030">[6]</ce:cross-ref>), the SuperQ approximation of Sec. <ce:cross-ref refid="se0040" id="crf1040">4</ce:cross-ref>, and the ISQ approximation of Sec. <ce:cross-ref refid="se0050" id="crf1050">5</ce:cross-ref>. The relative error for each of these approximations is given by<ce:display><ce:formula id="fm0470"><ce:label>(47)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.svg"><mml:mrow><mml:mi>ϵ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">‖</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">‖</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> The comparison of the different methods will be illustrated by plotting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si173.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>ϵ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>hooi</mml:mtext></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mtext>hooi</mml:mtext></mml:mrow></mml:msub></mml:math> in the figures below. For the ISQ method the iterations are initialized with the frames obtained using the interlaced SuperQ method. The ISQ method is iterated until the relative error <ce:italic>ϵ</ce:italic> has a relative change of less than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si174.svg"><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msup></mml:math> between two major iterations, where a major iteration corresponds to one update of all frames. For each iteration we tune the relaxation parameter <ce:italic>ω</ce:italic> to reduce the error. The number of major iterations varies between 20 and 100 with an average of around 50.</ce:para><ce:para id="pr0550">In a first example we consider random tensors <ce:italic>A</ce:italic> of order 4 with dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si175.svg"><mml:mn>10</mml:mn><mml:mo>×</mml:mo><mml:mn>10</mml:mn><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mo>×</mml:mo><mml:mn>100</mml:mn></mml:math>, whose rank is reduced by truncating the last two indices to dimension 10. The results shown in <ce:cross-ref refid="fg0060" id="crf1060">Fig. 6</ce:cross-ref><ce:float-anchor refid="fg0060"/> were computed for initial tensors filled with uniformly distributed elements in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si176.svg"><mml:mo stretchy="false">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math> (left panel) and normally distributed elements with mean <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si177.svg"><mml:mi>μ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> and standard deviation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si178.svg"><mml:mi>σ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:math> (right panel). In another example, shown in <ce:cross-ref refid="fg0070" id="crf1070">Fig. 7</ce:cross-ref><ce:float-anchor refid="fg0070"/>, all modes of random <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si179.svg"><mml:mn>30</mml:mn><mml:mo>×</mml:mo><mml:mn>30</mml:mn><mml:mo>×</mml:mo><mml:mn>30</mml:mn><mml:mo>×</mml:mo><mml:mn>30</mml:mn></mml:math> tensors are reduced to dimension 10. Again the random tensors are filled with elements from a uniform distribution (left) and a normal distribution (right).</ce:para><ce:para id="pr0560">In all examples, the hierarchy between the approximations is the same. In decreasing order of accuracy we find: HOOI, ISQ, HOSVD, SuperQ, and finally the Xie-method. We notice that, as we suggested in the derivation of Sec. <ce:cross-ref refid="se0040" id="crf1080">4</ce:cross-ref>, the SuperQ method performs better than the Xie-method. Both of them are superseded by HOSVD, which is logical as the latter does not have to satisfy the additional backward-forward symmetry constraint. However, a somewhat unexpected result is that in all examples, the ISQ method performs better than the standard HOSVD approximation, even though the former does satisfy the additional backward-forward symmetry constraint. In all cases HOOI performs best, as it is the best possible approximation of the given multi-rank.</ce:para></ce:section><ce:section id="se0070" role="conclusion"><ce:label>7</ce:label><ce:section-title id="st0080">Conclusions</ce:section-title><ce:para id="pr0570">In this paper we consider the reduction of the local truncation error in a single blocking step of the HOTRG procedure. We have discussed in detail the constraints imposed on the semi-orthogonal truncation frames in the HOTRG algorithm, where the backward and forward modes for each direction have to be projected on the same lower-dimensional subspace at each blocking step. We first introduced the SuperQ method, which minimizes a combined error on the backward and forward unfoldings for each individual direction. The method is computationally more efficient and generically yields a reduced local truncation error when compared to the original HOTRG truncation.</ce:para><ce:para id="pr0580">As a further improvement, we presented the iterative SuperQ method, where we formulate a constrained minimization problem, which leads to equations that have to be satisfied by the semi-orthogonal truncation frames in order to minimize the error on the lower-rank tensor approximation, while satisfying the backward-forward symmetry constraints. The method is inspired by the HOOI method, with the additional requirement that the same frames are used on the backward and forward modes of each direction. The equations form a coupled nonlinear eigenvalue problem, which we propose to solve using an iterative procedure, where decoupled linear eigenvalue problems are solved at each iteration step. Computing the optimal backward-forward symmetric truncation frames with the ISQ method is more expensive than the truncation applied in the original HOTRG method, as each iteration step requires new eigenvalue decompositions. 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B</sb:maintitle></sb:title><sb:volume-nr>95</sb:volume-nr></sb:series><sb:date>2017</sb:date></sb:issue><sb:article-number>045117</sb:article-number><ce:doi>10.1103/PhysRevB.95.045117</ce:doi></sb:host><sb:host><sb:e-host><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/preprint" xlink:href="arxiv:1509.07484" id="inf0070">arXiv:1509.07484</ce:inter-ref></sb:e-host></sb:host></sb:reference><ce:source-text id="srct0065">G. Evenbly, Algorithms for tensor network renormalization, Phys. Rev. B95 (2017) 045117. arXiv:1509.07484, doi:10.1103/PhysRevB.95.045117.</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> diff --git a/tests/units/elsevier/data/address-line-valid/main_rjjlr.xml b/tests/units/elsevier/data/address-line-valid/main_rjjlr.xml index 928b68c4..4034b408 100644 --- a/tests/units/elsevier/data/address-line-valid/main_rjjlr.xml +++ b/tests/units/elsevier/data/address-line-valid/main_rjjlr.xml @@ -1 +1 @@ -<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd" [<!ENTITY gr001 SYSTEM "gr001" NDATA IMAGE><!ENTITY gr002 SYSTEM "gr002" NDATA IMAGE><!ENTITY gr003 SYSTEM "gr003" NDATA IMAGE><!ENTITY gr004 SYSTEM "gr004" NDATA IMAGE><!ENTITY gr005 SYSTEM "gr005" NDATA IMAGE>]><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="sco" xml:lang="en"><item-info><jid>PLB</jid><aid>137649</aid><ce:article-number>137649</ce:article-number><ce:pii>S0370-2693(22)00783-3</ce:pii><ce:doi>10.1016/j.physletb.2022.137649</ce:doi><ce:copyright year="2023" type="other">The Author(s)</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>Experiments</ce:text></ce:doctopic></ce:doctopics><ce:preprint><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/preprint" xlink:href="arxiv:2204.10157" id="inf0010"/></ce:preprint><ce:associated-resource><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/research-data" xlink:href="https://www.hepdata.net/" id="inf0540">https://www.hepdata.net/</ce:inter-ref></ce:associated-resource></item-info><ce:floats><ce:figure id="fg0010"><ce:label>Fig. 1</ce:label><ce:caption id="cp0010"><ce:simple-para id="sp0010">Illustration of toward, away and transverse regions with respect to the leading particle in a collision.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0010">Fig. 1</ce:alt-text><ce:link locator="gr001" xlink:type="simple" xlink:href="pii:S0370269322007833/gr001" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0010"/></ce:figure><ce:figure id="fg0020"><ce:label>Fig. 2</ce:label><ce:caption id="cp0020"><ce:simple-para id="sp0020">Top panels: transverse momentum spectra of charged particles in the transverse region for different multiplicity classes in pp (left), p–Pb (middle) and Pb–Pb (right) collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra are measured at mid pseudorapidity (|<ce:italic>η</ce:italic>| < 0.8). Lower panels: Ratio of <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra in different multiplicity classes to the <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectrum in the 0−100% multiplicity class for the corresponding collision systems. The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0020">Fig. 2</ce:alt-text><ce:link locator="gr002" xlink:type="simple" xlink:href="pii:S0370269322007833/gr002" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0020"/></ce:figure><ce:figure id="fg0030"><ce:label>Fig. 3</ce:label><ce:caption id="cp0030"><ce:simple-para id="sp0030">Transverse momentum spectra of charged particles in Toward-Transverse, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> (top plot) and Away-Transverse, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> (bottom plot) regions for different multiplicity classes in pp (left), p–Pb (middle) and Pb–Pb (right) collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra are measured at mid pseudorapidity (|<ce:italic>η</ce:italic>| < 0.8). The lower panels of both plots show the ratio to minimum bias pp collisions. The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0030">Fig. 3</ce:alt-text><ce:link locator="gr003" xlink:type="simple" xlink:href="pii:S0370269322007833/gr003" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0030"/></ce:figure><ce:figure id="fg0040"><ce:label>Fig. 4</ce:label><ce:caption id="cp0040"><ce:simple-para id="sp0040">The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> (left) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> (right) as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> in 4 <<ce:italic>p</ce:italic><ce:inf>T</ce:inf>< 6 GeV/<ce:italic>c</ce:italic> for different multiplicity classes in pp, p–Pb and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. Pb–Pb results are shown assuming a flat background (filled markers), and assuming a <ce:italic>v</ce:italic><ce:inf>2</ce:inf>-modulated background (empty markers). The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0040">Fig. 4</ce:alt-text><ce:link locator="gr004" xlink:type="simple" xlink:href="pii:S0370269322007833/gr004" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0040"/></ce:figure><ce:figure id="fg0050"><ce:label>Fig. 5</ce:label><ce:caption id="cp0050"><ce:simple-para id="sp0050">Comparison of the measured the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> (left) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> (right) in 4 <<ce:italic>p</ce:italic><ce:inf>T</ce:inf>< 6 GeV/<ce:italic>c</ce:italic> with model predictions. The results are shown as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> for different multiplicity classes in pp (top panel), p–Pb (middle panel) and Pb–Pb (bottom panel) collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The red and magenta lines show the <ce:small-caps>PYTHIA</ce:small-caps> 8 (Monash) <ce:cross-ref refid="br0280" id="crf0010">[28]</ce:cross-ref> and <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr <ce:cross-ref refid="br0280" id="crf0020">[28]</ce:cross-ref> predictions, respectively. The blue lines show the EPOS-LHC <ce:cross-ref refid="br0210" id="crf0030">[21]</ce:cross-ref> results. The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0050">Fig. 5</ce:alt-text><ce:link locator="gr005" xlink:type="simple" xlink:href="pii:S0370269322007833/gr005" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0050"/></ce:figure><ce:table xmlns="http://www.elsevier.com/xml/common/cals/dtd" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" id="tbl0010" frame="topbot" rowsep="0" colsep="0"><ce:label>Table 1</ce:label><ce:caption id="cp0060"><ce:simple-para id="sp0060">Contributions to the relative (%) systematic uncertainty on the <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra of primary charged particles in pp, p–Pb, and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. Just for illustration, the range in the table corresponds to the lowest and highest relative systematic uncertainty in the considered <ce:italic>p</ce:italic><ce:inf>T</ce:inf> range. The individual contributions are summed in quadrature to obtain the total uncertainty.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0060">Table 1</ce:alt-text><tgroup cols="4"><colspec colnum="1" colname="col1" align="left"/><colspec colnum="2" colname="col2" align="left"/><colspec colnum="3" colname="col3" align="left"/><colspec colnum="4" colname="col4" align="left"/><thead valign="top"><row rowsep="1"><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Source of uncertainty</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">pp</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">p–Pb</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">Pb–Pb</entry></row></thead><tbody valign="top"><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Track selection</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.1–8.2</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.4–5.8</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.0–9.9</entry></row><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Particle composition</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.3–1.8</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.5–1.9</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.3–2.4</entry></row><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Secondary particles</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.0–0.4</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.0–2.4</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.0–1.9</entry></row><row rowsep="1"><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Matching efficiency</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.0–4.2</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.7–3.7</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.6–3.7</entry></row><row rowsep="1"><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Total</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.2–8.8</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.6–6.3</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.5–10.0</entry></row><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Total (<ce:italic>N</ce:italic><ce:inf>ch</ce:inf>-dependent)</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.0–4.5</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">1.7–4.0</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">1.1–3.7</entry></row></tbody></tgroup></ce:table></ce:floats><head><ce:title id="ti0010">Study of charged particle production at high <ce:italic>p</ce:italic><ce:inf>T</ce:inf> using event topology in pp, p–Pb and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV</ce:title><ce:author-group id="ag0010"><ce:collaboration id="co0010" collaboration-id="S0370269322007833-3bea72599603117cd9d18494a0279c47"><ce:text>ALICE Collaboration</ce:text><ce:cross-ref refid="fn0080" id="crf0040"><ce:sup>⋆</ce:sup></ce:cross-ref><ce:author-group id="ag0020"><ce:author orcid="0000-0002-9213-5329" id="au0010" author-id="S0370269322007833-e60c93a934b81cf9801254193264c6ee"><ce:given-name>S.</ce:given-name><ce:surname>Acharya</ce:surname><ce:cross-ref refid="aff1240" id="crf0050"><ce:sup>124</ce:sup></ce:cross-ref><ce:cross-ref refid="aff1310" id="crf0060"><ce:sup>131</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-0504-7428" id="au0020" author-id="S0370269322007833-0eab85892b6d74b18661e74a7987c599"><ce:given-name>D.</ce:given-name><ce:surname>Adamová</ce:surname><ce:cross-ref refid="aff0860" id="crf0070"><ce:sup>86</ce:sup></ce:cross-ref></ce:author><ce:author id="au0030" author-id="S0370269322007833-e83a30ae1d5f89088c60ca2ee154d714"><ce:given-name>A.</ce:given-name><ce:surname>Adler</ce:surname><ce:cross-ref refid="aff0690" id="crf0080"><ce:sup>69</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-9611-3696" id="au0040" author-id="S0370269322007833-ed2d58d89990c41bb43c091d01e5029a"><ce:given-name>G.</ce:given-name><ce:surname>Aglieri Rinella</ce:surname><ce:cross-ref refid="aff0320" id="crf0090"><ce:sup>32</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-0760-5075" id="au0050" author-id="S0370269322007833-0c7a7863b7384aa5fdf06a0f187949c8"><ce:given-name>M.</ce:given-name><ce:surname>Agnello</ce:surname><ce:cross-ref refid="aff0290" id="crf0100"><ce:sup>29</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0003-0348-9836" id="au0060" author-id="S0370269322007833-bbdbb014653d7bdacb111c613a0fcbe0"><ce:given-name>N.</ce:given-name><ce:surname>Agrawal</ce:surname><ce:cross-ref refid="aff0500" id="crf0110"><ce:sup>50</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0001-5241-7412" id="au0070" author-id="S0370269322007833-2059a9508121069139ea49dbf566539a"><ce:given-name>Z.</ce:given-name><ce:surname>Ahammed</ce:surname><ce:cross-ref refid="aff1310" id="crf0120"><ce:sup>131</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0003-0497-5705" id="au0080" author-id="S0370269322007833-397c2b4743f367cb4aceb69d448bb6c6"><ce:given-name>S.</ce:given-name><ce:surname>Ahmad</ce:surname><ce:cross-ref refid="aff0150" id="crf0130"><ce:sup>15</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0001-8847-489X" id="au0090" author-id="S0370269322007833-f7c3cbd2e9a0f545640d2202ad1ddbf3"><ce:given-name>S.U.</ce:given-name><ce:surname>Ahn</ce:surname><ce:cross-ref refid="aff0700" id="crf0140"><ce:sup>70</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-4417-1392" id="au0100" author-id="S0370269322007833-07322fd13596772e8f99876f081be003"><ce:given-name>I.</ce:given-name><ce:surname>Ahuja</ce:surname><ce:cross-ref refid="aff0370" id="crf0150"><ce:sup>37</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-7388-3022" id="au0110" author-id="S0370269322007833-e1ca35714b53c8e677e37d627a5bbf38"><ce:given-name>A.</ce:given-name><ce:surname>Akindinov</ce:surname><ce:cross-ref refid="aff1390" id="crf0160"><ce:sup>139</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-8071-4497" id="au0120" author-id="S0370269322007833-92548b9c2ec6b0a94be4b4532227aed9"><ce:given-name>M.</ce:given-name><ce:surname>Al-Turany</ce:surname><ce:cross-ref refid="aff0980" 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author-id="S0370269322007833-617c42c28cb1b1d3dab680558c293767"><ce:given-name>Y.</ce:given-name><ce:surname>Zhi</ce:surname><ce:cross-ref refid="aff0100" id="crf10800"><ce:sup>10</ce:sup></ce:cross-ref></ce:author><ce:author id="au10260" author-id="S0370269322007833-3038f7cd79250ffa05fa0aff7644d2b3"><ce:given-name>N.</ce:given-name><ce:surname>Zhigareva</ce:surname><ce:cross-ref refid="aff1390" id="crf10810"><ce:sup>139</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0009-0009-2528-906X" id="au10270" author-id="S0370269322007833-e9bc0465b352cd706eac4fc8b795a4c2"><ce:given-name>D.</ce:given-name><ce:surname>Zhou</ce:surname><ce:cross-ref refid="aff0060" id="crf10820"><ce:sup>6</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-7868-6706" id="au10280" author-id="S0370269322007833-4a0cdba9c352af9310759dd83e906db9"><ce:given-name>Y.</ce:given-name><ce:surname>Zhou</ce:surname><ce:cross-ref refid="aff0830" id="crf10830"><ce:sup>83</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0001-9358-5762" id="au10290" author-id="S0370269322007833-b564bd6149c66f06bbea2780670b8f50"><ce:given-name>J.</ce:given-name><ce:surname>Zhu</ce:surname><ce:cross-ref refid="aff0980" id="crf10840"><ce:sup>98</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0060" id="crf10850"><ce:sup>6</ce:sup></ce:cross-ref></ce:author><ce:author id="au10300" author-id="S0370269322007833-1fb0ab925cfcdaacb160193d82c1a978"><ce:given-name>Y.</ce:given-name><ce:surname>Zhu</ce:surname><ce:cross-ref refid="aff0060" id="crf10860"><ce:sup>6</ce:sup></ce:cross-ref></ce:author><ce:author id="au10310" author-id="S0370269322007833-2ad055472e2fac995111c51c08396d0c"><ce:given-name>G.</ce:given-name><ce:surname>Zinovjev</ce:surname><ce:cross-ref refid="aff0030" id="crf10870"><ce:sup>3</ce:sup></ce:cross-ref><ce:cross-ref refid="fn0010" id="crf10880"><ce:sup>I</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-7478-2493" id="au10320" author-id="S0370269322007833-92b5ecf5612e2e849fc1bba72c6cff99"><ce:given-name>N.</ce:given-name><ce:surname>Zurlo</ce:surname><ce:cross-ref refid="aff1300" id="crf10890"><ce:sup>130</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0540" id="crf10900"><ce:sup>54</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="aff0010" affiliation-id="S0370269322007833-79d30baa35325e84d46378ba6ce12c18"><ce:label>1</ce:label><ce:textfn>A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia</ce:textfn><sa:affiliation><sa:organization>A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation</sa:organization><sa:city>Yerevan</sa:city><sa:country>Armenia</sa:country></sa:affiliation><ce:source-text id="srct0005">A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia</ce:source-text></ce:affiliation><ce:affiliation id="aff0020" affiliation-id="S0370269322007833-65754d218cf7f84bf1e02306b80caca0"><ce:label>2</ce:label><ce:textfn>AGH University of Science and Technology, Cracow, Poland</ce:textfn><sa:affiliation><sa:organization>AGH University of Science and Technology</sa:organization><sa:city>Cracow</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0010">AGH University of Science and Technology, Cracow, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff0030" affiliation-id="S0370269322007833-e916a2f48a17bc32220b61ae0e9b8e05"><ce:label>3</ce:label><ce:textfn>Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine</ce:textfn><sa:affiliation><sa:organization>Bogolyubov Institute for Theoretical Physics</sa:organization><sa:organization>National Academy of Sciences of Ukraine</sa:organization><sa:city>Kiev</sa:city><sa:country>Ukraine</sa:country></sa:affiliation><ce:source-text id="srct0015">Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine</ce:source-text></ce:affiliation><ce:affiliation id="aff0040" affiliation-id="S0370269322007833-9869d95133b2275e34836b8ad2f235c3"><ce:label>4</ce:label><ce:textfn>Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Bose Institute</sa:organization><sa:organization>Department of Physics</sa:organization><sa:organization>Centre for Astroparticle Physics and Space Science (CAPSS)</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0020">Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0050" affiliation-id="S0370269322007833-b7f796ef6c934d79a497288cdec192f6"><ce:label>5</ce:label><ce:textfn>California Polytechnic State University, San Luis Obispo, CA, United States</ce:textfn><sa:affiliation><sa:organization>California Polytechnic State University</sa:organization><sa:city>San Luis Obispo</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0025">California Polytechnic State University, San Luis Obispo, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0060" affiliation-id="S0370269322007833-3dcd6ffc2e8f27d6ed2f6237a209a384"><ce:label>6</ce:label><ce:textfn>Central China Normal University, Wuhan, China</ce:textfn><sa:affiliation><sa:organization>Central China Normal University</sa:organization><sa:city>Wuhan</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0030">Central China Normal University, Wuhan, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0070" affiliation-id="S0370269322007833-110460c7f2fbb319ec6d52e7cc3fc1d5"><ce:label>7</ce:label><ce:textfn>Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba</ce:textfn><sa:affiliation><sa:organization>Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN)</sa:organization><sa:city>Havana</sa:city><sa:country>Cuba</sa:country></sa:affiliation><ce:source-text id="srct0035">Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba</ce:source-text></ce:affiliation><ce:affiliation id="aff0080" affiliation-id="S0370269322007833-641aa526558990d110c090840e6f3d0c"><ce:label>8</ce:label><ce:textfn>Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico</ce:textfn><sa:affiliation><sa:organization>Centro de Investigación y de Estudios Avanzados (CINVESTAV)</sa:organization><sa:city>Mexico City and Mérida</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0040">Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0090" affiliation-id="S0370269322007833-9c73ece39ab638447cd10f268e329b2e"><ce:label>9</ce:label><ce:textfn>Chicago State University, Chicago, IL, United States</ce:textfn><sa:affiliation><sa:organization>Chicago State University</sa:organization><sa:city>Chicago</sa:city><sa:state>IL</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0045">Chicago State University, Chicago, Illinois, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0100" affiliation-id="S0370269322007833-d7da93bf3f02be0a46bee74f439b5930"><ce:label>10</ce:label><ce:textfn>China Institute of Atomic Energy, Beijing, China</ce:textfn><sa:affiliation><sa:organization>China Institute of Atomic Energy</sa:organization><sa:city>Beijing</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0050">China Institute of Atomic Energy, Beijing, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0110" affiliation-id="S0370269322007833-d477851cd020cd39da135784676a96ff"><ce:label>11</ce:label><ce:textfn>Chungbuk National University, Cheongju, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Chungbuk National University</sa:organization><sa:city>Cheongju</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0055">Chungbuk National University, Cheongju, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0120" affiliation-id="S0370269322007833-b03a6e652a70daee257206602eb6f83f"><ce:label>12</ce:label><ce:textfn>Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Comenius University Bratislava</sa:organization><sa:organization>Faculty of Mathematics, Physics and Informatics</sa:organization><sa:city>Bratislava</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0060">Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0130" affiliation-id="S0370269322007833-78cc8333b14d598fc5e3c0230d1de22e"><ce:label>13</ce:label><ce:textfn>COMSATS University Islamabad, Islamabad, Pakistan</ce:textfn><sa:affiliation><sa:organization>COMSATS University Islamabad</sa:organization><sa:city>Islamabad</sa:city><sa:country>Pakistan</sa:country></sa:affiliation><ce:source-text id="srct0065">COMSATS University Islamabad, Islamabad, Pakistan</ce:source-text></ce:affiliation><ce:affiliation id="aff0140" affiliation-id="S0370269322007833-42161ea7466da89325c5b37601107c55"><ce:label>14</ce:label><ce:textfn>Creighton University, Omaha, NE, United States</ce:textfn><sa:affiliation><sa:organization>Creighton University</sa:organization><sa:city>Omaha</sa:city><sa:state>NE</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0070">Creighton University, Omaha, Nebraska, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0150" affiliation-id="S0370269322007833-bd5e6b818668501c1deea76e96cac833"><ce:label>15</ce:label><ce:textfn>Department of Physics, Aligarh Muslim University, Aligarh, India</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Aligarh Muslim University</sa:organization><sa:city>Aligarh</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0075">Department of Physics, Aligarh Muslim University, Aligarh, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0160" affiliation-id="S0370269322007833-ebe7875fb705d3c179953d196aa8b94b"><ce:label>16</ce:label><ce:textfn>Department of Physics, Pusan National University, Pusan, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Pusan National University</sa:organization><sa:city>Pusan</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0080">Department of Physics, Pusan National University, Pusan, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0170" affiliation-id="S0370269322007833-24ddc81b37e640977b88412ba28336a4"><ce:label>17</ce:label><ce:textfn>Department of Physics, Sejong University, Seoul, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Sejong University</sa:organization><sa:city>Seoul</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0085">Department of Physics, Sejong University, Seoul, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0180" affiliation-id="S0370269322007833-66245837acb0d7fb7ada51de0fd95043"><ce:label>18</ce:label><ce:textfn>Department of Physics, University of California, Berkeley, CA, United States</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>University of California</sa:organization><sa:city>Berkeley</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0090">Department of Physics, University of California, Berkeley, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0190" affiliation-id="S0370269322007833-63776eabafa9e32856b35562692a4488"><ce:label>19</ce:label><ce:textfn>Department of Physics, University of Oslo, Oslo, Norway</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>University of Oslo</sa:organization><sa:city>Oslo</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0095">Department of Physics, University of Oslo, Oslo, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0200" affiliation-id="S0370269322007833-020451a59d5e257505166bdb7847cdd7"><ce:label>20</ce:label><ce:textfn>Department of Physics and Technology, University of Bergen, Bergen, Norway</ce:textfn><sa:affiliation><sa:organization>Department of Physics and Technology</sa:organization><sa:organization>University of Bergen</sa:organization><sa:city>Bergen</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0100">Department of Physics and Technology, University of Bergen, Bergen, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0210" affiliation-id="S0370269322007833-f2de76852013198101844806b89d3836"><ce:label>21</ce:label><ce:textfn>Dipartimento di Fisica, Università di Pavia, Pavia, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica</sa:organization><sa:organization>Università di Pavia</sa:organization><sa:city>Pavia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0105">Dipartimento di Fisica, Università di Pavia, Pavia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0220" affiliation-id="S0370269322007833-81d87d2cd8c9f6aa2a1c7f4b13115ef3"><ce:label>22</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Cagliari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0110">Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0230" affiliation-id="S0370269322007833-849d0ba7888fcb2a09aad7ac1095ec15"><ce:label>23</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Trieste</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0115">Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0240" affiliation-id="S0370269322007833-68c63fd1cb9e8623af62118bbef39c9e"><ce:label>24</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0120">Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0250" affiliation-id="S0370269322007833-7a40dab487121429c63e59c3de15ccfa"><ce:label>25</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bologna</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0125">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0260" affiliation-id="S0370269322007833-72e3fe624de7d3d3223574366a22ef30"><ce:label>26</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Catania</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0130">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0270" affiliation-id="S0370269322007833-6145c67e3009fb117e82f5429b2af282"><ce:label>27</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Padova</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0135">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0280" affiliation-id="S0370269322007833-a001bc692b1f1f84377401c9e2632e51"><ce:label>28</ce:label><ce:textfn>Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università and Gruppo Collegato INFN, Salerno, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università</sa:organization><sa:organization>Gruppo Collegato INFN</sa:organization><sa:city>Salerno</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0140">Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università and Gruppo Collegato INFN, Salerno, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0290" affiliation-id="S0370269322007833-362983c57dbe79a62add2550bfe565dc"><ce:label>29</ce:label><ce:textfn>Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento DISAT del Politecnico</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0145">Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0300" affiliation-id="S0370269322007833-7a82e32411929fc5768d11b72609a4d8"><ce:label>30</ce:label><ce:textfn>Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Scienze MIFT</sa:organization><sa:organization>Università di Messina</sa:organization><sa:city>Messina</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0150">Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0310" affiliation-id="S0370269322007833-0bfba0b176e2b6e1b67e6564c87308b5"><ce:label>31</ce:label><ce:textfn>Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento Interateneo di Fisica ‘M. Merlin’</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0155">Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0320" affiliation-id="S0370269322007833-44f92095d23d6e3d2fb8e8fc998c51f2"><ce:label>32</ce:label><ce:textfn>European Organization for Nuclear Research (CERN), Geneva, Switzerland</ce:textfn><sa:affiliation><sa:organization>European Organization for Nuclear Research (CERN)</sa:organization><sa:city>Geneva</sa:city><sa:country>Switzerland</sa:country></sa:affiliation><ce:source-text id="srct0160">European Organization for Nuclear Research (CERN), Geneva, Switzerland</ce:source-text></ce:affiliation><ce:affiliation id="aff0330" affiliation-id="S0370269322007833-f220870dd6ed2747e8ec11b2ba624bf5"><ce:label>33</ce:label><ce:textfn>Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia</ce:textfn><sa:affiliation><sa:organization>Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture</sa:organization><sa:organization>University of Split</sa:organization><sa:city>Split</sa:city><sa:country>Croatia</sa:country></sa:affiliation><ce:source-text id="srct0165">Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia</ce:source-text></ce:affiliation><ce:affiliation id="aff0340" affiliation-id="S0370269322007833-5bf4b9d297f0544e6037addfb7689840"><ce:label>34</ce:label><ce:textfn>Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway</ce:textfn><sa:affiliation><sa:organization>Faculty of Engineering and Science</sa:organization><sa:organization>Western Norway University of Applied Sciences</sa:organization><sa:city>Bergen</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0170">Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0350" affiliation-id="S0370269322007833-56d412e1c3fe114d9a18225fa76274c9"><ce:label>35</ce:label><ce:textfn>Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Faculty of Nuclear Sciences and Physical Engineering</sa:organization><sa:organization>Czech Technical University in Prague</sa:organization><sa:city>Prague</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0175">Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0360" affiliation-id="S0370269322007833-de312990ec5b7745ed2a9609074bb450"><ce:label>36</ce:label><ce:textfn>Faculty of Physics, Sofia University, Sofia, Bulgaria</ce:textfn><sa:affiliation><sa:organization>Faculty of Physics</sa:organization><sa:organization>Sofia University</sa:organization><sa:city>Sofia</sa:city><sa:country>Bulgaria</sa:country></sa:affiliation><ce:source-text id="srct0180">Faculty of Physics, Sofia University, Sofia, Bulgaria</ce:source-text></ce:affiliation><ce:affiliation id="aff0370" affiliation-id="S0370269322007833-0879c60c6550c16bd60686725f5c6938"><ce:label>37</ce:label><ce:textfn>Faculty of Science, P.J. Šafárik University, Košice, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Faculty of Science</sa:organization><sa:organization>P.J. Šafárik University</sa:organization><sa:city>Košice</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0185">Faculty of Science, P.J. Šafárik University, Košice, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0380" affiliation-id="S0370269322007833-a18716885f5bc83f5d1aee8d0d80c7af"><ce:label>38</ce:label><ce:textfn>Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Frankfurt Institute for Advanced Studies</sa:organization><sa:organization>Johann Wolfgang Goethe-Universität Frankfurt</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0190">Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0390" affiliation-id="S0370269322007833-f0b0a2b18fef5547dcd39253b5714404"><ce:label>39</ce:label><ce:textfn>Fudan University, Shanghai, China</ce:textfn><sa:affiliation><sa:organization>Fudan University</sa:organization><sa:city>Shanghai</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0195">Fudan University, Shanghai, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0400" affiliation-id="S0370269322007833-a2398937a38c48ebf6ac3ff5e5d60b95"><ce:label>40</ce:label><ce:textfn>Gangneung-Wonju National University, Gangneung, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Gangneung-Wonju National University</sa:organization><sa:city>Gangneung</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0200">Gangneung-Wonju National University, Gangneung, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0410" affiliation-id="S0370269322007833-73bc80872cd116960a09bc477e035838"><ce:label>41</ce:label><ce:textfn>Gauhati University, Department of Physics, Guwahati, India</ce:textfn><sa:affiliation><sa:organization>Gauhati University</sa:organization><sa:organization>Department of Physics</sa:organization><sa:city>Guwahati</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0205">Gauhati University, Department of Physics, Guwahati, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0420" affiliation-id="S0370269322007833-9fd27eebf76465366fe59cb8d9620ae8"><ce:label>42</ce:label><ce:textfn>Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany</ce:textfn><sa:affiliation><sa:organization>Helmholtz-Institut für Strahlen- und Kernphysik</sa:organization><sa:organization>Rheinische Friedrich-Wilhelms-Universität Bonn</sa:organization><sa:city>Bonn</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0210">Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0430" affiliation-id="S0370269322007833-66b9f8889421b091fa908925e638d91e"><ce:label>43</ce:label><ce:textfn>Helsinki Institute of Physics (HIP), Helsinki, Finland</ce:textfn><sa:affiliation><sa:organization>Helsinki Institute of Physics (HIP)</sa:organization><sa:city>Helsinki</sa:city><sa:country>Finland</sa:country></sa:affiliation><ce:source-text id="srct0215">Helsinki Institute of Physics (HIP), Helsinki, Finland</ce:source-text></ce:affiliation><ce:affiliation id="aff0440" affiliation-id="S0370269322007833-6968018d6fe1bd12a4413918be70ab85"><ce:label>44</ce:label><ce:textfn>High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico</ce:textfn><sa:affiliation><sa:organization>High Energy Physics Group</sa:organization><sa:organization>Universidad Autónoma de Puebla</sa:organization><sa:city>Puebla</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0220">High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0450" affiliation-id="S0370269322007833-16bdf6e3577a480da99159f5d54452b1"><ce:label>45</ce:label><ce:textfn>Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>Horia Hulubei National Institute of Physics and Nuclear Engineering</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0225">Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff0460" affiliation-id="S0370269322007833-78df0ad3e050545612fd8d71a4776a19"><ce:label>46</ce:label><ce:textfn>Indian Institute of Technology Bombay (IIT), Mumbai, India</ce:textfn><sa:affiliation><sa:organization>Indian Institute of Technology Bombay (IIT)</sa:organization><sa:city>Mumbai</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0230">Indian Institute of Technology Bombay (IIT), Mumbai, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0470" affiliation-id="S0370269322007833-b8ad6c9375b7a89b768adb13f27427b4"><ce:label>47</ce:label><ce:textfn>Indian Institute of Technology Indore, Indore, India</ce:textfn><sa:affiliation><sa:organization>Indian Institute of Technology Indore</sa:organization><sa:city>Indore</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0235">Indian Institute of Technology Indore, Indore, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0480" affiliation-id="S0370269322007833-25658fba725d22058ae2a8649ceeb084"><ce:label>48</ce:label><ce:textfn>INFN, Laboratori Nazionali di Frascati, Frascati, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Laboratori Nazionali di Frascati</sa:organization><sa:city>Frascati</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0240">INFN, Laboratori Nazionali di Frascati, Frascati, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0490" affiliation-id="S0370269322007833-5d9dee68bdf16e34f2f3f01d930f367d"><ce:label>49</ce:label><ce:textfn>INFN, Sezione di Bari, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Bari</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0245">INFN, Sezione di Bari, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0500" affiliation-id="S0370269322007833-340d750afed4ff705cf1dd72bf688ca4"><ce:label>50</ce:label><ce:textfn>INFN, Sezione di Bologna, Bologna, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Bologna</sa:organization><sa:city>Bologna</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0250">INFN, Sezione di Bologna, Bologna, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0510" affiliation-id="S0370269322007833-436e8c989d6c10cae74944a81714a4e2"><ce:label>51</ce:label><ce:textfn>INFN, Sezione di Cagliari, Cagliari, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Cagliari</sa:organization><sa:city>Cagliari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0255">INFN, Sezione di Cagliari, Cagliari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0520" affiliation-id="S0370269322007833-f0607c03a8b381da2bb83a11f8d899c8"><ce:label>52</ce:label><ce:textfn>INFN, Sezione di Catania, Catania, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Catania</sa:organization><sa:city>Catania</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0260">INFN, Sezione di Catania, Catania, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0530" affiliation-id="S0370269322007833-cb33711bf32ecc9cc4697cbbea10fa88"><ce:label>53</ce:label><ce:textfn>INFN, Sezione di Padova, Padova, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Padova</sa:organization><sa:city>Padova</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0265">INFN, Sezione di Padova, Padova, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0540" affiliation-id="S0370269322007833-5c70f16aa0d389aa33e2589db6fcd5d6"><ce:label>54</ce:label><ce:textfn>INFN, Sezione di Pavia, Pavia, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Pavia</sa:organization><sa:city>Pavia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0270">INFN, Sezione di Pavia, Pavia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0550" affiliation-id="S0370269322007833-f1faa0de67f6d8d1ece0914257c7635c"><ce:label>55</ce:label><ce:textfn>INFN, Sezione di Torino, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Torino</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0275">INFN, Sezione di Torino, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0560" affiliation-id="S0370269322007833-0035d662cbe7eef98bed21545fd04324"><ce:label>56</ce:label><ce:textfn>INFN, Sezione di Trieste, Trieste, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Trieste</sa:organization><sa:city>Trieste</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0280">INFN, Sezione di Trieste, Trieste, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0570" affiliation-id="S0370269322007833-22e3a6d8462b39bb6c155bce0c9b21cd"><ce:label>57</ce:label><ce:textfn>Inha University, Incheon, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Inha University</sa:organization><sa:city>Incheon</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0285">Inha University, Incheon, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0580" affiliation-id="S0370269322007833-d4941ebd67ab9bbf96b1e6a906f4397c"><ce:label>58</ce:label><ce:textfn>Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands</ce:textfn><sa:affiliation><sa:organization>Institute for Gravitational and Subatomic Physics (GRASP)</sa:organization><sa:organization>Utrecht University/Nikhef</sa:organization><sa:city>Utrecht</sa:city><sa:country>Netherlands</sa:country></sa:affiliation><ce:source-text id="srct0290">Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands</ce:source-text></ce:affiliation><ce:affiliation id="aff0590" affiliation-id="S0370269322007833-f6d8becf2ef35885577b8164313401cb"><ce:label>59</ce:label><ce:textfn>Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Institute of Experimental Physics</sa:organization><sa:organization>Slovak Academy of Sciences</sa:organization><sa:city>Košice</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0295">Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0600" affiliation-id="S0370269322007833-6fa96ff1fcc4aaff09633c1217f06ff0"><ce:label>60</ce:label><ce:textfn>Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India</ce:textfn><sa:affiliation><sa:organization>Institute of Physics</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Bhubaneswar</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0300">Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0610" affiliation-id="S0370269322007833-9f77e70dcbde10c6ed3d37b3b0071107"><ce:label>61</ce:label><ce:textfn>Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Institute of Physics of the Czech Academy of Sciences</sa:organization><sa:city>Prague</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0305">Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0620" affiliation-id="S0370269322007833-3309ea65bfe3ae2f96b232545cb67043"><ce:label>62</ce:label><ce:textfn>Institute of Space Science (ISS), Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>Institute of Space Science (ISS)</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0310">Institute of Space Science (ISS), Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff0630" affiliation-id="S0370269322007833-2759820a41f8b0ece8c42aa319465ed5"><ce:label>63</ce:label><ce:textfn>Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Institut für Kernphysik</sa:organization><sa:organization>Johann Wolfgang Goethe-Universität Frankfurt</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0315">Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0640" affiliation-id="S0370269322007833-b18018b82a469dc4e4eb94f595cf4812"><ce:label>64</ce:label><ce:textfn>Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:textfn><sa:affiliation><sa:organization>Instituto de Ciencias Nucleares</sa:organization><sa:organization>Universidad Nacional Autónoma de México</sa:organization><sa:city>Mexico City</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0320">Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0650" affiliation-id="S0370269322007833-2aa194eda46d62198ea9b929240200b8"><ce:label>65</ce:label><ce:textfn>Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil</ce:textfn><sa:affiliation><sa:organization>Instituto de Física</sa:organization><sa:organization>Universidade Federal do Rio Grande do Sul (UFRGS)</sa:organization><sa:city>Porto Alegre</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0325">Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff0660" affiliation-id="S0370269322007833-9e9b95fa2c082308cb0efc3488541c67"><ce:label>66</ce:label><ce:textfn>Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:textfn><sa:affiliation><sa:organization>Instituto de Física</sa:organization><sa:organization>Universidad Nacional Autónoma de México</sa:organization><sa:city>Mexico City</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0330">Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0670" affiliation-id="S0370269322007833-4ef9aeae6b2f74e366edd12aafbea4cf"><ce:label>67</ce:label><ce:textfn>iThemba LABS, National Research Foundation, Somerset West, South Africa</ce:textfn><sa:affiliation><sa:organization>iThemba LABS</sa:organization><sa:organization>National Research Foundation</sa:organization><sa:city>Somerset West</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0335">iThemba LABS, National Research Foundation, Somerset West, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff0680" affiliation-id="S0370269322007833-2f4c4db0447d27fd2e43117b90bb74d4"><ce:label>68</ce:label><ce:textfn>Jeonbuk National University, Jeonju, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Jeonbuk National University</sa:organization><sa:city>Jeonju</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0340">Jeonbuk National University, Jeonju, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0690" affiliation-id="S0370269322007833-81ec7d4c8c69486a57ef1ca6a567076c"><ce:label>69</ce:label><ce:textfn>Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik</sa:organization><sa:organization>Fachbereich Informatik und Mathematik</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0345">Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0700" affiliation-id="S0370269322007833-c1d849875c0de0db6476f9cc96b07dd2"><ce:label>70</ce:label><ce:textfn>Korea Institute of Science and Technology Information, Daejeon, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Korea Institute of Science and Technology Information</sa:organization><sa:city>Daejeon</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0350">Korea Institute of Science and Technology Information, Daejeon, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0710" affiliation-id="S0370269322007833-c5cd420fa3d6b7c79dc3ab9e7c2dfb06"><ce:label>71</ce:label><ce:textfn>KTO Karatay University, Konya, Turkey</ce:textfn><sa:affiliation><sa:organization>KTO Karatay University</sa:organization><sa:city>Konya</sa:city><sa:country>Turkey</sa:country></sa:affiliation><ce:source-text id="srct0355">KTO Karatay University, Konya, Turkey</ce:source-text></ce:affiliation><ce:affiliation id="aff0720" affiliation-id="S0370269322007833-0bdd954451acb47f491c8c4996fa87da"><ce:label>72</ce:label><ce:textfn>Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France</ce:textfn><sa:affiliation><sa:organization>Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie</sa:organization><sa:city>Orsay</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0360">Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France</ce:source-text></ce:affiliation><ce:affiliation id="aff0730" affiliation-id="S0370269322007833-1487532e5bfe30325bc10c0583f7c38e"><ce:label>73</ce:label><ce:textfn>Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France</ce:textfn><sa:affiliation><sa:organization>Laboratoire de Physique Subatomique et de Cosmologie</sa:organization><sa:organization>Université Grenoble-Alpes</sa:organization><sa:organization>CNRS-IN2P3</sa:organization><sa:city>Grenoble</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0365">Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France</ce:source-text></ce:affiliation><ce:affiliation id="aff0740" affiliation-id="S0370269322007833-dd5512fbf2faf90b56635e0b411d44a2"><ce:label>74</ce:label><ce:textfn>Lawrence Berkeley National Laboratory, Berkeley, CA, United States</ce:textfn><sa:affiliation><sa:organization>Lawrence Berkeley National Laboratory</sa:organization><sa:city>Berkeley</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0370">Lawrence Berkeley National Laboratory, Berkeley, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0750" affiliation-id="S0370269322007833-ed03eefd58007822249c697d882deffc"><ce:label>75</ce:label><ce:textfn>Lund University Department of Physics, Division of Particle Physics, Lund, Sweden</ce:textfn><sa:affiliation><sa:organization>Lund University Department of Physics</sa:organization><sa:organization>Division of Particle Physics</sa:organization><sa:city>Lund</sa:city><sa:country>Sweden</sa:country></sa:affiliation><ce:source-text id="srct0375">Lund University Department of Physics, Division of Particle Physics, Lund, Sweden</ce:source-text></ce:affiliation><ce:affiliation id="aff0760" affiliation-id="S0370269322007833-5b47ca06644b7f88e2267e9accbe867b"><ce:label>76</ce:label><ce:textfn>Nagasaki Institute of Applied Science, Nagasaki, Japan</ce:textfn><sa:affiliation><sa:organization>Nagasaki Institute of Applied Science</sa:organization><sa:city>Nagasaki</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0380">Nagasaki Institute of Applied Science, Nagasaki, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0770" affiliation-id="S0370269322007833-61cd55f0b96e9831d6063318d9967d43"><ce:label>77</ce:label><ce:textfn>Nara Women's University (NWU), Nara, Japan</ce:textfn><sa:affiliation><sa:organization>Nara Women's University (NWU)</sa:organization><sa:city>Nara</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0385">Nara Women's University (NWU), Nara, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0780" affiliation-id="S0370269322007833-7f50d99eb340e408fbc323cbce0c8905"><ce:label>78</ce:label><ce:textfn>National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece</ce:textfn><sa:affiliation><sa:organization>National and Kapodistrian University of Athens</sa:organization><sa:organization>School of Science</sa:organization><sa:organization>Department of Physics</sa:organization><sa:city>Athens</sa:city><sa:country>Greece</sa:country></sa:affiliation><ce:source-text id="srct0390">National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece</ce:source-text></ce:affiliation><ce:affiliation id="aff0790" affiliation-id="S0370269322007833-58192de9f95a93cb8bf485b9a5647bf5"><ce:label>79</ce:label><ce:textfn>National Centre for Nuclear Research, Warsaw, Poland</ce:textfn><sa:affiliation><sa:organization>National Centre for Nuclear Research</sa:organization><sa:city>Warsaw</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0395">National Centre for Nuclear Research, Warsaw, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff0800" affiliation-id="S0370269322007833-5c73c93f2c9aba1b7fcfa30741ce17ca"><ce:label>80</ce:label><ce:textfn>National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India</ce:textfn><sa:affiliation><sa:organization>National Institute of Science Education and Research</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Jatni</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0400">National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0810" affiliation-id="S0370269322007833-1e229a9758219f877a7f903c2dde9c2b"><ce:label>81</ce:label><ce:textfn>National Nuclear Research Center, Baku, Azerbaijan</ce:textfn><sa:affiliation><sa:organization>National Nuclear Research Center</sa:organization><sa:city>Baku</sa:city><sa:country>Azerbaijan</sa:country></sa:affiliation><ce:source-text id="srct0405">National Nuclear Research Center, Baku, Azerbaijan</ce:source-text></ce:affiliation><ce:affiliation id="aff0820" affiliation-id="S0370269322007833-ab3fe404fa49046cc1d6ed593a4d5eb8"><ce:label>82</ce:label><ce:textfn>National Research and Innovation Agency - BRIN, Jakarta, Indonesia</ce:textfn><sa:affiliation><sa:organization>National Research and Innovation Agency - BRIN</sa:organization><sa:city>Jakarta</sa:city><sa:country>Indonesia</sa:country></sa:affiliation><ce:source-text id="srct0410">National Research and Innovation Agency - BRIN, Jakarta, Indonesia</ce:source-text></ce:affiliation><ce:affiliation id="aff0830" affiliation-id="S0370269322007833-3e7cb9222cae4e75df0d93d2ea96329f"><ce:label>83</ce:label><ce:textfn>Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark</ce:textfn><sa:affiliation><sa:organization>Niels Bohr Institute</sa:organization><sa:organization>University of Copenhagen</sa:organization><sa:city>Copenhagen</sa:city><sa:country>Denmark</sa:country></sa:affiliation><ce:source-text id="srct0415">Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark</ce:source-text></ce:affiliation><ce:affiliation id="aff0840" affiliation-id="S0370269322007833-11b144edfb6d992e108d17991cc7681b"><ce:label>84</ce:label><ce:textfn>Nikhef, National institute for subatomic physics, Amsterdam, Netherlands</ce:textfn><sa:affiliation><sa:organization>Nikhef, National institute for subatomic physics</sa:organization><sa:city>Amsterdam</sa:city><sa:country>Netherlands</sa:country></sa:affiliation><ce:source-text id="srct0420">Nikhef, National institute for subatomic physics, Amsterdam, Netherlands</ce:source-text></ce:affiliation><ce:affiliation id="aff0850" affiliation-id="S0370269322007833-7ffc71d9a245ae3927bb4d373ff478ed"><ce:label>85</ce:label><ce:textfn>Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Group</sa:organization><sa:organization>STFC Daresbury Laboratory</sa:organization><sa:city>Daresbury</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0425">Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff0860" affiliation-id="S0370269322007833-4ea73349ab4e79a7b6570dd0fd42cd13"><ce:label>86</ce:label><ce:textfn>Nuclear Physics Institute of the Czech Academy of Sciences, Husinec-Řež, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Institute of the Czech Academy of Sciences</sa:organization><sa:city>Husinec-Řež</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0430">Nuclear Physics Institute of the Czech Academy of Sciences, Husinec-Řež, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0870" affiliation-id="S0370269322007833-43c0b745ad096562d858417552718575"><ce:label>87</ce:label><ce:textfn>Oak Ridge National Laboratory, Oak Ridge, TN, United States</ce:textfn><sa:affiliation><sa:organization>Oak Ridge National Laboratory</sa:organization><sa:city>Oak Ridge</sa:city><sa:state>TN</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0435">Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0880" affiliation-id="S0370269322007833-e4e32a64c7e436b0f2a688687dedfe5e"><ce:label>88</ce:label><ce:textfn>Ohio State University, Columbus, OH, United States</ce:textfn><sa:affiliation><sa:organization>Ohio State University</sa:organization><sa:city>Columbus</sa:city><sa:state>OH</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0440">Ohio State University, Columbus, Ohio, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0890" affiliation-id="S0370269322007833-3c4eb171ede7112c64f1bc2c164f4732"><ce:label>89</ce:label><ce:textfn>Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia</ce:textfn><sa:affiliation><sa:organization>Physics department</sa:organization><sa:organization>Faculty of science, University of Zagreb</sa:organization><sa:city>Zagreb</sa:city><sa:country>Croatia</sa:country></sa:affiliation><ce:source-text id="srct0445">Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia</ce:source-text></ce:affiliation><ce:affiliation id="aff0900" affiliation-id="S0370269322007833-30d6dbb96747c914840798f88bf250cb"><ce:label>90</ce:label><ce:textfn>Physics Department, Panjab University, Chandigarh, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>Panjab University</sa:organization><sa:city>Chandigarh</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0450">Physics Department, Panjab University, Chandigarh, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0910" affiliation-id="S0370269322007833-712bc80495e97050498604da603cb5cc"><ce:label>91</ce:label><ce:textfn>Physics Department, University of Jammu, Jammu, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>University of Jammu</sa:organization><sa:city>Jammu</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0455">Physics Department, University of Jammu, Jammu, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0920" affiliation-id="S0370269322007833-500771749243cadcbaeeac3f726e6a62"><ce:label>92</ce:label><ce:textfn>Physics Department, University of Rajasthan, Jaipur, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>University of Rajasthan</sa:organization><sa:city>Jaipur</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0460">Physics Department, University of Rajasthan, Jaipur, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0930" affiliation-id="S0370269322007833-87f0c1fe6b138b8300b7695dc3391666"><ce:label>93</ce:label><ce:textfn>Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan</ce:textfn><sa:affiliation><sa:organization>Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2)</sa:organization><sa:organization>Hiroshima University</sa:organization><sa:city>Hiroshima</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0465">Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0940" affiliation-id="S0370269322007833-0886f70da565231fd0e43398021fd604"><ce:label>94</ce:label><ce:textfn>Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany</ce:textfn><sa:affiliation><sa:organization>Physikalisches Institut</sa:organization><sa:organization>Eberhard-Karls-Universität Tübingen</sa:organization><sa:city>Tübingen</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0470">Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0950" affiliation-id="S0370269322007833-1c67678e124982de924355cdd6d6b91b"><ce:label>95</ce:label><ce:textfn>Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany</ce:textfn><sa:affiliation><sa:organization>Physikalisches Institut</sa:organization><sa:organization>Ruprecht-Karls-Universität Heidelberg</sa:organization><sa:city>Heidelberg</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0475">Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0960" affiliation-id="S0370269322007833-7a5765cc3928149c8b77128cd52462ae"><ce:label>96</ce:label><ce:textfn>Physik Department, Technische Universität München, Munich, Germany</ce:textfn><sa:affiliation><sa:organization>Physik Department</sa:organization><sa:organization>Technische Universität München</sa:organization><sa:city>Munich</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0480">Physik Department, Technische Universität München, Munich, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0970" affiliation-id="S0370269322007833-0a06f4f5fb60a25c8f1936ed71c96cfe"><ce:label>97</ce:label><ce:textfn>Politecnico di Bari and Sezione INFN, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>Politecnico di Bari</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0485">Politecnico di Bari and Sezione INFN, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0980" affiliation-id="S0370269322007833-3451d358e90b759225813c9b9a6f098c"><ce:label>98</ce:label><ce:textfn>Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany</ce:textfn><sa:affiliation><sa:organization>Research Division</sa:organization><sa:organization>ExtreMe Matter Institute EMMI</sa:organization><sa:organization>GSI Helmholtzzentrum für Schwerionenforschung GmbH</sa:organization><sa:city>Darmstadt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0490">Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0990" affiliation-id="S0370269322007833-2771a2ae295804be11a4a937b894a546"><ce:label>99</ce:label><ce:textfn>Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Saha Institute of Nuclear Physics</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0495">Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1000" affiliation-id="S0370269322007833-b602cde70213041ca40ec5720e4e3e75"><ce:label>100</ce:label><ce:textfn>School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom</ce:textfn><sa:affiliation><sa:organization>School of Physics and Astronomy</sa:organization><sa:organization>University of Birmingham</sa:organization><sa:city>Birmingham</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0500">School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff1010" affiliation-id="S0370269322007833-1a1f0f3ae33ba0323ad4da08a216f4c3"><ce:label>101</ce:label><ce:textfn>Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru</ce:textfn><sa:affiliation><sa:organization>Sección Física</sa:organization><sa:organization>Departamento de Ciencias</sa:organization><sa:organization>Pontificia Universidad Católica del Perú</sa:organization><sa:city>Lima</sa:city><sa:country>Peru</sa:country></sa:affiliation><ce:source-text id="srct0505">Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru</ce:source-text></ce:affiliation><ce:affiliation id="aff1020" affiliation-id="S0370269322007833-24347400090c9a1efb87e7537af7461b"><ce:label>102</ce:label><ce:textfn>Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria</ce:textfn><sa:affiliation><sa:organization>Stefan Meyer Institut für Subatomare Physik (SMI)</sa:organization><sa:city>Vienna</sa:city><sa:country>Austria</sa:country></sa:affiliation><ce:source-text id="srct0510">Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria</ce:source-text></ce:affiliation><ce:affiliation id="aff1030" affiliation-id="S0370269322007833-317714469d8c208c77efd6f026942633"><ce:label>103</ce:label><ce:textfn>SUBATECH, IMT Atlantique, Nantes Université, CNRS-IN2P3, Nantes, France</ce:textfn><sa:affiliation><sa:organization>SUBATECH</sa:organization><sa:organization>IMT Atlantique</sa:organization><sa:organization>Nantes Université</sa:organization><sa:organization>CNRS-IN2P3</sa:organization><sa:city>Nantes</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0515">SUBATECH, IMT Atlantique, Nantes Université, CNRS-IN2P3, Nantes, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1040" affiliation-id="S0370269322007833-c3972f6af24eef12cc2ae3a53d6a7623"><ce:label>104</ce:label><ce:textfn>Suranaree University of Technology, Nakhon Ratchasima, Thailand</ce:textfn><sa:affiliation><sa:organization>Suranaree University of Technology</sa:organization><sa:city>Nakhon Ratchasima</sa:city><sa:country>Thailand</sa:country></sa:affiliation><ce:source-text id="srct0520">Suranaree University of Technology, Nakhon Ratchasima, Thailand</ce:source-text></ce:affiliation><ce:affiliation id="aff1050" affiliation-id="S0370269322007833-9186de958ef0b51277f8f34872e5c71b"><ce:label>105</ce:label><ce:textfn>Technical University of Košice, Košice, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Technical University of Košice</sa:organization><sa:city>Košice</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0525">Technical University of Košice, Košice, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff1060" affiliation-id="S0370269322007833-425836ada61fe0d3ee4ebf5b87598919"><ce:label>106</ce:label><ce:textfn>The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland</ce:textfn><sa:affiliation><sa:organization>The Henryk Niewodniczanski Institute of Nuclear Physics</sa:organization><sa:organization>Polish Academy of Sciences</sa:organization><sa:city>Cracow</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0530">The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff1070" affiliation-id="S0370269322007833-302a66584ead3636c3e74b86b51cf474"><ce:label>107</ce:label><ce:textfn>The University of Texas at Austin, Austin, TX, United States</ce:textfn><sa:affiliation><sa:organization>The University of Texas at Austin</sa:organization><sa:city>Austin</sa:city><sa:state>TX</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0535">The University of Texas at Austin, Austin, Texas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1080" affiliation-id="S0370269322007833-48ea3700cc067946e6c3032832f0ce0a"><ce:label>108</ce:label><ce:textfn>Universidad Autónoma de Sinaloa, Culiacán, Mexico</ce:textfn><sa:affiliation><sa:organization>Universidad Autónoma de Sinaloa</sa:organization><sa:city>Culiacán</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0540">Universidad Autónoma de Sinaloa, Culiacán, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff1090" affiliation-id="S0370269322007833-310945a13cafb9845fe651fc7aa661b6"><ce:label>109</ce:label><ce:textfn>Universidade de São Paulo (USP), São Paulo, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade de São Paulo (USP)</sa:organization><sa:city>São Paulo</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0545">Universidade de São Paulo (USP), São Paulo, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1100" affiliation-id="S0370269322007833-226f6a475b7a1634e1530e2077d7590e"><ce:label>110</ce:label><ce:textfn>Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade Estadual de Campinas (UNICAMP)</sa:organization><sa:city>Campinas</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0550">Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1110" affiliation-id="S0370269322007833-7582533209ee7a368c4d249782e93c03"><ce:label>111</ce:label><ce:textfn>Universidade Federal do ABC, Santo Andre, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade Federal do ABC</sa:organization><sa:city>Santo Andre</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0555">Universidade Federal do ABC, Santo Andre, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1120" affiliation-id="S0370269322007833-f37f0b36132124cce5539b1f9c518079"><ce:label>112</ce:label><ce:textfn>University of Cape Town, Cape Town, South Africa</ce:textfn><sa:affiliation><sa:organization>University of Cape Town</sa:organization><sa:city>Cape Town</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0560">University of Cape Town, Cape Town, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff1130" affiliation-id="S0370269322007833-58fe35a5cc9d91eea34fd6e19293599b"><ce:label>113</ce:label><ce:textfn>University of Houston, Houston, TX, United States</ce:textfn><sa:affiliation><sa:organization>University of Houston</sa:organization><sa:city>Houston</sa:city><sa:state>TX</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0565">University of Houston, Houston, Texas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1140" affiliation-id="S0370269322007833-e9be2b1885c8285b7cb7b6534ca93d98"><ce:label>114</ce:label><ce:textfn>University of Jyväskylä, Jyväskylä, Finland</ce:textfn><sa:affiliation><sa:organization>University of Jyväskylä</sa:organization><sa:city>Jyväskylä</sa:city><sa:country>Finland</sa:country></sa:affiliation><ce:source-text id="srct0570">University of Jyväskylä, Jyväskylä, Finland</ce:source-text></ce:affiliation><ce:affiliation id="aff1150" affiliation-id="S0370269322007833-6e2074f6fd0b88987501b22e5d74f9c9"><ce:label>115</ce:label><ce:textfn>University of Kansas, Lawrence, KS, United States</ce:textfn><sa:affiliation><sa:organization>University of Kansas</sa:organization><sa:city>Lawrence</sa:city><sa:state>KS</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0575">University of Kansas, Lawrence, Kansas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1160" affiliation-id="S0370269322007833-f538fc5a59ec1d1c8f86d2a83cb4ced6"><ce:label>116</ce:label><ce:textfn>University of Liverpool, Liverpool, United Kingdom</ce:textfn><sa:affiliation><sa:organization>University of Liverpool</sa:organization><sa:city>Liverpool</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0580">University of Liverpool, Liverpool, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff1170" affiliation-id="S0370269322007833-5507e3344bf7b46ba698c2c045a8aba6"><ce:label>117</ce:label><ce:textfn>University of Science and Technology of China, Hefei, China</ce:textfn><sa:affiliation><sa:organization>University of Science and Technology of China</sa:organization><sa:city>Hefei</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0585">University of Science and Technology of China, Hefei, China</ce:source-text></ce:affiliation><ce:affiliation id="aff1180" affiliation-id="S0370269322007833-5825f1740b6b974441dd90d9c8ad0a53"><ce:label>118</ce:label><ce:textfn>University of South-Eastern Norway, Kongsberg, Norway</ce:textfn><sa:affiliation><sa:organization>University of South-Eastern Norway</sa:organization><sa:city>Kongsberg</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0590">University of South-Eastern Norway, Kongsberg, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff1190" affiliation-id="S0370269322007833-d6620449e5365c80224ff22fe1ca4e4a"><ce:label>119</ce:label><ce:textfn>University of Tennessee, Knoxville, TN, United States</ce:textfn><sa:affiliation><sa:organization>University of Tennessee</sa:organization><sa:city>Knoxville</sa:city><sa:state>TN</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0595">University of Tennessee, Knoxville, Tennessee, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1200" affiliation-id="S0370269322007833-6d43022152ccaa62119e4287bf3b5270"><ce:label>120</ce:label><ce:textfn>University of the Witwatersrand, Johannesburg, South Africa</ce:textfn><sa:affiliation><sa:organization>University of the Witwatersrand</sa:organization><sa:city>Johannesburg</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0600">University of the Witwatersrand, Johannesburg, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff1210" affiliation-id="S0370269322007833-9b9071fc23073da0e4317b3863c91b9e"><ce:label>121</ce:label><ce:textfn>University of Tokyo, Tokyo, Japan</ce:textfn><sa:affiliation><sa:organization>University of Tokyo</sa:organization><sa:city>Tokyo</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0605">University of Tokyo, Tokyo, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff1220" affiliation-id="S0370269322007833-da3fb88e8779358429638ee8f8dc4cb5"><ce:label>122</ce:label><ce:textfn>University of Tsukuba, Tsukuba, Japan</ce:textfn><sa:affiliation><sa:organization>University of Tsukuba</sa:organization><sa:city>Tsukuba</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0610">University of Tsukuba, Tsukuba, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff1230" affiliation-id="S0370269322007833-9fc1715eadb7b2ba79b1a87cef8015cd"><ce:label>123</ce:label><ce:textfn>University Politehnica of Bucharest, Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>University Politehnica of Bucharest</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0615">University Politehnica of Bucharest, Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff1240" affiliation-id="S0370269322007833-43e2ac82e11c4ca2fabbc7e457c74b37"><ce:label>124</ce:label><ce:textfn>Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France</ce:textfn><sa:affiliation><sa:organization>Université Clermont Auvergne</sa:organization><sa:organization>CNRS/IN2P3</sa:organization><sa:organization>LPC</sa:organization><sa:city>Clermont-Ferrand</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0620">Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1250" affiliation-id="S0370269322007833-08f13150d05b32d440951efec6f80d29"><ce:label>125</ce:label><ce:textfn>Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France</ce:textfn><sa:affiliation><sa:organization>Université de Lyon</sa:organization><sa:organization>CNRS/IN2P3</sa:organization><sa:organization>Institut de Physique des 2 Infinis de Lyon</sa:organization><sa:city>Lyon</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0625">Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1260" affiliation-id="S0370269322007833-766d0075165899c16752f5f7ff1b8771"><ce:label>126</ce:label><ce:textfn>Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France</ce:textfn><sa:affiliation><sa:organization>Université de Strasbourg</sa:organization><sa:organization>CNRS</sa:organization><sa:organization>IPHC UMR 7178</sa:organization><sa:city>Strasbourg</sa:city><sa:postal-code>F-67000</sa:postal-code><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0630">Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1270" affiliation-id="S0370269322007833-36117066550bb6faaf9708b1b0ea670e"><ce:label>127</ce:label><ce:textfn>Université Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France</ce:textfn><sa:affiliation><sa:organization>Université Paris-Saclay Centre d'Etudes de Saclay (CEA)</sa:organization><sa:organization>IRFU</sa:organization><sa:organization>Départment de Physique Nucléaire (DPhN)</sa:organization><sa:city>Saclay</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0635">Université Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1280" affiliation-id="S0370269322007833-47ef552cdaf90d05d791574b14a8dcf9"><ce:label>128</ce:label><ce:textfn>Università degli Studi di Foggia, Foggia, Italy</ce:textfn><sa:affiliation><sa:organization>Università degli Studi di Foggia</sa:organization><sa:city>Foggia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0640">Università degli Studi di Foggia, Foggia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1290" affiliation-id="S0370269322007833-79fbcbe2a42e9099094360b5bf7bd602"><ce:label>129</ce:label><ce:textfn>Università del Piemonte Orientale, Vercelli, Italy</ce:textfn><sa:affiliation><sa:organization>Università del Piemonte Orientale</sa:organization><sa:city>Vercelli</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0645">Università del Piemonte Orientale, Vercelli, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1300" affiliation-id="S0370269322007833-83cd0c0929f483f88e3d2cb59f064287"><ce:label>130</ce:label><ce:textfn>Università di Brescia, Brescia, Italy</ce:textfn><sa:affiliation><sa:organization>Università di Brescia</sa:organization><sa:city>Brescia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0650">Università di Brescia, Brescia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1310" affiliation-id="S0370269322007833-f1ae52f852d4d7d99988b3e872f887e4"><ce:label>131</ce:label><ce:textfn>Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Variable Energy Cyclotron Centre</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0655">Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1320" affiliation-id="S0370269322007833-bfd4fe0d3b8675ea55d96ce64033d817"><ce:label>132</ce:label><ce:textfn>Warsaw University of Technology, Warsaw, Poland</ce:textfn><sa:affiliation><sa:organization>Warsaw University of Technology</sa:organization><sa:city>Warsaw</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0660">Warsaw University of Technology, Warsaw, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff1330" affiliation-id="S0370269322007833-4e11d38f810a3540206ffb5151a35d3b"><ce:label>133</ce:label><ce:textfn>Wayne State University, Detroit, MI, United States</ce:textfn><sa:affiliation><sa:organization>Wayne State University</sa:organization><sa:city>Detroit</sa:city><sa:state>MI</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0665">Wayne State University, Detroit, Michigan, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1340" affiliation-id="S0370269322007833-0b76517a6de7ff0579ec14780f79d81b"><ce:label>134</ce:label><ce:textfn>Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany</ce:textfn><sa:affiliation><sa:organization>Westfälische Wilhelms-Universität Münster</sa:organization><sa:organization>Institut für Kernphysik</sa:organization><sa:city>Münster</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0670">Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1350" affiliation-id="S0370269322007833-6b1ae45228f1f040c55767dc107b589d"><ce:label>135</ce:label><ce:textfn>Wigner Research Centre for Physics, Budapest, Hungary</ce:textfn><sa:affiliation><sa:organization>Wigner Research Centre for Physics</sa:organization><sa:city>Budapest</sa:city><sa:country>Hungary</sa:country></sa:affiliation><ce:source-text id="srct0675">Wigner Research Centre for Physics, Budapest, Hungary</ce:source-text></ce:affiliation><ce:affiliation id="aff1360" affiliation-id="S0370269322007833-96e79e7381eab77664732c3553e247e8"><ce:label>136</ce:label><ce:textfn>Yale University, New Haven, CT, United States</ce:textfn><sa:affiliation><sa:organization>Yale University</sa:organization><sa:city>New Haven</sa:city><sa:state>CT</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0680">Yale University, New Haven, Connecticut, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1370" affiliation-id="S0370269322007833-f3db17ceaf6ea4190e2176a6a129c96b"><ce:label>137</ce:label><ce:textfn>Yonsei University, Seoul, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Yonsei University</sa:organization><sa:city>Seoul</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0685">Yonsei University, Seoul, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff1380" affiliation-id="S0370269322007833-2e935afd39a812eb118b95cc53ef2ac3"><ce:label>138</ce:label><ce:textfn>Zentrum für Technologie und Transfer (ZTT), Worms, Germany</ce:textfn><sa:affiliation><sa:organization>Zentrum für Technologie und Transfer (ZTT)</sa:organization><sa:city>Worms</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0690">Zentrum für Technologie und Transfer (ZTT), Worms, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1390" affiliation-id="S0370269322007833-7972cea8d4a6e5b7150baf4a74e03970"><ce:label>139</ce:label><ce:textfn>Affiliated with an institute covered by a cooperation agreement with CERN</ce:textfn><sa:affiliation><sa:address-line>Affiliated with an institute covered by a cooperation agreement with CERN</sa:address-line></sa:affiliation><ce:source-text id="srct0695">Affiliated with an institute covered by a cooperation agreement with CERN</ce:source-text></ce:affiliation><ce:affiliation id="aff1400" affiliation-id="S0370269322007833-3c2eaa2a494293a99583fccbf5d95e9f"><ce:label>140</ce:label><ce:textfn>Affiliated with an international laboratory covered by a cooperation agreement with CERN</ce:textfn><sa:affiliation><sa:address-line>Affiliated with an international laboratory covered by a cooperation agreement with CERN</sa:address-line></sa:affiliation><ce:source-text id="srct0700">Affiliated with an international laboratory covered by a cooperation agreement with CERN</ce:source-text></ce:affiliation><ce:footnote id="fn0010"><ce:label>I</ce:label><ce:note-para id="np0010">Deceased.</ce:note-para></ce:footnote><ce:footnote id="fn0020"><ce:label>II</ce:label><ce:note-para id="np0020">Also at: Max-Planck-Institut für Physik, Munich, Germany.</ce:note-para></ce:footnote><ce:footnote id="fn0030"><ce:label>III</ce:label><ce:note-para id="np0030">Also at: Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA), Bologna, Italy.</ce:note-para></ce:footnote><ce:footnote id="fn0040"><ce:label>IV</ce:label><ce:note-para id="np0040">Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy.</ce:note-para></ce:footnote><ce:footnote id="fn0050"><ce:label>V</ce:label><ce:note-para id="np0050">Also at: Department of Applied Physics, Aligarh Muslim University, Aligarh, India.</ce:note-para></ce:footnote><ce:footnote id="fn0060"><ce:label>VI</ce:label><ce:note-para id="np0060">Also at: Institute of Theoretical Physics, University of Wroclaw, Poland.</ce:note-para></ce:footnote><ce:footnote id="fn0070"><ce:label>VII</ce:label><ce:note-para id="np0070">Also at: An institution covered by a cooperation agreement with CERN.</ce:note-para></ce:footnote></ce:author-group></ce:collaboration><ce:footnote id="fn0080"><ce:label>⋆</ce:label><ce:note-para id="np0080"><ce:italic>E-mail address:</ce:italic> <ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/text/html" xlink:href="mailto:alice-publications@cern.ch" id="inf0020">alice-publications@cern.ch</ce:inter-ref>.</ce:note-para></ce:footnote></ce:author-group><ce:date-received day="5" month="5" year="2022"/><ce:date-revised day="2" month="12" year="2022"/><ce:date-accepted day="23" month="12" year="2022"/><ce:miscellaneous id="ms0010">Editor: M. Pierini</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0070">This letter reports measurements which characterize the underlying event associated with hard scatterings at mid-pseudorapidity (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math>) in pp, p–Pb and Pb–Pb collisions at centre-of-mass energy per nucleon pair, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The measurements are performed with ALICE at the LHC. Different multiplicity classes are defined based on the event activity measured at forward rapidities. The hard scatterings are identified by the leading particle defined as the charged particle with the largest transverse momentum (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) in the collision and having 8 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mo linebreak="badbreak" linebreakstyle="after"><</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra of associated particles (0.5 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) are measured in different azimuthal regions defined with respect to the leading particle direction: toward, transverse, and away. The associated charged particle yields in the transverse region are subtracted from those of the away and toward regions. The remaining jet-like yields are reported as a function of the multiplicity measured in the transverse region. The measurements show a suppression of the jet-like yield in the away region and an enhancement of high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> associated particles in the toward region in central Pb–Pb collisions, as compared to minimum-bias pp collisions. These observations are consistent with previous measurements that used two-particle correlations, and with an interpretation in terms of parton energy loss in a high-density quark gluon plasma. These yield modifications vanish in peripheral Pb–Pb collisions and are not observed in either high-multiplicity pp or p–Pb collisions.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0080">Data availability</ce:section-title><ce:para id="pr0200">This manuscript has associated data in a HEPData repository at <ce:inter-ref xlink:href="https://www.hepdata.net/" xlink:role="http://www.elsevier.com/xml/linking-roles/research-data" id="inf0550">https://www.hepdata.net/</ce:inter-ref>.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">In proton-proton (pp) collisions, jets, originating from partonic scatterings with large momentum transfer, are accompanied by particles produced by initial- and final-state radiation (ISR and FSR, respectively), as well as, by a plethora of other mechanisms. These include proton break-up, and, in a scenario incorporating multi-parton interactions (MPI) <ce:cross-refs refid="br0010 br0020" id="crs0010">[1,2]</ce:cross-refs>, several semi-hard parton-parton scatterings in a single pp collision. These jet-accompanying particles experimentally make up the underlying event (UE) and are commonly studied via azimuthal separations from the jets to minimise the influence of hard scatterings. The present study follows the strategy originally introduced by the CDF collaboration <ce:cross-ref refid="br0030" id="crf10910">[3]</ce:cross-ref>. First, the leading charged particle in the event is found, i.e., the charged particle with the highest transverse momentum in the collision (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math>). Secondly, the associated particles (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math>) are measured in three topological regions depending on their azimuthal angle relative to the leading particle, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>φ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msup><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">assoc</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">|</mml:mo></mml:math>, see <ce:cross-ref refid="fg0010" id="crf10920">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/>.</ce:para><ce:para id="pr0020">The toward region contains the primary jet within the acceptance of the detector, while the away region contains the back-scattered particles of the recoil jet <ce:cross-ref refid="br0040" id="crf10930">[4]</ce:cross-ref>. In contrast, the transverse region is dominated by the underlying-event dynamics, but it also includes contributions from ISR and FSR <ce:cross-ref refid="br0050" id="crf10940">[5]</ce:cross-ref>.</ce:para><ce:para id="pr0030">The measurements performed at RHIC and LHC in small systems (pp, p–A, and d–A collisions) have shown for high particle multiplicities similar phenomena as were originally observed only in A–A collisions and have been attributed there to the formation of the strongly interacting quark gluon plasma <ce:cross-refs refid="br0060 br0070" id="crs0020">[6,7]</ce:cross-refs>, namely, long range angular correlations and collectivity <ce:cross-ref refid="br0080" id="crf10950">[8]</ce:cross-ref>. The origin of these effects in small systems is still an open question; on one hand, hydrodynamical calculations describe some aspects of the data <ce:cross-ref refid="br0090" id="crf10960">[9]</ce:cross-ref>; on the other hand, mechanisms like colour reconnection <ce:cross-ref refid="br0100" id="crf10970">[10]</ce:cross-ref>, rope hadronisation <ce:cross-ref refid="br0110" id="crf10980">[11]</ce:cross-ref>, and string shoving <ce:cross-ref refid="br0120" id="crf10990">[12]</ce:cross-ref> can produce collective-like effects in Monte Carlo event generators such as <ce:small-caps>PYTHIA</ce:small-caps> 8 <ce:cross-ref refid="br0130" id="crf11000">[13]</ce:cross-ref>. Thus, investigating pp collisions as a function of the charged particle multiplicity has become ever more pertinent <ce:cross-refs refid="br0090 br0140 br0150 br0160 br0170 br0180" id="crs0030">[9,14–18]</ce:cross-refs>. The interpretation of the results from the analysis of high-multiplicity pp collisions is challenging due to the selection biases of the sample towards events in which partonic scatterings with large momentum transfer (hard scatterings) occurred. To mitigate this inherent bias, Martin et al. <ce:cross-ref refid="br0190" id="crf11010">[19]</ce:cross-ref> suggested to use the charged-particle multiplicity in the transverse region (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>) as a classifier of the activity in the collisions, since the correlation between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math> and the hardest scattering in the collision is small. The ALICE collaboration has reported the first <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math> spectra measured in pp collisions at centre-of-mass energy, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:msqrt><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>13</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV <ce:cross-ref refid="br0200" id="crf11020">[20]</ce:cross-ref>. Event generators, such as <ce:small-caps>PYTHIA</ce:small-caps> 8 <ce:cross-ref refid="br0130" id="crf11030">[13]</ce:cross-ref> and EPOS-LHC <ce:cross-ref refid="br0210" id="crf11040">[21]</ce:cross-ref>, do not provide a good description of the measured distribution of the ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>/<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> is the event-averaged charged-particle multiplicity in the transverse region, underestimating in particular the number of collisions with large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:math>. In the framework of MPI-based models, like those implemented in <ce:small-caps>PYTHIA</ce:small-caps> 8 and <ce:small-caps>HERWIG</ce:small-caps> 7 <ce:cross-ref refid="br0220" id="crf11050">[22]</ce:cross-ref>, the probability for a hard scattering in the collision increases with decreasing impact parameter<ce:cross-ref refid="fn0090" id="crf11060"><ce:sup>VIII</ce:sup></ce:cross-ref><ce:footnote id="fn0090"><ce:label>VIII</ce:label><ce:note-para id="np0090">In event generators like <ce:small-caps>PYTHIA</ce:small-caps> 8 the impact parameter profile is described by an overlap matter distribution of the two incoming hadrons.</ce:note-para></ce:footnote> between the colliding protons. Thus, requiring a high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> particle (e.g., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>8</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) in a given pp collision biases the selection of collisions towards those with a smaller impact parameter <ce:cross-ref refid="br0230" id="crf11070">[23]</ce:cross-ref>, which in turn biases the selection towards pp collisions with more MPI <ce:cross-ref refid="br0200" id="crf11080">[20]</ce:cross-ref>. This feature of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>-based analysis is important for the isolation of potential MPI and colour reconnection effects, which according to <ce:small-caps>PYTHIA</ce:small-caps> 8, produce effects resembling collective behaviour <ce:cross-ref refid="br0100" id="crf11090">[10]</ce:cross-ref>. By construction, MPI and colour reconnection effects are expected to be more relevant in the transverse region than in the away and toward regions <ce:cross-ref refid="br0240" id="crf11100">[24]</ce:cross-ref>. It is worth mentioning that the MPI picture has been used to explain the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in p–Pb collisions and peripheral Pb–Pb collisions <ce:cross-refs refid="br0250 br0260 br0270" id="crs0040">[25–27]</ce:cross-refs>. Studies, as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>, are therefore important to the understanding of the effects observed in high-multiplicity pp collisions. Last but not least, measurements of UE observables are also important to tune event generators <ce:cross-ref refid="br0280" id="crf11110">[28]</ce:cross-ref> that include hard partonic scatterings and MPI.</ce:para><ce:para id="pr0040">This letter reports the inclusive charged-particle transverse momentum spectra in pp, p–Pb and Pb–Pb collisions at centre-of-mass energy per nucleon pair <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV containing a high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> leading particle within the kinematic intervals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>8</mml:mn><mml:mo>≤</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math>. This guarantees the selection of collisions in which the average activity in the transverse region is roughly flat as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math> <ce:cross-ref refid="br0200" id="crf11120">[20]</ce:cross-ref>, and therefore, any additional selection on the charged particle multiplicity will only modulate the UE activity. The measurements are performed considering different event classes defined in terms of the multiplicity registered in the forward detectors. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra of associated charged particles (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mn>0.5</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math>) are measured in the toward, away, and transverse regions as a function of the average charged particle multiplicity in the transverse region. To further investigate the possible modification of the particles produced in the hard scattering in pp, p–Pb, and Pb–Pb collisions, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> distributions in the toward (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) and away (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) regions obtained after the subtraction of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in the transverse region (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) are also reported. The subtracted yields (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) are further normalised to those measured in minimum-bias (MB) pp collisions,<ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pp</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">MB</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pp</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">MB</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>X</ce:italic> indicates the collision system and the event multiplicity class. In this way, the hard process <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in the toward and away regions are isolated, and thus allowing us to study possible modifications to the produced particles due to medium effects in high-multiplicity pp, p–Pb, and Pb–Pb collisions. In heavy-ion collisions, this ratio is sensitive to the same effects which were studied using the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math> quantity <ce:cross-refs refid="br0290 br0300 br0310" id="crs0050">[29–31]</ce:cross-refs>, where jets produced in the early stage of the collision propagate through the hot and dense quark–gluon plasma. Their interaction with the coloured medium lead to parton-energy loss (jet quenching) <ce:cross-ref refid="br0320" id="crf11130">[32]</ce:cross-ref> which, for example, results in the suppression of the charged-particle yield at high <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> <ce:cross-ref refid="br0330" id="crf11140">[33]</ce:cross-ref>, and the suppression of the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> yield in the away region <ce:cross-refs refid="br0290 br0300" id="crs0060">[29,30]</ce:cross-refs>. It is worth mentioning that jet quenching effects have not been observed so far in small systems <ce:cross-refs refid="br0330 br0340" id="crs0070">[33,34]</ce:cross-refs>.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">Experiment and data analysis</ce:section-title><ce:para id="pr0050">This analysis is based on the data recorded by the ALICE apparatus during the pp and Pb–Pb runs at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV in 2015, and the p–Pb run at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV in 2016. The present study uses the V0 detector, and the Silicon Pixel Detector (SPD) for triggering and background rejection. The V0 consists of two arrays of scintillating tiles placed on each side of the interaction point covering the full azimuthal acceptance and the pseudorapidity intervals of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:mn>2.8</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5.1</mml:mn></mml:math> (V0A) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.7</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1.7</mml:mn></mml:math> (V0C). The SPD is the innermost part of the Inner Tracking System (ITS) and it is the closest detector to the interaction point. It consists of two cylindrical silicon pixel layers at radial distances of 3.9 and 7.6 cm from the beam line and the pseudorapidity coverages of the two layers are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1.4</mml:mn></mml:math>, respectively. The data were collected using a minimum-bias trigger, which required a signal in both V0A and V0C detectors. The offline event selection was optimised to reject beam-induced background in all collision systems by utilising the timing signals in the two V0 detectors. In Pb–Pb collisions, the beam-induced background is further suppressed by correlating the timing signals of the neutron zero degree calorimeters, which are positioned on both sides of the interaction point at 112.5<ce:hsp sp="0.20"/>m distance along the beam axis <ce:cross-ref refid="br0350" id="crf11150">[35]</ce:cross-ref>. The signals from the zero degree calorimeters are also used to suppress the contamination from electromagnetic interactions. This is performed by requesting the coincidence of the signals coming from both side zero degree calorimeters by which the background due to single nucleus electromagnetic dissociation processes is excluded. A criterion based on the offline reconstruction of multiple primary vertices in the SPD is applied to reduce the pileup caused by multiple interactions in the same bunch crossing <ce:cross-ref refid="br0360" id="crf11160">[36]</ce:cross-ref>. The results presented in this letter are for minimum-bias triggered pp collisions having at least one charged particle in the pseudorapidity interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1</mml:mn></mml:math> (INEL>0). The INEL>0 event class corresponds to about 75% of the total inelastic cross section <ce:cross-ref refid="br0370" id="crf11170">[37]</ce:cross-ref>. For pp and Pb–Pb collisions, the sample is subdivided into different multiplicity classes based on the total charge deposited in both V0 sub-detectors, which is termed as V0M amplitude <ce:cross-ref refid="br0380" id="crf11180">[38]</ce:cross-ref>. For p–Pb collisions, the sample is subdivided based on the total charge deposited in V0A sub-detector (V0A amplitude) <ce:cross-ref refid="br0390" id="crf11190">[39]</ce:cross-ref>, which is located in the Pb-going direction. The V0A estimator has been implemented in previous measurements that used p–Pb data (see e.g. <ce:cross-ref refid="br0400" id="crf11200">[40]</ce:cross-ref>). This allows for comparisons with other observables for similar V0A multiplicity classes. To ensure that a hard scattering took place in the collision, events are required to have a trigger particle within <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>8</mml:mn><mml:mo>≤</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math> GeV/<ce:italic>c</ce:italic>. In this <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math> interval, the momentum resolution effects are negligible on the extracted yields, and therefore, no <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math> resolution correction is applied. The total number of analysed collisions before the trigger particle selection are about 10<ce:sup>8</ce:sup>, 10<ce:sup>8</ce:sup>, and 10<ce:sup>7</ce:sup> for pp, p–Pb, and Pb–Pb collisions, respectively.</ce:para><ce:para id="pr0060">The transverse momentum of particles is determined from measurements in the central barrel with the ITS and the Time Projection Chamber (TPC). The ITS is a tracking detector which consists of six cylindrical layers of silicon detectors. The TPC is a cylindrical drift detector which covers a radial distance of 85-247<ce:hsp sp="0.20"/>cm from the beam axis and it has longitudinal dimension extending from about -250<ce:hsp sp="0.20"/>cm to +250<ce:hsp sp="0.20"/>cm around the nominal interaction point. Primary charged particles are measured in the pseudorapidity range of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math> and with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0.5</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, where <ce:italic>η</ce:italic> is measured in the laboratory frame for the three collision systems. The configuration for p–Pb collisions with protons at 4<ce:hsp sp="0.20"/>TeV energy colliding with Pb ions that have per-nucleon energies of (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mi>Z</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>A</mml:mi></mml:math>) × 4<ce:hsp sp="0.20"/>TeV ∼ 1.58<ce:hsp sp="0.20"/>TeV results in a shift in the rapidity of the nucleon–nucleon centre-of-mass system by 0.465 in the direction of the proton beam (negative z-direction). Here <ce:italic>Z</ce:italic> and <ce:italic>A</ce:italic> are the atomic and mass numbers of the Pb ion, respectively. Therefore, the detector coverage <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math> corresponds to roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>0.3</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cms</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1.3</mml:mn></mml:math> for p–Pb collisions. The particles with mean proper lifetime larger than 1<ce:hsp sp="0.20"/>cm/<ce:italic>c</ce:italic>, which are either produced directly in the interaction or from decays of particles with mean proper lifetime smaller than 1<ce:hsp sp="0.20"/>cm/<ce:italic>c</ce:italic> are termed as primary particles <ce:cross-ref refid="br0410" id="crf11210">[41]</ce:cross-ref>. The track selection follows a procedure similar to the one described in Ref. <ce:cross-ref refid="br0420" id="crf11220">[42]</ce:cross-ref> and only few specific details are reported here. Tracks (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">tracks</mml:mi></mml:mrow></mml:msub></mml:math>) are required to have two hits in the ITS, out of which at least one should be in either of the two innermost layers. The geometrical track length <ce:italic>L</ce:italic> is calculated in the TPC readout plane, excluding the information from the pads at the sector boundaries (≈3<ce:hsp sp="0.20"/>cm from the sector edges). The trajectory lengths built from radial segments, i.e. the crossed TPC pad rows, traversed in the TPC by a particle are required to be larger than 85% of the geometrical track length. The pad rows are made of at least 3 neighbouring individual observations (clusters), and their height varies from 7.5<ce:hsp sp="0.20"/>mm to 15<ce:hsp sp="0.20"/>mm <ce:cross-ref refid="br0430" id="crf11230">[43]</ce:cross-ref>. The trajectory lengths built from clusters (one cluster per pad row) is required to be larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mn>0.7</mml:mn><mml:mo>×</mml:mo><mml:mi>L</mml:mi></mml:math>. The fraction of TPC clusters shared with another track is required to be lower than 0.4. The fit quality for the ITS and TPC track points must satisfy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">hits</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>36</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">clusters</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>4</mml:mn></mml:math>, respectively, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">hits</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">clusters</mml:mi></mml:mrow></mml:msub></mml:math> are the numbers of hits in the ITS and the number of clusters in the TPC, respectively. Only tracks with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>36</mml:mn></mml:math> are included in the analysis, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> is calculated comparing the track parameters from the combined ITS and TPC track reconstruction to that derived only from the TPC and constrained to the interaction point. The definition of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> can be found in Ref. <ce:cross-ref refid="br0440" id="crf11240">[44]</ce:cross-ref>. To reduce the contamination from secondary particles, tracks are accepted if their distance-of-closest-approach (DCA) to the reconstructed primary interaction vertex satisfies in the longitudinal (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow></mml:msub></mml:math>) and transverse (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">xy</mml:mi></mml:mrow></mml:msub></mml:math>) directions the conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math><ce:hsp sp="0.20"/>cm and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">xy</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.018</mml:mn></mml:math><ce:hsp sp="0.20"/>cm + 0.035<ce:hsp sp="0.20"/>(cm×GeV/<ce:italic>c</ce:italic>)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>.</ce:para><ce:para id="pr0070">The measurement of the transverse momentum spectra of charged particles follows the standard procedure of the ALICE collaboration <ce:cross-refs refid="br0420 br0450" id="crs0080">[42,45]</ce:cross-refs>. The raw yields are corrected for efficiency and contamination from secondary particles. The efficiency correction is calculated from Monte Carlo simulations with GEANT3 <ce:cross-ref refid="br0460" id="crf11250">[46]</ce:cross-ref> transport code, which made use of PYTHIA 8 (Monash) <ce:cross-ref refid="br0280" id="crf11260">[28]</ce:cross-ref>, EPOS-LHC <ce:cross-ref refid="br0210" id="crf11270">[21]</ce:cross-ref> and HIJING <ce:cross-ref refid="br0470" id="crf11280">[47]</ce:cross-ref> event generators for pp, p–Pb and Pb–Pb collisions, respectively and incorporated a detailed description of the detector material, geometry and response. Since the event generators do not reproduce the relative abundances of different particle species in the real data, the efficiency obtained from the simulations is re-weighted considering the particle composition from data as outlined in <ce:cross-ref refid="br0420" id="crf11290">[42]</ce:cross-ref>. A multi-component template fit based on the DCA distributions from the simulation is used for the estimation of secondary contamination <ce:cross-ref refid="br0420" id="crf11300">[42]</ce:cross-ref>.</ce:para><ce:para id="pr0080">The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra for the toward and away regions include contributions from the jet fragmentation, ISR, and FSR, as well as, the contribution from the underlying event. In order to increase the sensitivity to the hardest process of the event, the particle yields measured in the transverse region are subtracted from the corresponding yields in both the toward and away regions: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>. This approach assumes that the background (UE, ISR, and FSR) in the toward and away regions is similar to the activity in the transverse region. However, one has to keep in mind that in Pb–Pb collisions two-particle correlations are affected by anisotropic transverse flow. In particular, the main contribution is due to the elliptic flow, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>, which is the second order coefficient in the Fourier expansion of the azimuthal distribution of the particle momenta <ce:cross-ref refid="br0480" id="crf11310">[48]</ce:cross-ref>. This elliptic azimuthal anisotropy modulates the background according to:<ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi>B</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>φ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant="normal">cos</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>φ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> is approximately given by the product of anisotropic flow coefficients for trigger and associated particles at their respective momenta i.e. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant="normal">assoc</mml:mi></mml:mrow></mml:msubsup></mml:math>. The existing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> measurements over a broad transverse momentum range <ce:cross-ref refid="br0490" id="crf11320">[49]</ce:cross-ref> suggest that the effect of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulation of background should be more relevant in semi-central Pb–Pb collisions. The effect is expected to be important at low and intermediate transverse momenta and decreases for high transverse momentum particles <ce:cross-ref refid="br0300" id="crf11330">[30]</ce:cross-ref>. In the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> region of interest for the jet quenching studies, namely <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>4</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, the effect of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulation is estimated to be small (about 5%) for Pb–Pb collisions. Given that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> effect is larger in Pb–Pb collisions than in pp and p–Pb collisions, no correction for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulation is applied for pp and p–Pb collisions since its effect is smaller than the other sources of systematic uncertainty.</ce:para><ce:para id="pr0090">The results are shown as a function of the average number of charged particles in the transverse region <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math>. The values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> are extracted in each multiplicity class from the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">tracks</mml:mi></mml:mrow></mml:msub></mml:math> distributions in the transverse region that are corrected for detector effects using a Bayesian unfolding <ce:cross-ref refid="br0500" id="crf11340">[50]</ce:cross-ref>. The Bayesian unfolding requires the multiplicity response matrix, which is built from the correlation between the measured multiplicity and the multiplicity at generator level (without detector effects) in the transverse region. This has been obtained from MC simulations which include the propagation of particles through the detector using GEANT 3. As a crosscheck, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> values are also calculated by integrating the transverse momentum distributions in the interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mn>0.5</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>8</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>. The difference between the results from the two strategies is assigned as the systematic uncertainty on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math>, where the effects related to the discrepancy between data and MC in the particle composition and secondary contamination are considered. This uncertainty amounts up to 3.5%, 4% and 6.5% for pp, p–Pb and Pb–Pb collisions, respectively.</ce:para><ce:para id="pr0100">The systematic uncertainties related to the track selection criteria were studied by repeating the analysis varying one-by-one the track selection criteria <ce:cross-refs refid="br0420 br0450" id="crs0090">[42,45]</ce:cross-refs>. In particular, the upper limits of the track fit quality parameters in the ITS (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">hits</mml:mi></mml:mrow></mml:msub></mml:math>) and in the TPC (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">clusters</mml:mi></mml:mrow></mml:msub></mml:math>) were varied in the ranges of 25–49 and 3–5, respectively. The maximum fraction of shared TPC clusters was varied between 0.2 to 1 and the maximum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow></mml:msub></mml:math> was varied between 1 and 5<ce:hsp sp="0.20"/>cm <ce:cross-ref refid="br0420" id="crf11350">[42]</ce:cross-ref>. We have also quantified the impact of not including the ITS hit requirement in the track selection. The systematic uncertainty on the primary particle composition was estimated using a procedure similar to the one described in <ce:cross-ref refid="br0420" id="crf11360">[42]</ce:cross-ref>. To quantify the uncertainty due to the imperfect simulation of the detector response, the track matching between the TPC and the ITS information in the data and in the simulation were compared. To achieve this, the fraction of secondary particles was rescaled according to fits to the measured DCA distributions. After this rescaling, the agreement between data and model was found to be within 3% for all collision systems. This value was assigned as an additional systematic uncertainty <ce:cross-ref refid="br0420" id="crf11370">[42]</ce:cross-ref>. The systematic uncertainty on the secondary particle contamination considers the imperfection of the method (multi-component template fit) used to extract the correction. The fit ranges were varied and the fit was repeated using templates with two (primaries, secondaries) or three (primaries, secondaries from material, secondaries from weak decays) components. The maximum spread among these variations was assigned as the systematic uncertainty on the secondary contamination. This contribution dominates at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>. The density of materials used in simulations of the experimental setup was varied by ± 4.5% <ce:cross-ref refid="br0350" id="crf11380">[35]</ce:cross-ref>, resulting in a negligible systematic uncertainty in the considered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> range of 0.5 to 6.0 GeV/<ce:italic>c</ce:italic>. For the estimation of total systematic uncertainty, all the above listed contributions were summed in quadrature. The systematic uncertainties are independent of the difference between the azimuthal angle of the associated particle and that of the trigger particle. The estimated systematic uncertainties on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra significantly depend on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>, while the dependence on the multiplicity classes is mild. The ranges of systematic uncertainties in the three considered collision systems are reported in <ce:cross-ref refid="tbl0010" id="crf11390">Table 1</ce:cross-ref><ce:float-anchor refid="tbl0010"/> for the various sources described above.</ce:para></ce:section><ce:section id="se0030" role="results"><ce:label>3</ce:label><ce:section-title id="st0040">Results and discussion</ce:section-title><ce:para id="pr0110">The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra measured in the transverse region for pp, p–Pb, and Pb–Pb collisions are shown in <ce:cross-ref refid="fg0020" id="crf11400">Fig. 2</ce:cross-ref><ce:float-anchor refid="fg0020"/> (top panel). Results are presented for different multiplicity classes. The ratios between the spectra in the individual multiplicity classes and the MB (0−100%) one are shown in the bottom panel. In the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:mn>0.5</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, the ratios for the highest multiplicity class (0−5%) are larger than unity and show an increasing trend with increasing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:mo linebreak="badbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) followed at higher <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> by a slow decrease. Instead, for the lowest multiplicity classes (40−60% and 60−90%) the ratios are lower than unity and follow an opposite trend with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>, decreasing at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> and increasing for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>3</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>. The behaviour of the ratios as a function of the event activity is reminiscent of analogous ratios as a function of the number of MPI in pp collisions simulated with <ce:small-caps>PYTHIA</ce:small-caps> 8, including colour reconnection <ce:cross-ref refid="br0510" id="crf11410">[51]</ce:cross-ref>. In particular, at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:math> GeV/<ce:italic>c</ce:italic> the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectrum of pp collisions with large MPI activity exhibits an enhancement with respect to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectrum of MB pp collisions. The effect was not observed before in data because, in contrast to the present analysis, the jet contribution was included in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra <ce:cross-ref refid="br0450" id="crf11420">[45]</ce:cross-ref>.</ce:para><ce:para id="pr0120">The top (bottom) panel of <ce:cross-ref refid="fg0030" id="crf11430">Fig. 3</ce:cross-ref><ce:float-anchor refid="fg0030"/> shows the charged particle yields for the toward (away) region after the subtraction of the yields measured in the transverse region in pp, p–Pb and Pb–Pb collisions. Results are compared with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra measured for MB pp collisions (0−100% V0M pp event class) quantified with the ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math>, as defined in Eq. <ce:cross-ref refid="fm0010" id="crf11440">(1)</ce:cross-ref>. At low transverse momenta, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>4</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is close to unity in pp and p–Pb collisions. In contrast, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> in Pb–Pb collisions exhibits a strong multiplicity dependence over the whole measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> interval. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> magnitude is larger for semi-peripheral Pb–Pb collisions, the maximum is observed for 20−40% Pb–Pb collisions, and is smaller for the most central and most peripheral classes. Given that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> contribution is not subtracted from the jet-like yields reported in <ce:cross-ref refid="fg0030" id="crf11450">Fig. 3</ce:cross-ref>, the centrality dependence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> follows the behaviour of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> as a function of collision centrality and particle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> in Pb–Pb collisions at LHC energies <ce:cross-ref refid="br0520" id="crf11460">[52]</ce:cross-ref>.</ce:para><ce:para id="pr0130"><ce:cross-ref refid="fg0040" id="crf11660">Fig. 4</ce:cross-ref><ce:float-anchor refid="fg0040"/> shows the measured values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> in the transverse momentum interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mn>4</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> as a function of the average multiplicity in the transverse region for all the multiplicity classes considered in pp, p–Pb and Pb–Pb collisions. The figure shows that, within uncertainties, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are close to unity for all the multiplicity classes measured in pp and p–Pb collisions. This indicates that effects induced by possible energy loss in these systems are not observed within uncertainties. This result is consistent with previous studies of nuclear modification factor <ce:cross-ref refid="br0330" id="crf11480">[33]</ce:cross-ref> and hadron-jet recoil measurements <ce:cross-ref refid="br0340" id="crf11490">[34]</ce:cross-ref>. By contrast, for Pb–Pb collisions the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are compatible to unity for peripheral collisions, and show a gradual enhancement (reduction) with the increase in multiplicity for the toward (away) region. The behaviour is the same for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values measured either assuming a flat background or a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>-modulated background. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>-modulated background was estimated following the approach depicted in Eq. <ce:cross-ref refid="fm0020" id="crf11500">(2)</ce:cross-ref> and using the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> data reported in <ce:cross-ref refid="br0490" id="crf11510">[49]</ce:cross-ref>. This behaviour is qualitatively similar to the di-hadron correlation results reported by the STAR and ALICE collaborations <ce:cross-refs refid="br0290 br0300" id="crs0100">[29,30]</ce:cross-refs>. In Pb–Pb collisions, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> provides information about the fragmenting jet leaving the medium, while on the away side, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> reflects the survival probability of the recoiling parton during passage through the medium. Thus a suppression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> would indicate that fewer partons survive the passage through the medium and is expected from the strong in-medium energy loss. On the other hand, the enhancement observed in the toward region is also subject to medium effects. The ratio is sensitive to a) a possible change of the fragmentation functions, b) a possible modification of the quark to gluon jet ratio in the final state due to different coupling with medium, and c) a possible bias on the parton spectrum due to trigger particle selection. Moreover, given that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is sensitive to the same effects as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math>, the interpretation of the results is similar to that reported in <ce:cross-ref refid="br0300" id="crf11520">[30]</ce:cross-ref>. It is likely that all three effects play a role <ce:cross-ref refid="br0300" id="crf11530">[30]</ce:cross-ref>. A detailed quantification of the contribution of each effect is beyond the scope of the present paper.</ce:para><ce:para id="pr0140">In order to get further insight into the effect, the measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are compared in <ce:cross-ref refid="fg0050" id="crf11540">Fig. 5</ce:cross-ref><ce:float-anchor refid="fg0050"/> with model predictions. Following the similar treatment of the experimental data, for the models, the total sample is subdivided into different V0M classes and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> is calculated for each class. For high-multiplicity pp collisions, although <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is close to unity, a small trend with multiplicity is visible, which is not seen at similar multiplicities (20−90% V0A) in p–Pb data. To understand the source of these slight deviations from unity, the data are compared with the predictions from the <ce:small-caps>PYTHIA</ce:small-caps> 8 (Monash tune <ce:cross-ref refid="br0280" id="crf11550">[28]</ce:cross-ref>) and EPOS-LHC <ce:cross-ref refid="br0210" id="crf11560">[21]</ce:cross-ref> event generators. In PYTHIA, the hadronization of quarks is simulated using the Lund string fragmentation model <ce:cross-ref refid="br0530" id="crf11570">[53]</ce:cross-ref>. Various PYTHIA tunes have been developed through extensive comparison of Monte Carlo distributions with the minimum-bias data from different experiments. The Monash tune of <ce:small-caps>PYTHIA</ce:small-caps> 8 is tuned to LHC data and uses an updated set of hadronization parameters compared to the previous tunes <ce:cross-ref refid="br0280" id="crf11580">[28]</ce:cross-ref>. EPOS-LHC is built on the Parton-Based Gribov Regge Theory. Utilising the colour exchange mechanism of string excitation, the model is tuned to LHC data <ce:cross-ref refid="br0210" id="crf11590">[21]</ce:cross-ref>. In this model, a part of the collision system which has high parton densities becomes a “core” region that evolves hydrodynamically as a quark–gluon plasma and it is surrounded by a more dilute “corona” for which fragmentation occurs in the vacuum. The upper panel of <ce:cross-ref refid="fg0050" id="crf11600">Fig. 5</ce:cross-ref> shows <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> for different multiplicity classes. The observed deviations from unity are reproduced by <ce:small-caps>PYTHIA</ce:small-caps> 8 for both the toward and away regions. Given that <ce:small-caps>PYTHIA</ce:small-caps> 8 does not incorporate any jet quenching mechanism, the origin of the effect in high <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> collisions is related to a remaining bias towards harder fragmentation and more activity from initial and final state radiation <ce:cross-ref refid="br0540" id="crf11610">[54]</ce:cross-ref>. These effects enhance the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> yield in the toward region, and produce a broadening in the away region <ce:cross-ref refid="br0550" id="crf11620">[55]</ce:cross-ref>. The EPOS-LHC results in the away region are similar to both data and <ce:small-caps>PYTHIA</ce:small-caps> 8. However, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> EPOS-LHC exhibits a trend with a maximum at intermediate multiplicity and a reduction toward low and high multiplicities, which is not consistent with the measurements.</ce:para><ce:para id="pr0150">The middle and bottom panels of <ce:cross-ref refid="fg0050" id="crf11630">Fig. 5</ce:cross-ref> show <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> measured for p–Pb and Pb–Pb collisions, respectively. The data are compared to <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr <ce:cross-ref refid="br0560" id="crf11640">[56]</ce:cross-ref> and EPOS-LHC predictions. The Angantyr model in <ce:small-caps>PYTHIA</ce:small-caps> 8 extrapolates the dynamics from pp collisions to p–Pb and Pb–Pb collisions, generalising the formalism adopted for pp collisions by including a description of the nucleon positions within the colliding nuclei and utilising the Glauber model to calculate the number of interacting nucleons and binary nucleon–nucleon collisions. <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr, which does not include jet quenching effects, predicts <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values consistent with unity for all the multiplicity classes in Pb–Pb collisions. Whereas for p–Pb collisions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is consistent with unity, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> is slightly below unity. In EPOS-LHC, a certain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> cutoff is defined in such a way that, above this cutoff, a particle loses part of its momentum in the core but survives as an independent particle produced by a flux tube. Soft particles, which are below the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> cutoff, get completely absorbed and form the core. This sort of energy loss mechanism implemented in EPOS-LHC depends on the system size <ce:cross-refs refid="br0210 br0570 br0580" id="crs0110">[21,57,58]</ce:cross-refs>. <ce:cross-ref refid="fg0050" id="crf11650">Fig. 5</ce:cross-ref> (middle) shows that for p–Pb collisions, EPOS-LHC does not describe either the magnitude or the trend of the multiplicity dependence of the measured ratio in the toward region, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math>. However, the model is in reasonable agreement with data in the away region. For Pb–Pb collisions, EPOS-LHC predicts a significant enhancement of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> for low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> ranges and deviates significantly from the experimental results.</ce:para><ce:para id="pr0160">In summary, while the data from Pb–Pb collisions are in qualitative agreement with expectations from parton energy loss due to the presence of a hot and dense medium, pp and p–Pb data do not show any hint of medium effects in the multiplicity range which is reported.</ce:para></ce:section><ce:section id="se0040"><ce:label>4</ce:label><ce:section-title id="st0050">Summary</ce:section-title><ce:para id="pr0170">The transverse momentum spectra (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mn>0.5</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) of primary charged particles in three azimuthal regions (toward, away and transverse) defined with respect to the direction of the particle with the highest transverse momentum in the event (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>8</mml:mn><mml:mo>≤</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) are reported. The spectra are studied in intervals of the multiplicity measured at forward pseudorapidities for pp, p–Pb, and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in the transverse region are subtracted from those of the away and toward regions. This is based on the assumption that the transverse side provides a good estimation of the underlying event contribution in both the toward and away regions. However, for the interpretation of the results one has to keep in mind that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulates the background and this effect is important for semi-central Pb–Pb collisions and for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>4</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> the effect is less than 5% in central and peripheral Pb–Pb collisions. Ratios to MB pp (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math>), i.e., the multiplicity dependent yields normalised to the yield measured in MB pp collisions, are reported. At low transverse momentum (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>), within 20%, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are multiplicity independent for both the toward and away regions in pp and p–Pb collisions. In contrast, in Pb–Pb collisions for both toward and away regions the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values exhibit a centrality dependence which is expected given the residual presence of elliptic flow. In the highest transverse momentum interval (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mn>4</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>), the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values in pp collisions are closer to unity but they exhibit a small reduction (increase) towards high V0 activity in pp collisions. This trend is well reproduced by <ce:small-caps>PYTHIA</ce:small-caps> 8. In the model, it is due to a selection bias towards pp collisions with harder fragmentation and larger activity from initial and final state radiation. For p–Pb collisions, within uncertainties, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are consistent with unity and do not show a multiplicity dependence. <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr fairly describes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math>, but it underestimates by about 10% the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> values in the low multiplicity classes (40−90% V0A event class). For Pb–Pb collisions, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are close to unity for peripheral collisions, and show a gradual increase (reduction) in the toward (away) region with increasing multiplicity. A similar observable, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math>, based on the per-trigger yield of associated particles in di-hadron correlation has been studied for central and peripheral Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The behaviour of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> exhibits the same features as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math>: in central collisions, on the away-side, a suppression is observed as expected from strong in-medium energy loss. In the toward region, an enhancement is observed. <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr predicts <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:math> for all multiplicity intervals, and it does not reproduce the observed away-side suppression or toward-side enhancement. Generally, EPOS-LHC does not describe the measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> ratios.</ce:para><ce:para id="pr0180">In summary, within the multiplicity reach reported in this paper, no jet quenching effects are observed in pp and p–Pb collisions within uncertainties. Further studies are required to extend the present analysis to higher multiplicities, which are currently limited by the event selection based on the forward V0 detector. The analysis of future pp and p–Pb collisions with much larger integrated luminosity may remove this limitation.</ce:para> </ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0090">Declaration of Competing Interest</ce:section-title><ce:para id="pr0210">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0060">Acknowledgements</ce:section-title><ce:para id="pr0190">The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: <ce:grant-sponsor id="gsp0010">A.I. 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National Research and Innovation Agency - <ce:grant-sponsor id="gsp0380" sponsor-id="https://doi.org/10.13039/100020473">BRIN</ce:grant-sponsor>, Indonesia; Istituto Nazionale di Fisica Nucleare (<ce:grant-sponsor id="gsp0390" sponsor-id="https://doi.org/10.13039/501100004007">INFN</ce:grant-sponsor>), Italy; Japanese <ce:grant-sponsor id="gsp0400" sponsor-id="https://doi.org/10.13039/501100001700">Ministry of Education, Culture, Sports, Science and Technology</ce:grant-sponsor> (MEXT) and <ce:grant-sponsor id="gsp0410" sponsor-id="https://doi.org/10.13039/501100001691">Japan Society for the Promotion of Science</ce:grant-sponsor> (JSPS) KAKENHI, Japan; Consejo Nacional de Ciencia (<ce:grant-sponsor id="gsp0420" sponsor-id="https://doi.org/10.13039/501100003141">CONACYT</ce:grant-sponsor>) y Tecnología, through <ce:grant-sponsor id="gsp0430" sponsor-id="https://doi.org/10.13039/501100007709">Fondo de Cooperación Internacional en Ciencia y Tecnología</ce:grant-sponsor> (FONCICYT) and <ce:grant-sponsor id="gsp0440" sponsor-id="https://doi.org/10.13039/501100006087">Dirección General de Asuntos del Personal Académico</ce:grant-sponsor> (DGAPA), Mexico; <ce:grant-sponsor id="gsp0450" sponsor-id="https://doi.org/10.13039/501100003246">Nederlandse Organisatie voor Wetenschappelijk Onderzoek</ce:grant-sponsor> (NWO), Netherlands; The <ce:grant-sponsor id="gsp0460" sponsor-id="https://doi.org/10.13039/501100005416">Research Council of Norway</ce:grant-sponsor>, Norway; <ce:grant-sponsor id="gsp0470">Commission on Science and Technology for Sustainable Development in the South</ce:grant-sponsor> (COMSATS), Pakistan; <ce:grant-sponsor id="gsp0480" sponsor-id="https://doi.org/10.13039/501100011871">Pontificia Universidad Católica del Perú</ce:grant-sponsor>, Peru; <ce:grant-sponsor id="gsp0490">Ministry of Education and Science</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0500" sponsor-id="https://doi.org/10.13039/501100004281">National Science Centre</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0510">WUT ID-UB</ce:grant-sponsor>, Poland; <ce:grant-sponsor id="gsp0520" sponsor-id="https://doi.org/10.13039/501100003708">Korea Institute of Science and Technology Information</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0530" sponsor-id="https://doi.org/10.13039/501100003725">National Research Foundation of Korea</ce:grant-sponsor> (NRF), Republic of Korea; <ce:grant-sponsor id="gsp0540">Ministry of Education and Scientific Research</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0550" sponsor-id="https://doi.org/10.13039/501100019278">Institute of Atomic Physics</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0560" sponsor-id="https://doi.org/10.13039/501100015622">Ministry of Research and Innovation</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0570" sponsor-id="https://doi.org/10.13039/501100019278">Institute of Atomic Physics</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0580">University Politehnica of Bucharest</ce:grant-sponsor>, Romania; <ce:grant-sponsor id="gsp0590" sponsor-id="https://doi.org/10.13039/501100003193">Ministry of Education, Science, Research and Sport of the Slovak Republic</ce:grant-sponsor>, Slovakia; <ce:grant-sponsor id="gsp0600">National Research Foundation of South Africa</ce:grant-sponsor>, South Africa; <ce:grant-sponsor id="gsp0610" sponsor-id="https://doi.org/10.13039/501100004359">Swedish Research Council</ce:grant-sponsor> (VR) and <ce:grant-sponsor id="gsp0620">Knut & Alice Wallenberg Foundation</ce:grant-sponsor> (KAW), Sweden; <ce:grant-sponsor id="gsp0630" sponsor-id="https://doi.org/10.13039/100012470">European Organization for Nuclear Research</ce:grant-sponsor>, Switzerland; <ce:grant-sponsor id="gsp0640" sponsor-id="https://doi.org/10.13039/501100004352">Suranaree University of Technology</ce:grant-sponsor> (SUT), <ce:grant-sponsor id="gsp0650" sponsor-id="https://doi.org/10.13039/501100004192">National Science and Technology Development Agency</ce:grant-sponsor> (NSTDA), <ce:grant-sponsor id="gsp0660" sponsor-id="https://doi.org/10.13039/501100017170">Thailand Science Research and Innovation</ce:grant-sponsor> (TSRI) and <ce:grant-sponsor id="gsp0670">National Science, Research and Innovation Fund</ce:grant-sponsor> (NSRF), Thailand; <ce:grant-sponsor id="gsp0680" sponsor-id="https://doi.org/10.13039/100020381">Turkish Energy, Nuclear and Mineral Research Agency</ce:grant-sponsor> (TENMAK), Turkey; <ce:grant-sponsor id="gsp0690" sponsor-id="https://doi.org/10.13039/501100004742">National Academy of Sciences of Ukraine</ce:grant-sponsor>, Ukraine; <ce:grant-sponsor id="gsp0700" sponsor-id="https://doi.org/10.13039/501100000271">Science and Technology Facilities Council</ce:grant-sponsor> (STFC), United Kingdom; National Science Foundation of the United States of America (<ce:grant-sponsor id="gsp0710" sponsor-id="https://doi.org/10.13039/100000001">NSF</ce:grant-sponsor>) and <ce:grant-sponsor id="gsp0720" sponsor-id="https://doi.org/10.13039/100000015">United States Department of Energy</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0730" sponsor-id="https://doi.org/10.13039/100006209">Office of Nuclear Physics</ce:grant-sponsor> (DOE NP), United States of America. In addition, individual groups or members have received support from: Marie Skłodowska Curie, Strong 2020 - <ce:grant-sponsor id="gsp0740" sponsor-id="https://doi.org/10.13039/100010661">Horizon 2020</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0750" sponsor-id="https://doi.org/10.13039/501100000781">European Research Council</ce:grant-sponsor> (grant nos. <ce:grant-number refid="gsp0750">824093</ce:grant-number>, <ce:grant-number refid="gsp0750">896850</ce:grant-number>, <ce:grant-number refid="gsp0750">950692</ce:grant-number>), <ce:grant-sponsor id="gsp0760" sponsor-id="https://doi.org/10.13039/501100000780">European Union</ce:grant-sponsor>; <ce:grant-sponsor id="gsp0770" sponsor-id="https://doi.org/10.13039/501100002341">Academy of Finland</ce:grant-sponsor> (Center of Excellence in Quark Matter) (grant nos. <ce:grant-number refid="gsp0770">346327</ce:grant-number>, <ce:grant-number refid="gsp0770">346328</ce:grant-number>), Finland; <ce:grant-sponsor id="gsp0780">Programa de Apoyos para la Superación del Personal Académico</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0790" sponsor-id="https://doi.org/10.13039/501100005739">UNAM</ce:grant-sponsor>, Mexico.</ce:para></ce:acknowledgment></body><tail><ce:bibliography id="bl0010"><ce:section-title id="st0070">References</ce:section-title><ce:bibliography-sec id="bs0010"><ce:bib-reference id="br0010"><ce:label>[1]</ce:label><sb:reference id="bibD17CEA27EFFA6FC144ED8C3DC8A6A4C2s1"><sb:contribution><sb:authors><sb:author><ce:given-name>T.</ce:given-name><ce:surname>Sjöstrand</ce:surname></sb:author><sb:author><ce:given-name>M.</ce:given-name><ce:surname>van Zijl</ce:surname></sb:author></sb:authors><sb:title><sb:maintitle>A multiple interaction model for the event structure in hadron collisions</sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Phys. 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Peigne, “Collisional Energy Loss of a Fast Parton in a QGP”, AIP Conf. Proc. 1038 no. 1, (2008) 139–148, arXiv:0806.0242 [hep-ph].</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> \ No newline at end of file +<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd" [<!ENTITY gr001 SYSTEM "gr001" NDATA IMAGE><!ENTITY gr002 SYSTEM "gr002" NDATA IMAGE><!ENTITY gr003 SYSTEM "gr003" NDATA IMAGE><!ENTITY gr004 SYSTEM "gr004" NDATA IMAGE><!ENTITY gr005 SYSTEM "gr005" NDATA IMAGE>]><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="sco" xml:lang="en"><item-info><jid>PLB</jid><aid>137649</aid><ce:article-number>137649</ce:article-number><ce:pii>S0370-2693(22)00783-3</ce:pii><ce:doi>10.1016/j.physletb.2022.137649</ce:doi><ce:copyright year="2023" type="other">The Author(s)</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>Experiments</ce:text></ce:doctopic></ce:doctopics><ce:preprint><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/preprint" xlink:href="arxiv:2204.10157" id="inf0010"/></ce:preprint><ce:associated-resource><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/research-data" xlink:href="https://www.hepdata.net/" id="inf0540">https://www.hepdata.net/</ce:inter-ref></ce:associated-resource></item-info><ce:floats><ce:figure id="fg0010"><ce:label>Fig. 1</ce:label><ce:caption id="cp0010"><ce:simple-para id="sp0010">Illustration of toward, away and transverse regions with respect to the leading particle in a collision.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0010">Fig. 1</ce:alt-text><ce:link locator="gr001" xlink:type="simple" xlink:href="pii:S0370269322007833/gr001" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0010"/></ce:figure><ce:figure id="fg0020"><ce:label>Fig. 2</ce:label><ce:caption id="cp0020"><ce:simple-para id="sp0020">Top panels: transverse momentum spectra of charged particles in the transverse region for different multiplicity classes in pp (left), p–Pb (middle) and Pb–Pb (right) collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra are measured at mid pseudorapidity (|<ce:italic>η</ce:italic>| < 0.8). Lower panels: Ratio of <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra in different multiplicity classes to the <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectrum in the 0−100% multiplicity class for the corresponding collision systems. The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0020">Fig. 2</ce:alt-text><ce:link locator="gr002" xlink:type="simple" xlink:href="pii:S0370269322007833/gr002" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0020"/></ce:figure><ce:figure id="fg0030"><ce:label>Fig. 3</ce:label><ce:caption id="cp0030"><ce:simple-para id="sp0030">Transverse momentum spectra of charged particles in Toward-Transverse, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> (top plot) and Away-Transverse, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> (bottom plot) regions for different multiplicity classes in pp (left), p–Pb (middle) and Pb–Pb (right) collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra are measured at mid pseudorapidity (|<ce:italic>η</ce:italic>| < 0.8). The lower panels of both plots show the ratio to minimum bias pp collisions. The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0030">Fig. 3</ce:alt-text><ce:link locator="gr003" xlink:type="simple" xlink:href="pii:S0370269322007833/gr003" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0030"/></ce:figure><ce:figure id="fg0040"><ce:label>Fig. 4</ce:label><ce:caption id="cp0040"><ce:simple-para id="sp0040">The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> (left) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> (right) as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> in 4 <<ce:italic>p</ce:italic><ce:inf>T</ce:inf>< 6 GeV/<ce:italic>c</ce:italic> for different multiplicity classes in pp, p–Pb and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. Pb–Pb results are shown assuming a flat background (filled markers), and assuming a <ce:italic>v</ce:italic><ce:inf>2</ce:inf>-modulated background (empty markers). The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0040">Fig. 4</ce:alt-text><ce:link locator="gr004" xlink:type="simple" xlink:href="pii:S0370269322007833/gr004" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0040"/></ce:figure><ce:figure id="fg0050"><ce:label>Fig. 5</ce:label><ce:caption id="cp0050"><ce:simple-para id="sp0050">Comparison of the measured the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> (left) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> (right) in 4 <<ce:italic>p</ce:italic><ce:inf>T</ce:inf>< 6 GeV/<ce:italic>c</ce:italic> with model predictions. The results are shown as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> for different multiplicity classes in pp (top panel), p–Pb (middle panel) and Pb–Pb (bottom panel) collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The red and magenta lines show the <ce:small-caps>PYTHIA</ce:small-caps> 8 (Monash) <ce:cross-ref refid="br0280" id="crf0010">[28]</ce:cross-ref> and <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr <ce:cross-ref refid="br0280" id="crf0020">[28]</ce:cross-ref> predictions, respectively. The blue lines show the EPOS-LHC <ce:cross-ref refid="br0210" id="crf0030">[21]</ce:cross-ref> results. The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0050">Fig. 5</ce:alt-text><ce:link locator="gr005" xlink:type="simple" xlink:href="pii:S0370269322007833/gr005" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0050"/></ce:figure><ce:table xmlns="http://www.elsevier.com/xml/common/cals/dtd" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" id="tbl0010" frame="topbot" rowsep="0" colsep="0"><ce:label>Table 1</ce:label><ce:caption id="cp0060"><ce:simple-para id="sp0060">Contributions to the relative (%) systematic uncertainty on the <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra of primary charged particles in pp, p–Pb, and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. Just for illustration, the range in the table corresponds to the lowest and highest relative systematic uncertainty in the considered <ce:italic>p</ce:italic><ce:inf>T</ce:inf> range. The individual contributions are summed in quadrature to obtain the total uncertainty.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0060">Table 1</ce:alt-text><tgroup cols="4"><colspec colnum="1" colname="col1" align="left"/><colspec colnum="2" colname="col2" align="left"/><colspec colnum="3" colname="col3" align="left"/><colspec colnum="4" colname="col4" align="left"/><thead valign="top"><row rowsep="1"><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Source of uncertainty</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">pp</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">p–Pb</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">Pb–Pb</entry></row></thead><tbody valign="top"><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Track selection</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.1–8.2</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.4–5.8</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.0–9.9</entry></row><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Particle composition</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.3–1.8</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.5–1.9</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.3–2.4</entry></row><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Secondary particles</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.0–0.4</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.0–2.4</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.0–1.9</entry></row><row rowsep="1"><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Matching efficiency</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.0–4.2</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.7–3.7</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.6–3.7</entry></row><row rowsep="1"><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Total</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.2–8.8</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.6–6.3</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.5–10.0</entry></row><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Total (<ce:italic>N</ce:italic><ce:inf>ch</ce:inf>-dependent)</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.0–4.5</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">1.7–4.0</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">1.1–3.7</entry></row></tbody></tgroup></ce:table></ce:floats><head><ce:title id="ti0010">Study of charged particle production at high <ce:italic>p</ce:italic><ce:inf>T</ce:inf> using event topology in pp, p–Pb and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV</ce:title><ce:author-group id="ag0010"><ce:collaboration id="co0010" collaboration-id="S0370269322007833-3bea72599603117cd9d18494a0279c47"><ce:text>ALICE Collaboration</ce:text><ce:cross-ref refid="fn0080" id="crf0040"><ce:sup>⋆</ce:sup></ce:cross-ref><ce:author-group id="ag0020"><ce:author orcid="0000-0002-9213-5329" id="au0010" author-id="S0370269322007833-e60c93a934b81cf9801254193264c6ee"><ce:given-name>S.</ce:given-name><ce:surname>Acharya</ce:surname><ce:cross-ref refid="aff1240" id="crf0050"><ce:sup>124</ce:sup></ce:cross-ref><ce:cross-ref refid="aff1310" id="crf0060"><ce:sup>131</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-0504-7428" id="au0020" author-id="S0370269322007833-0eab85892b6d74b18661e74a7987c599"><ce:given-name>D.</ce:given-name><ce:surname>Adamová</ce:surname><ce:cross-ref refid="aff0860" id="crf0070"><ce:sup>86</ce:sup></ce:cross-ref></ce:author><ce:author id="au0030" author-id="S0370269322007833-e83a30ae1d5f89088c60ca2ee154d714"><ce:given-name>A.</ce:given-name><ce:surname>Adler</ce:surname><ce:cross-ref refid="aff0690" id="crf0080"><ce:sup>69</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-9611-3696" id="au0040" author-id="S0370269322007833-ed2d58d89990c41bb43c091d01e5029a"><ce:given-name>G.</ce:given-name><ce:surname>Aglieri Rinella</ce:surname><ce:cross-ref refid="aff0320" id="crf0090"><ce:sup>32</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-0760-5075" id="au0050" author-id="S0370269322007833-0c7a7863b7384aa5fdf06a0f187949c8"><ce:given-name>M.</ce:given-name><ce:surname>Agnello</ce:surname><ce:cross-ref refid="aff0290" 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id="crf10830"><ce:sup>83</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0001-9358-5762" id="au10290" author-id="S0370269322007833-b564bd6149c66f06bbea2780670b8f50"><ce:given-name>J.</ce:given-name><ce:surname>Zhu</ce:surname><ce:cross-ref refid="aff0980" id="crf10840"><ce:sup>98</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0060" id="crf10850"><ce:sup>6</ce:sup></ce:cross-ref></ce:author><ce:author id="au10300" author-id="S0370269322007833-1fb0ab925cfcdaacb160193d82c1a978"><ce:given-name>Y.</ce:given-name><ce:surname>Zhu</ce:surname><ce:cross-ref refid="aff0060" id="crf10860"><ce:sup>6</ce:sup></ce:cross-ref></ce:author><ce:author id="au10310" author-id="S0370269322007833-2ad055472e2fac995111c51c08396d0c"><ce:given-name>G.</ce:given-name><ce:surname>Zinovjev</ce:surname><ce:cross-ref refid="aff0030" id="crf10870"><ce:sup>3</ce:sup></ce:cross-ref><ce:cross-ref refid="fn0010" id="crf10880"><ce:sup>I</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-7478-2493" id="au10320" author-id="S0370269322007833-92b5ecf5612e2e849fc1bba72c6cff99"><ce:given-name>N.</ce:given-name><ce:surname>Zurlo</ce:surname><ce:cross-ref refid="aff1300" id="crf10890"><ce:sup>130</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0540" id="crf10900"><ce:sup>54</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="aff0010" affiliation-id="S0370269322007833-79d30baa35325e84d46378ba6ce12c18"><ce:label>1</ce:label><ce:textfn>A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia</ce:textfn><sa:affiliation><sa:organization>A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation</sa:organization><sa:city>Yerevan</sa:city><sa:country>Armenia</sa:country></sa:affiliation><ce:source-text id="srct0005">A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia</ce:source-text></ce:affiliation><ce:affiliation id="aff0020" affiliation-id="S0370269322007833-65754d218cf7f84bf1e02306b80caca0"><ce:label>2</ce:label><ce:textfn>AGH University of Science and Technology, Cracow, Poland</ce:textfn><sa:affiliation><sa:organization>AGH University of Science and Technology</sa:organization><sa:city>Cracow</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0010">AGH University of Science and Technology, Cracow, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff0030" affiliation-id="S0370269322007833-e916a2f48a17bc32220b61ae0e9b8e05"><ce:label>3</ce:label><ce:textfn>Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine</ce:textfn><sa:affiliation><sa:organization>Bogolyubov Institute for Theoretical Physics</sa:organization><sa:organization>National Academy of Sciences of Ukraine</sa:organization><sa:city>Kiev</sa:city><sa:country>Ukraine</sa:country></sa:affiliation><ce:source-text id="srct0015">Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine</ce:source-text></ce:affiliation><ce:affiliation id="aff0040" affiliation-id="S0370269322007833-9869d95133b2275e34836b8ad2f235c3"><ce:label>4</ce:label><ce:textfn>Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Bose Institute</sa:organization><sa:organization>Department of Physics</sa:organization><sa:organization>Centre for Astroparticle Physics and Space Science (CAPSS)</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0020">Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0050" affiliation-id="S0370269322007833-b7f796ef6c934d79a497288cdec192f6"><ce:label>5</ce:label><ce:textfn>California Polytechnic State University, San Luis Obispo, CA, United States</ce:textfn><sa:affiliation><sa:organization>California Polytechnic State University</sa:organization><sa:city>San Luis Obispo</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0025">California Polytechnic State University, San Luis Obispo, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0060" affiliation-id="S0370269322007833-3dcd6ffc2e8f27d6ed2f6237a209a384"><ce:label>6</ce:label><ce:textfn>Central China Normal University, Wuhan, China</ce:textfn><sa:affiliation><sa:organization>Central China Normal University</sa:organization><sa:city>Wuhan</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0030">Central China Normal University, Wuhan, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0070" affiliation-id="S0370269322007833-110460c7f2fbb319ec6d52e7cc3fc1d5"><ce:label>7</ce:label><ce:textfn>Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba</ce:textfn><sa:affiliation><sa:organization>Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN)</sa:organization><sa:city>Havana</sa:city><sa:country>Cuba</sa:country></sa:affiliation><ce:source-text id="srct0035">Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba</ce:source-text></ce:affiliation><ce:affiliation id="aff0080" affiliation-id="S0370269322007833-641aa526558990d110c090840e6f3d0c"><ce:label>8</ce:label><ce:textfn>Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico</ce:textfn><sa:affiliation><sa:organization>Centro de Investigación y de Estudios Avanzados (CINVESTAV)</sa:organization><sa:city>Mexico City and Mérida</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0040">Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0090" affiliation-id="S0370269322007833-9c73ece39ab638447cd10f268e329b2e"><ce:label>9</ce:label><ce:textfn>Chicago State University, Chicago, IL, United States</ce:textfn><sa:affiliation><sa:organization>Chicago State University</sa:organization><sa:city>Chicago</sa:city><sa:state>IL</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0045">Chicago State University, Chicago, Illinois, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0100" affiliation-id="S0370269322007833-d7da93bf3f02be0a46bee74f439b5930"><ce:label>10</ce:label><ce:textfn>China Institute of Atomic Energy, Beijing, China</ce:textfn><sa:affiliation><sa:organization>China Institute of Atomic Energy</sa:organization><sa:city>Beijing</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0050">China Institute of Atomic Energy, Beijing, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0110" affiliation-id="S0370269322007833-d477851cd020cd39da135784676a96ff"><ce:label>11</ce:label><ce:textfn>Chungbuk National University, Cheongju, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Chungbuk National University</sa:organization><sa:city>Cheongju</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0055">Chungbuk National University, Cheongju, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0120" affiliation-id="S0370269322007833-b03a6e652a70daee257206602eb6f83f"><ce:label>12</ce:label><ce:textfn>Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Comenius University Bratislava</sa:organization><sa:organization>Faculty of Mathematics, Physics and Informatics</sa:organization><sa:city>Bratislava</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0060">Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0130" affiliation-id="S0370269322007833-78cc8333b14d598fc5e3c0230d1de22e"><ce:label>13</ce:label><ce:textfn>COMSATS University Islamabad, Islamabad, Pakistan</ce:textfn><sa:affiliation><sa:organization>COMSATS University Islamabad</sa:organization><sa:city>Islamabad</sa:city><sa:country>Pakistan</sa:country></sa:affiliation><ce:source-text id="srct0065">COMSATS University Islamabad, Islamabad, Pakistan</ce:source-text></ce:affiliation><ce:affiliation id="aff0140" affiliation-id="S0370269322007833-42161ea7466da89325c5b37601107c55"><ce:label>14</ce:label><ce:textfn>Creighton University, Omaha, NE, United States</ce:textfn><sa:affiliation><sa:organization>Creighton University</sa:organization><sa:city>Omaha</sa:city><sa:state>NE</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0070">Creighton University, Omaha, Nebraska, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0150" affiliation-id="S0370269322007833-bd5e6b818668501c1deea76e96cac833"><ce:label>15</ce:label><ce:textfn>Department of Physics, Aligarh Muslim University, Aligarh, India</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Aligarh Muslim University</sa:organization><sa:city>Aligarh</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0075">Department of Physics, Aligarh Muslim University, Aligarh, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0160" affiliation-id="S0370269322007833-ebe7875fb705d3c179953d196aa8b94b"><ce:label>16</ce:label><ce:textfn>Department of Physics, Pusan National University, Pusan, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Pusan National University</sa:organization><sa:city>Pusan</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0080">Department of Physics, Pusan National University, Pusan, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0170" affiliation-id="S0370269322007833-24ddc81b37e640977b88412ba28336a4"><ce:label>17</ce:label><ce:textfn>Department of Physics, Sejong University, Seoul, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Sejong University</sa:organization><sa:city>Seoul</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0085">Department of Physics, Sejong University, Seoul, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0180" affiliation-id="S0370269322007833-66245837acb0d7fb7ada51de0fd95043"><ce:label>18</ce:label><ce:textfn>Department of Physics, University of California, Berkeley, CA, United States</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>University of California</sa:organization><sa:city>Berkeley</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0090">Department of Physics, University of California, Berkeley, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0190" affiliation-id="S0370269322007833-63776eabafa9e32856b35562692a4488"><ce:label>19</ce:label><ce:textfn>Department of Physics, University of Oslo, Oslo, Norway</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>University of Oslo</sa:organization><sa:city>Oslo</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0095">Department of Physics, University of Oslo, Oslo, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0200" affiliation-id="S0370269322007833-020451a59d5e257505166bdb7847cdd7"><ce:label>20</ce:label><ce:textfn>Department of Physics and Technology, University of Bergen, Bergen, Norway</ce:textfn><sa:affiliation><sa:organization>Department of Physics and Technology</sa:organization><sa:organization>University of Bergen</sa:organization><sa:city>Bergen</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0100">Department of Physics and Technology, University of Bergen, Bergen, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0210" affiliation-id="S0370269322007833-f2de76852013198101844806b89d3836"><ce:label>21</ce:label><ce:textfn>Dipartimento di Fisica, Università di Pavia, Pavia, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica</sa:organization><sa:organization>Università di Pavia</sa:organization><sa:city>Pavia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0105">Dipartimento di Fisica, Università di Pavia, Pavia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0220" affiliation-id="S0370269322007833-81d87d2cd8c9f6aa2a1c7f4b13115ef3"><ce:label>22</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Cagliari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0110">Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0230" affiliation-id="S0370269322007833-849d0ba7888fcb2a09aad7ac1095ec15"><ce:label>23</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Trieste</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0115">Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0240" affiliation-id="S0370269322007833-68c63fd1cb9e8623af62118bbef39c9e"><ce:label>24</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0120">Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0250" affiliation-id="S0370269322007833-7a40dab487121429c63e59c3de15ccfa"><ce:label>25</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bologna</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0125">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0260" affiliation-id="S0370269322007833-72e3fe624de7d3d3223574366a22ef30"><ce:label>26</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Catania</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0130">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0270" affiliation-id="S0370269322007833-6145c67e3009fb117e82f5429b2af282"><ce:label>27</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Padova</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0135">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0280" affiliation-id="S0370269322007833-a001bc692b1f1f84377401c9e2632e51"><ce:label>28</ce:label><ce:textfn>Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università and Gruppo Collegato INFN, Salerno, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università</sa:organization><sa:organization>Gruppo Collegato INFN</sa:organization><sa:city>Salerno</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0140">Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università and Gruppo Collegato INFN, Salerno, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0290" affiliation-id="S0370269322007833-362983c57dbe79a62add2550bfe565dc"><ce:label>29</ce:label><ce:textfn>Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento DISAT del Politecnico</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0145">Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0300" affiliation-id="S0370269322007833-7a82e32411929fc5768d11b72609a4d8"><ce:label>30</ce:label><ce:textfn>Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Scienze MIFT</sa:organization><sa:organization>Università di Messina</sa:organization><sa:city>Messina</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0150">Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0310" affiliation-id="S0370269322007833-0bfba0b176e2b6e1b67e6564c87308b5"><ce:label>31</ce:label><ce:textfn>Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento Interateneo di Fisica ‘M. Merlin’</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0155">Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0320" affiliation-id="S0370269322007833-44f92095d23d6e3d2fb8e8fc998c51f2"><ce:label>32</ce:label><ce:textfn>European Organization for Nuclear Research (CERN), Geneva, Switzerland</ce:textfn><sa:affiliation><sa:organization>European Organization for Nuclear Research (CERN)</sa:organization><sa:city>Geneva</sa:city><sa:country>Switzerland</sa:country></sa:affiliation><ce:source-text id="srct0160">European Organization for Nuclear Research (CERN), Geneva, Switzerland</ce:source-text></ce:affiliation><ce:affiliation id="aff0330" affiliation-id="S0370269322007833-f220870dd6ed2747e8ec11b2ba624bf5"><ce:label>33</ce:label><ce:textfn>Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia</ce:textfn><sa:affiliation><sa:organization>Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture</sa:organization><sa:organization>University of Split</sa:organization><sa:city>Split</sa:city><sa:country>Croatia</sa:country></sa:affiliation><ce:source-text id="srct0165">Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia</ce:source-text></ce:affiliation><ce:affiliation id="aff0340" affiliation-id="S0370269322007833-5bf4b9d297f0544e6037addfb7689840"><ce:label>34</ce:label><ce:textfn>Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway</ce:textfn><sa:affiliation><sa:organization>Faculty of Engineering and Science</sa:organization><sa:organization>Western Norway University of Applied Sciences</sa:organization><sa:city>Bergen</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0170">Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0350" affiliation-id="S0370269322007833-56d412e1c3fe114d9a18225fa76274c9"><ce:label>35</ce:label><ce:textfn>Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Faculty of Nuclear Sciences and Physical Engineering</sa:organization><sa:organization>Czech Technical University in Prague</sa:organization><sa:city>Prague</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0175">Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0360" affiliation-id="S0370269322007833-de312990ec5b7745ed2a9609074bb450"><ce:label>36</ce:label><ce:textfn>Faculty of Physics, Sofia University, Sofia, Bulgaria</ce:textfn><sa:affiliation><sa:organization>Faculty of Physics</sa:organization><sa:organization>Sofia University</sa:organization><sa:city>Sofia</sa:city><sa:country>Bulgaria</sa:country></sa:affiliation><ce:source-text id="srct0180">Faculty of Physics, Sofia University, Sofia, Bulgaria</ce:source-text></ce:affiliation><ce:affiliation id="aff0370" affiliation-id="S0370269322007833-0879c60c6550c16bd60686725f5c6938"><ce:label>37</ce:label><ce:textfn>Faculty of Science, P.J. Šafárik University, Košice, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Faculty of Science</sa:organization><sa:organization>P.J. Šafárik University</sa:organization><sa:city>Košice</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0185">Faculty of Science, P.J. Šafárik University, Košice, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0380" affiliation-id="S0370269322007833-a18716885f5bc83f5d1aee8d0d80c7af"><ce:label>38</ce:label><ce:textfn>Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Frankfurt Institute for Advanced Studies</sa:organization><sa:organization>Johann Wolfgang Goethe-Universität Frankfurt</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0190">Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0390" affiliation-id="S0370269322007833-f0b0a2b18fef5547dcd39253b5714404"><ce:label>39</ce:label><ce:textfn>Fudan University, Shanghai, China</ce:textfn><sa:affiliation><sa:organization>Fudan University</sa:organization><sa:city>Shanghai</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0195">Fudan University, Shanghai, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0400" affiliation-id="S0370269322007833-a2398937a38c48ebf6ac3ff5e5d60b95"><ce:label>40</ce:label><ce:textfn>Gangneung-Wonju National University, Gangneung, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Gangneung-Wonju National University</sa:organization><sa:city>Gangneung</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0200">Gangneung-Wonju National University, Gangneung, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0410" affiliation-id="S0370269322007833-73bc80872cd116960a09bc477e035838"><ce:label>41</ce:label><ce:textfn>Gauhati University, Department of Physics, Guwahati, India</ce:textfn><sa:affiliation><sa:organization>Gauhati University</sa:organization><sa:organization>Department of Physics</sa:organization><sa:city>Guwahati</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0205">Gauhati University, Department of Physics, Guwahati, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0420" affiliation-id="S0370269322007833-9fd27eebf76465366fe59cb8d9620ae8"><ce:label>42</ce:label><ce:textfn>Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany</ce:textfn><sa:affiliation><sa:organization>Helmholtz-Institut für Strahlen- und Kernphysik</sa:organization><sa:organization>Rheinische Friedrich-Wilhelms-Universität Bonn</sa:organization><sa:city>Bonn</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0210">Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0430" affiliation-id="S0370269322007833-66b9f8889421b091fa908925e638d91e"><ce:label>43</ce:label><ce:textfn>Helsinki Institute of Physics (HIP), Helsinki, Finland</ce:textfn><sa:affiliation><sa:organization>Helsinki Institute of Physics (HIP)</sa:organization><sa:city>Helsinki</sa:city><sa:country>Finland</sa:country></sa:affiliation><ce:source-text id="srct0215">Helsinki Institute of Physics (HIP), Helsinki, Finland</ce:source-text></ce:affiliation><ce:affiliation id="aff0440" affiliation-id="S0370269322007833-6968018d6fe1bd12a4413918be70ab85"><ce:label>44</ce:label><ce:textfn>High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico</ce:textfn><sa:affiliation><sa:organization>High Energy Physics Group</sa:organization><sa:organization>Universidad Autónoma de Puebla</sa:organization><sa:city>Puebla</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0220">High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0450" affiliation-id="S0370269322007833-16bdf6e3577a480da99159f5d54452b1"><ce:label>45</ce:label><ce:textfn>Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>Horia Hulubei National Institute of Physics and Nuclear Engineering</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0225">Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff0460" affiliation-id="S0370269322007833-78df0ad3e050545612fd8d71a4776a19"><ce:label>46</ce:label><ce:textfn>Indian Institute of Technology Bombay (IIT), Mumbai, India</ce:textfn><sa:affiliation><sa:organization>Indian Institute of Technology Bombay (IIT)</sa:organization><sa:city>Mumbai</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0230">Indian Institute of Technology Bombay (IIT), Mumbai, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0470" affiliation-id="S0370269322007833-b8ad6c9375b7a89b768adb13f27427b4"><ce:label>47</ce:label><ce:textfn>Indian Institute of Technology Indore, Indore, India</ce:textfn><sa:affiliation><sa:organization>Indian Institute of Technology Indore</sa:organization><sa:city>Indore</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0235">Indian Institute of Technology Indore, Indore, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0480" affiliation-id="S0370269322007833-25658fba725d22058ae2a8649ceeb084"><ce:label>48</ce:label><ce:textfn>INFN, Laboratori Nazionali di Frascati, Frascati, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Laboratori Nazionali di Frascati</sa:organization><sa:city>Frascati</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0240">INFN, Laboratori Nazionali di Frascati, Frascati, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0490" affiliation-id="S0370269322007833-5d9dee68bdf16e34f2f3f01d930f367d"><ce:label>49</ce:label><ce:textfn>INFN, Sezione di Bari, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Bari</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0245">INFN, Sezione di Bari, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0500" affiliation-id="S0370269322007833-340d750afed4ff705cf1dd72bf688ca4"><ce:label>50</ce:label><ce:textfn>INFN, Sezione di Bologna, Bologna, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Bologna</sa:organization><sa:city>Bologna</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0250">INFN, Sezione di Bologna, Bologna, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0510" affiliation-id="S0370269322007833-436e8c989d6c10cae74944a81714a4e2"><ce:label>51</ce:label><ce:textfn>INFN, Sezione di Cagliari, Cagliari, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Cagliari</sa:organization><sa:city>Cagliari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0255">INFN, Sezione di Cagliari, Cagliari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0520" affiliation-id="S0370269322007833-f0607c03a8b381da2bb83a11f8d899c8"><ce:label>52</ce:label><ce:textfn>INFN, Sezione di Catania, Catania, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Catania</sa:organization><sa:city>Catania</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0260">INFN, Sezione di Catania, Catania, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0530" affiliation-id="S0370269322007833-cb33711bf32ecc9cc4697cbbea10fa88"><ce:label>53</ce:label><ce:textfn>INFN, Sezione di Padova, Padova, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Padova</sa:organization><sa:city>Padova</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0265">INFN, Sezione di Padova, Padova, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0540" affiliation-id="S0370269322007833-5c70f16aa0d389aa33e2589db6fcd5d6"><ce:label>54</ce:label><ce:textfn>INFN, Sezione di Pavia, Pavia, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Pavia</sa:organization><sa:city>Pavia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0270">INFN, Sezione di Pavia, Pavia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0550" affiliation-id="S0370269322007833-f1faa0de67f6d8d1ece0914257c7635c"><ce:label>55</ce:label><ce:textfn>INFN, Sezione di Torino, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Torino</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0275">INFN, Sezione di Torino, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0560" affiliation-id="S0370269322007833-0035d662cbe7eef98bed21545fd04324"><ce:label>56</ce:label><ce:textfn>INFN, Sezione di Trieste, Trieste, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Trieste</sa:organization><sa:city>Trieste</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0280">INFN, Sezione di Trieste, Trieste, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0570" affiliation-id="S0370269322007833-22e3a6d8462b39bb6c155bce0c9b21cd"><ce:label>57</ce:label><ce:textfn>Inha University, Incheon, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Inha University</sa:organization><sa:city>Incheon</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0285">Inha University, Incheon, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0580" affiliation-id="S0370269322007833-d4941ebd67ab9bbf96b1e6a906f4397c"><ce:label>58</ce:label><ce:textfn>Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands</ce:textfn><sa:affiliation><sa:organization>Institute for Gravitational and Subatomic Physics (GRASP)</sa:organization><sa:organization>Utrecht University/Nikhef</sa:organization><sa:city>Utrecht</sa:city><sa:country>Netherlands</sa:country></sa:affiliation><ce:source-text id="srct0290">Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands</ce:source-text></ce:affiliation><ce:affiliation id="aff0590" affiliation-id="S0370269322007833-f6d8becf2ef35885577b8164313401cb"><ce:label>59</ce:label><ce:textfn>Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Institute of Experimental Physics</sa:organization><sa:organization>Slovak Academy of Sciences</sa:organization><sa:city>Košice</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0295">Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0600" affiliation-id="S0370269322007833-6fa96ff1fcc4aaff09633c1217f06ff0"><ce:label>60</ce:label><ce:textfn>Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India</ce:textfn><sa:affiliation><sa:organization>Institute of Physics</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Bhubaneswar</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0300">Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0610" affiliation-id="S0370269322007833-9f77e70dcbde10c6ed3d37b3b0071107"><ce:label>61</ce:label><ce:textfn>Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Institute of Physics of the Czech Academy of Sciences</sa:organization><sa:city>Prague</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0305">Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0620" affiliation-id="S0370269322007833-3309ea65bfe3ae2f96b232545cb67043"><ce:label>62</ce:label><ce:textfn>Institute of Space Science (ISS), Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>Institute of Space Science (ISS)</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0310">Institute of Space Science (ISS), Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff0630" affiliation-id="S0370269322007833-2759820a41f8b0ece8c42aa319465ed5"><ce:label>63</ce:label><ce:textfn>Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Institut für Kernphysik</sa:organization><sa:organization>Johann Wolfgang Goethe-Universität Frankfurt</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0315">Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0640" affiliation-id="S0370269322007833-b18018b82a469dc4e4eb94f595cf4812"><ce:label>64</ce:label><ce:textfn>Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:textfn><sa:affiliation><sa:organization>Instituto de Ciencias Nucleares</sa:organization><sa:organization>Universidad Nacional Autónoma de México</sa:organization><sa:city>Mexico City</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0320">Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0650" affiliation-id="S0370269322007833-2aa194eda46d62198ea9b929240200b8"><ce:label>65</ce:label><ce:textfn>Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil</ce:textfn><sa:affiliation><sa:organization>Instituto de Física</sa:organization><sa:organization>Universidade Federal do Rio Grande do Sul (UFRGS)</sa:organization><sa:city>Porto Alegre</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0325">Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff0660" affiliation-id="S0370269322007833-9e9b95fa2c082308cb0efc3488541c67"><ce:label>66</ce:label><ce:textfn>Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:textfn><sa:affiliation><sa:organization>Instituto de Física</sa:organization><sa:organization>Universidad Nacional Autónoma de México</sa:organization><sa:city>Mexico City</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0330">Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0670" affiliation-id="S0370269322007833-4ef9aeae6b2f74e366edd12aafbea4cf"><ce:label>67</ce:label><ce:textfn>iThemba LABS, National Research Foundation, Somerset West, South Africa</ce:textfn><sa:affiliation><sa:organization>iThemba LABS</sa:organization><sa:organization>National Research Foundation</sa:organization><sa:city>Somerset West</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0335">iThemba LABS, National Research Foundation, Somerset West, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff0680" affiliation-id="S0370269322007833-2f4c4db0447d27fd2e43117b90bb74d4"><ce:label>68</ce:label><ce:textfn>Jeonbuk National University, Jeonju, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Jeonbuk National University</sa:organization><sa:city>Jeonju</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0340">Jeonbuk National University, Jeonju, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0690" affiliation-id="S0370269322007833-81ec7d4c8c69486a57ef1ca6a567076c"><ce:label>69</ce:label><ce:textfn>Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik</sa:organization><sa:organization>Fachbereich Informatik und Mathematik</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0345">Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0700" affiliation-id="S0370269322007833-c1d849875c0de0db6476f9cc96b07dd2"><ce:label>70</ce:label><ce:textfn>Korea Institute of Science and Technology Information, Daejeon, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Korea Institute of Science and Technology Information</sa:organization><sa:city>Daejeon</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0350">Korea Institute of Science and Technology Information, Daejeon, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0710" affiliation-id="S0370269322007833-c5cd420fa3d6b7c79dc3ab9e7c2dfb06"><ce:label>71</ce:label><ce:textfn>KTO Karatay University, Konya, Turkey</ce:textfn><sa:affiliation><sa:organization>KTO Karatay University</sa:organization><sa:city>Konya</sa:city><sa:country>Turkey</sa:country></sa:affiliation><ce:source-text id="srct0355">KTO Karatay University, Konya, Turkey</ce:source-text></ce:affiliation><ce:affiliation id="aff0720" affiliation-id="S0370269322007833-0bdd954451acb47f491c8c4996fa87da"><ce:label>72</ce:label><ce:textfn>Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France</ce:textfn><sa:affiliation><sa:organization>Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie</sa:organization><sa:city>Orsay</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0360">Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France</ce:source-text></ce:affiliation><ce:affiliation id="aff0730" affiliation-id="S0370269322007833-1487532e5bfe30325bc10c0583f7c38e"><ce:label>73</ce:label><ce:textfn>Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France</ce:textfn><sa:affiliation><sa:organization>Laboratoire de Physique Subatomique et de Cosmologie</sa:organization><sa:organization>Université Grenoble-Alpes</sa:organization><sa:organization>CNRS-IN2P3</sa:organization><sa:city>Grenoble</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0365">Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France</ce:source-text></ce:affiliation><ce:affiliation id="aff0740" affiliation-id="S0370269322007833-dd5512fbf2faf90b56635e0b411d44a2"><ce:label>74</ce:label><ce:textfn>Lawrence Berkeley National Laboratory, Berkeley, CA, United States</ce:textfn><sa:affiliation><sa:organization>Lawrence Berkeley National Laboratory</sa:organization><sa:city>Berkeley</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0370">Lawrence Berkeley National Laboratory, Berkeley, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0750" affiliation-id="S0370269322007833-ed03eefd58007822249c697d882deffc"><ce:label>75</ce:label><ce:textfn>Lund University Department of Physics, Division of Particle Physics, Lund, Sweden</ce:textfn><sa:affiliation><sa:organization>Lund University Department of Physics</sa:organization><sa:organization>Division of Particle Physics</sa:organization><sa:city>Lund</sa:city><sa:country>Sweden</sa:country></sa:affiliation><ce:source-text id="srct0375">Lund University Department of Physics, Division of Particle Physics, Lund, Sweden</ce:source-text></ce:affiliation><ce:affiliation id="aff0760" affiliation-id="S0370269322007833-5b47ca06644b7f88e2267e9accbe867b"><ce:label>76</ce:label><ce:textfn>Nagasaki Institute of Applied Science, Nagasaki, Japan</ce:textfn><sa:affiliation><sa:organization>Nagasaki Institute of Applied Science</sa:organization><sa:city>Nagasaki</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0380">Nagasaki Institute of Applied Science, Nagasaki, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0770" affiliation-id="S0370269322007833-61cd55f0b96e9831d6063318d9967d43"><ce:label>77</ce:label><ce:textfn>Nara Women's University (NWU), Nara, Japan</ce:textfn><sa:affiliation><sa:organization>Nara Women's University (NWU)</sa:organization><sa:city>Nara</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0385">Nara Women's University (NWU), Nara, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0780" affiliation-id="S0370269322007833-7f50d99eb340e408fbc323cbce0c8905"><ce:label>78</ce:label><ce:textfn>National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece</ce:textfn><sa:affiliation><sa:organization>National and Kapodistrian University of Athens</sa:organization><sa:organization>School of Science</sa:organization><sa:organization>Department of Physics</sa:organization><sa:city>Athens</sa:city><sa:country>Greece</sa:country></sa:affiliation><ce:source-text id="srct0390">National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece</ce:source-text></ce:affiliation><ce:affiliation id="aff0790" affiliation-id="S0370269322007833-58192de9f95a93cb8bf485b9a5647bf5"><ce:label>79</ce:label><ce:textfn>National Centre for Nuclear Research, Warsaw, Poland</ce:textfn><sa:affiliation><sa:organization>National Centre for Nuclear Research</sa:organization><sa:city>Warsaw</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0395">National Centre for Nuclear Research, Warsaw, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff0800" affiliation-id="S0370269322007833-5c73c93f2c9aba1b7fcfa30741ce17ca"><ce:label>80</ce:label><ce:textfn>National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India</ce:textfn><sa:affiliation><sa:organization>National Institute of Science Education and Research</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Jatni</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0400">National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0810" affiliation-id="S0370269322007833-1e229a9758219f877a7f903c2dde9c2b"><ce:label>81</ce:label><ce:textfn>National Nuclear Research Center, Baku, Azerbaijan</ce:textfn><sa:affiliation><sa:organization>National Nuclear Research Center</sa:organization><sa:city>Baku</sa:city><sa:country>Azerbaijan</sa:country></sa:affiliation><ce:source-text id="srct0405">National Nuclear Research Center, Baku, Azerbaijan</ce:source-text></ce:affiliation><ce:affiliation id="aff0820" affiliation-id="S0370269322007833-ab3fe404fa49046cc1d6ed593a4d5eb8"><ce:label>82</ce:label><ce:textfn>National Research and Innovation Agency - BRIN, Jakarta, Indonesia</ce:textfn><sa:affiliation><sa:organization>National Research and Innovation Agency - BRIN</sa:organization><sa:city>Jakarta</sa:city><sa:country>Indonesia</sa:country></sa:affiliation><ce:source-text id="srct0410">National Research and Innovation Agency - BRIN, Jakarta, Indonesia</ce:source-text></ce:affiliation><ce:affiliation id="aff0830" affiliation-id="S0370269322007833-3e7cb9222cae4e75df0d93d2ea96329f"><ce:label>83</ce:label><ce:textfn>Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark</ce:textfn><sa:affiliation><sa:organization>Niels Bohr Institute</sa:organization><sa:organization>University of Copenhagen</sa:organization><sa:city>Copenhagen</sa:city><sa:country>Denmark</sa:country></sa:affiliation><ce:source-text id="srct0415">Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark</ce:source-text></ce:affiliation><ce:affiliation id="aff0840" affiliation-id="S0370269322007833-11b144edfb6d992e108d17991cc7681b"><ce:label>84</ce:label><ce:textfn>Nikhef, National institute for subatomic physics, Amsterdam, Netherlands</ce:textfn><sa:affiliation><sa:organization>Nikhef, National institute for subatomic physics</sa:organization><sa:city>Amsterdam</sa:city><sa:country>Netherlands</sa:country></sa:affiliation><ce:source-text id="srct0420">Nikhef, National institute for subatomic physics, Amsterdam, Netherlands</ce:source-text></ce:affiliation><ce:affiliation id="aff0850" affiliation-id="S0370269322007833-7ffc71d9a245ae3927bb4d373ff478ed"><ce:label>85</ce:label><ce:textfn>Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Group</sa:organization><sa:organization>STFC Daresbury Laboratory</sa:organization><sa:city>Daresbury</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0425">Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff0860" affiliation-id="S0370269322007833-4ea73349ab4e79a7b6570dd0fd42cd13"><ce:label>86</ce:label><ce:textfn>Nuclear Physics Institute of the Czech Academy of Sciences, Husinec-Řež, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Institute of the Czech Academy of Sciences</sa:organization><sa:city>Husinec-Řež</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0430">Nuclear Physics Institute of the Czech Academy of Sciences, Husinec-Řež, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0870" affiliation-id="S0370269322007833-43c0b745ad096562d858417552718575"><ce:label>87</ce:label><ce:textfn>Oak Ridge National Laboratory, Oak Ridge, TN, United States</ce:textfn><sa:affiliation><sa:organization>Oak Ridge National Laboratory</sa:organization><sa:city>Oak Ridge</sa:city><sa:state>TN</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0435">Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0880" affiliation-id="S0370269322007833-e4e32a64c7e436b0f2a688687dedfe5e"><ce:label>88</ce:label><ce:textfn>Ohio State University, Columbus, OH, United States</ce:textfn><sa:affiliation><sa:organization>Ohio State University</sa:organization><sa:city>Columbus</sa:city><sa:state>OH</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0440">Ohio State University, Columbus, Ohio, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0890" affiliation-id="S0370269322007833-3c4eb171ede7112c64f1bc2c164f4732"><ce:label>89</ce:label><ce:textfn>Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia</ce:textfn><sa:affiliation><sa:organization>Physics department</sa:organization><sa:organization>Faculty of science, University of Zagreb</sa:organization><sa:city>Zagreb</sa:city><sa:country>Croatia</sa:country></sa:affiliation><ce:source-text id="srct0445">Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia</ce:source-text></ce:affiliation><ce:affiliation id="aff0900" affiliation-id="S0370269322007833-30d6dbb96747c914840798f88bf250cb"><ce:label>90</ce:label><ce:textfn>Physics Department, Panjab University, Chandigarh, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>Panjab University</sa:organization><sa:city>Chandigarh</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0450">Physics Department, Panjab University, Chandigarh, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0910" affiliation-id="S0370269322007833-712bc80495e97050498604da603cb5cc"><ce:label>91</ce:label><ce:textfn>Physics Department, University of Jammu, Jammu, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>University of Jammu</sa:organization><sa:city>Jammu</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0455">Physics Department, University of Jammu, Jammu, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0920" affiliation-id="S0370269322007833-500771749243cadcbaeeac3f726e6a62"><ce:label>92</ce:label><ce:textfn>Physics Department, University of Rajasthan, Jaipur, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>University of Rajasthan</sa:organization><sa:city>Jaipur</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0460">Physics Department, University of Rajasthan, Jaipur, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0930" affiliation-id="S0370269322007833-87f0c1fe6b138b8300b7695dc3391666"><ce:label>93</ce:label><ce:textfn>Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan</ce:textfn><sa:affiliation><sa:organization>Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2)</sa:organization><sa:organization>Hiroshima University</sa:organization><sa:city>Hiroshima</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0465">Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0940" affiliation-id="S0370269322007833-0886f70da565231fd0e43398021fd604"><ce:label>94</ce:label><ce:textfn>Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany</ce:textfn><sa:affiliation><sa:organization>Physikalisches Institut</sa:organization><sa:organization>Eberhard-Karls-Universität Tübingen</sa:organization><sa:city>Tübingen</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0470">Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0950" affiliation-id="S0370269322007833-1c67678e124982de924355cdd6d6b91b"><ce:label>95</ce:label><ce:textfn>Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany</ce:textfn><sa:affiliation><sa:organization>Physikalisches Institut</sa:organization><sa:organization>Ruprecht-Karls-Universität Heidelberg</sa:organization><sa:city>Heidelberg</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0475">Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0960" affiliation-id="S0370269322007833-7a5765cc3928149c8b77128cd52462ae"><ce:label>96</ce:label><ce:textfn>Physik Department, Technische Universität München, Munich, Germany</ce:textfn><sa:affiliation><sa:organization>Physik Department</sa:organization><sa:organization>Technische Universität München</sa:organization><sa:city>Munich</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0480">Physik Department, Technische Universität München, Munich, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0970" affiliation-id="S0370269322007833-0a06f4f5fb60a25c8f1936ed71c96cfe"><ce:label>97</ce:label><ce:textfn>Politecnico di Bari and Sezione INFN, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>Politecnico di Bari</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0485">Politecnico di Bari and Sezione INFN, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0980" affiliation-id="S0370269322007833-3451d358e90b759225813c9b9a6f098c"><ce:label>98</ce:label><ce:textfn>Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany</ce:textfn><sa:affiliation><sa:organization>Research Division</sa:organization><sa:organization>ExtreMe Matter Institute EMMI</sa:organization><sa:organization>GSI Helmholtzzentrum für Schwerionenforschung GmbH</sa:organization><sa:city>Darmstadt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0490">Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0990" affiliation-id="S0370269322007833-2771a2ae295804be11a4a937b894a546"><ce:label>99</ce:label><ce:textfn>Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Saha Institute of Nuclear Physics</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0495">Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1000" affiliation-id="S0370269322007833-b602cde70213041ca40ec5720e4e3e75"><ce:label>100</ce:label><ce:textfn>School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom</ce:textfn><sa:affiliation><sa:organization>School of Physics and Astronomy</sa:organization><sa:organization>University of Birmingham</sa:organization><sa:city>Birmingham</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0500">School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff1010" affiliation-id="S0370269322007833-1a1f0f3ae33ba0323ad4da08a216f4c3"><ce:label>101</ce:label><ce:textfn>Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru</ce:textfn><sa:affiliation><sa:organization>Sección Física</sa:organization><sa:organization>Departamento de Ciencias</sa:organization><sa:organization>Pontificia Universidad Católica del Perú</sa:organization><sa:city>Lima</sa:city><sa:country>Peru</sa:country></sa:affiliation><ce:source-text id="srct0505">Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru</ce:source-text></ce:affiliation><ce:affiliation id="aff1020" affiliation-id="S0370269322007833-24347400090c9a1efb87e7537af7461b"><ce:label>102</ce:label><ce:textfn>Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria</ce:textfn><sa:affiliation><sa:organization>Stefan Meyer Institut für Subatomare Physik (SMI)</sa:organization><sa:city>Vienna</sa:city><sa:country>Austria</sa:country></sa:affiliation><ce:source-text id="srct0510">Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria</ce:source-text></ce:affiliation><ce:affiliation id="aff1030" affiliation-id="S0370269322007833-317714469d8c208c77efd6f026942633"><ce:label>103</ce:label><ce:textfn>SUBATECH, IMT Atlantique, Nantes Université, CNRS-IN2P3, Nantes, France</ce:textfn><sa:affiliation><sa:organization>SUBATECH</sa:organization><sa:organization>IMT Atlantique</sa:organization><sa:organization>Nantes Université</sa:organization><sa:organization>CNRS-IN2P3</sa:organization><sa:city>Nantes</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0515">SUBATECH, IMT Atlantique, Nantes Université, CNRS-IN2P3, Nantes, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1040" affiliation-id="S0370269322007833-c3972f6af24eef12cc2ae3a53d6a7623"><ce:label>104</ce:label><ce:textfn>Suranaree University of Technology, Nakhon Ratchasima, Thailand</ce:textfn><sa:affiliation><sa:organization>Suranaree University of Technology</sa:organization><sa:city>Nakhon Ratchasima</sa:city><sa:country>Thailand</sa:country></sa:affiliation><ce:source-text id="srct0520">Suranaree University of Technology, Nakhon Ratchasima, Thailand</ce:source-text></ce:affiliation><ce:affiliation id="aff1050" affiliation-id="S0370269322007833-9186de958ef0b51277f8f34872e5c71b"><ce:label>105</ce:label><ce:textfn>Technical University of Košice, Košice, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Technical University of Košice</sa:organization><sa:city>Košice</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0525">Technical University of Košice, Košice, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff1060" affiliation-id="S0370269322007833-425836ada61fe0d3ee4ebf5b87598919"><ce:label>106</ce:label><ce:textfn>The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland</ce:textfn><sa:affiliation><sa:organization>The Henryk Niewodniczanski Institute of Nuclear Physics</sa:organization><sa:organization>Polish Academy of Sciences</sa:organization><sa:city>Cracow</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0530">The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff1070" affiliation-id="S0370269322007833-302a66584ead3636c3e74b86b51cf474"><ce:label>107</ce:label><ce:textfn>The University of Texas at Austin, Austin, TX, United States</ce:textfn><sa:affiliation><sa:organization>The University of Texas at Austin</sa:organization><sa:city>Austin</sa:city><sa:state>TX</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0535">The University of Texas at Austin, Austin, Texas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1080" affiliation-id="S0370269322007833-48ea3700cc067946e6c3032832f0ce0a"><ce:label>108</ce:label><ce:textfn>Universidad Autónoma de Sinaloa, Culiacán, Mexico</ce:textfn><sa:affiliation><sa:organization>Universidad Autónoma de Sinaloa</sa:organization><sa:city>Culiacán</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0540">Universidad Autónoma de Sinaloa, Culiacán, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff1090" affiliation-id="S0370269322007833-310945a13cafb9845fe651fc7aa661b6"><ce:label>109</ce:label><ce:textfn>Universidade de São Paulo (USP), São Paulo, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade de São Paulo (USP)</sa:organization><sa:city>São Paulo</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0545">Universidade de São Paulo (USP), São Paulo, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1100" affiliation-id="S0370269322007833-226f6a475b7a1634e1530e2077d7590e"><ce:label>110</ce:label><ce:textfn>Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade Estadual de Campinas (UNICAMP)</sa:organization><sa:city>Campinas</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0550">Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1110" affiliation-id="S0370269322007833-7582533209ee7a368c4d249782e93c03"><ce:label>111</ce:label><ce:textfn>Universidade Federal do ABC, Santo Andre, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade Federal do ABC</sa:organization><sa:city>Santo Andre</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0555">Universidade Federal do ABC, Santo Andre, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1120" affiliation-id="S0370269322007833-f37f0b36132124cce5539b1f9c518079"><ce:label>112</ce:label><ce:textfn>University of Cape Town, Cape Town, South Africa</ce:textfn><sa:affiliation><sa:organization>University of Cape Town</sa:organization><sa:city>Cape Town</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0560">University of Cape Town, Cape Town, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff1130" affiliation-id="S0370269322007833-58fe35a5cc9d91eea34fd6e19293599b"><ce:label>113</ce:label><ce:textfn>University of Houston, Houston, TX, United States</ce:textfn><sa:affiliation><sa:organization>University of Houston</sa:organization><sa:city>Houston</sa:city><sa:state>TX</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0565">University of Houston, Houston, Texas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1140" affiliation-id="S0370269322007833-e9be2b1885c8285b7cb7b6534ca93d98"><ce:label>114</ce:label><ce:textfn>University of Jyväskylä, Jyväskylä, Finland</ce:textfn><sa:affiliation><sa:organization>University of Jyväskylä</sa:organization><sa:city>Jyväskylä</sa:city><sa:country>Finland</sa:country></sa:affiliation><ce:source-text id="srct0570">University of Jyväskylä, Jyväskylä, Finland</ce:source-text></ce:affiliation><ce:affiliation id="aff1150" affiliation-id="S0370269322007833-6e2074f6fd0b88987501b22e5d74f9c9"><ce:label>115</ce:label><ce:textfn>University of Kansas, Lawrence, KS, United States</ce:textfn><sa:affiliation><sa:organization>University of Kansas</sa:organization><sa:city>Lawrence</sa:city><sa:state>KS</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0575">University of Kansas, Lawrence, Kansas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1160" affiliation-id="S0370269322007833-f538fc5a59ec1d1c8f86d2a83cb4ced6"><ce:label>116</ce:label><ce:textfn>University of Liverpool, Liverpool, United Kingdom</ce:textfn><sa:affiliation><sa:organization>University of Liverpool</sa:organization><sa:city>Liverpool</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0580">University of Liverpool, Liverpool, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff1170" affiliation-id="S0370269322007833-5507e3344bf7b46ba698c2c045a8aba6"><ce:label>117</ce:label><ce:textfn>University of Science and Technology of China, Hefei, China</ce:textfn><sa:affiliation><sa:organization>University of Science and Technology of China</sa:organization><sa:city>Hefei</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0585">University of Science and Technology of China, Hefei, China</ce:source-text></ce:affiliation><ce:affiliation id="aff1180" affiliation-id="S0370269322007833-5825f1740b6b974441dd90d9c8ad0a53"><ce:label>118</ce:label><ce:textfn>University of South-Eastern Norway, Kongsberg, Norway</ce:textfn><sa:affiliation><sa:organization>University of South-Eastern Norway</sa:organization><sa:city>Kongsberg</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0590">University of South-Eastern Norway, Kongsberg, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff1190" affiliation-id="S0370269322007833-d6620449e5365c80224ff22fe1ca4e4a"><ce:label>119</ce:label><ce:textfn>University of Tennessee, Knoxville, TN, United States</ce:textfn><sa:affiliation><sa:organization>University of Tennessee</sa:organization><sa:city>Knoxville</sa:city><sa:state>TN</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0595">University of Tennessee, Knoxville, Tennessee, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1200" affiliation-id="S0370269322007833-6d43022152ccaa62119e4287bf3b5270"><ce:label>120</ce:label><ce:textfn>University of the Witwatersrand, Johannesburg, South Africa</ce:textfn><sa:affiliation><sa:organization>University of the Witwatersrand</sa:organization><sa:city>Johannesburg</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0600">University of the Witwatersrand, Johannesburg, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff1210" affiliation-id="S0370269322007833-9b9071fc23073da0e4317b3863c91b9e"><ce:label>121</ce:label><ce:textfn>University of Tokyo, Tokyo, Japan</ce:textfn><sa:affiliation><sa:organization>University of Tokyo</sa:organization><sa:city>Tokyo</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0605">University of Tokyo, Tokyo, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff1220" affiliation-id="S0370269322007833-da3fb88e8779358429638ee8f8dc4cb5"><ce:label>122</ce:label><ce:textfn>University of Tsukuba, Tsukuba, Japan</ce:textfn><sa:affiliation><sa:organization>University of Tsukuba</sa:organization><sa:city>Tsukuba</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0610">University of Tsukuba, Tsukuba, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff1230" affiliation-id="S0370269322007833-9fc1715eadb7b2ba79b1a87cef8015cd"><ce:label>123</ce:label><ce:textfn>University Politehnica of Bucharest, Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>University Politehnica of Bucharest</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0615">University Politehnica of Bucharest, Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff1240" affiliation-id="S0370269322007833-43e2ac82e11c4ca2fabbc7e457c74b37"><ce:label>124</ce:label><ce:textfn>Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France</ce:textfn><sa:affiliation><sa:organization>Université Clermont Auvergne</sa:organization><sa:organization>CNRS/IN2P3</sa:organization><sa:organization>LPC</sa:organization><sa:city>Clermont-Ferrand</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0620">Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1250" affiliation-id="S0370269322007833-08f13150d05b32d440951efec6f80d29"><ce:label>125</ce:label><ce:textfn>Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France</ce:textfn><sa:affiliation><sa:organization>Université de Lyon</sa:organization><sa:organization>CNRS/IN2P3</sa:organization><sa:organization>Institut de Physique des 2 Infinis de Lyon</sa:organization><sa:city>Lyon</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0625">Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1260" affiliation-id="S0370269322007833-766d0075165899c16752f5f7ff1b8771"><ce:label>126</ce:label><ce:textfn>Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France</ce:textfn><sa:affiliation><sa:organization>Université de Strasbourg</sa:organization><sa:organization>CNRS</sa:organization><sa:organization>IPHC UMR 7178</sa:organization><sa:city>Strasbourg</sa:city><sa:postal-code>F-67000</sa:postal-code><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0630">Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1270" affiliation-id="S0370269322007833-36117066550bb6faaf9708b1b0ea670e"><ce:label>127</ce:label><ce:textfn>Université Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France</ce:textfn><sa:affiliation><sa:organization>Université Paris-Saclay Centre d'Etudes de Saclay (CEA)</sa:organization><sa:organization>IRFU</sa:organization><sa:organization>Départment de Physique Nucléaire (DPhN)</sa:organization><sa:city>Saclay</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0635">Université Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1280" affiliation-id="S0370269322007833-47ef552cdaf90d05d791574b14a8dcf9"><ce:label>128</ce:label><ce:textfn>Università degli Studi di Foggia, Foggia, Italy</ce:textfn><sa:affiliation><sa:organization>Università degli Studi di Foggia</sa:organization><sa:city>Foggia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0640">Università degli Studi di Foggia, Foggia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1290" affiliation-id="S0370269322007833-79fbcbe2a42e9099094360b5bf7bd602"><ce:label>129</ce:label><ce:textfn>Università del Piemonte Orientale, Vercelli, Italy</ce:textfn><sa:affiliation><sa:organization>Università del Piemonte Orientale</sa:organization><sa:city>Vercelli</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0645">Università del Piemonte Orientale, Vercelli, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1300" affiliation-id="S0370269322007833-83cd0c0929f483f88e3d2cb59f064287"><ce:label>130</ce:label><ce:textfn>Università di Brescia, Brescia, Italy</ce:textfn><sa:affiliation><sa:organization>Università di Brescia</sa:organization><sa:city>Brescia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0650">Università di Brescia, Brescia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1310" affiliation-id="S0370269322007833-f1ae52f852d4d7d99988b3e872f887e4"><ce:label>131</ce:label><ce:textfn>Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Variable Energy Cyclotron Centre</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0655">Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1320" affiliation-id="S0370269322007833-bfd4fe0d3b8675ea55d96ce64033d817"><ce:label>132</ce:label><ce:textfn>Warsaw University of Technology, Warsaw, Poland</ce:textfn><sa:affiliation><sa:organization>Warsaw University of Technology</sa:organization><sa:city>Warsaw</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0660">Warsaw University of Technology, Warsaw, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff1330" affiliation-id="S0370269322007833-4e11d38f810a3540206ffb5151a35d3b"><ce:label>133</ce:label><ce:textfn>Wayne State University, Detroit, MI, United States</ce:textfn><sa:affiliation><sa:organization>Wayne State University</sa:organization><sa:city>Detroit</sa:city><sa:state>MI</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0665">Wayne State University, Detroit, Michigan, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1340" affiliation-id="S0370269322007833-0b76517a6de7ff0579ec14780f79d81b"><ce:label>134</ce:label><ce:textfn>Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany</ce:textfn><sa:affiliation><sa:organization>Westfälische Wilhelms-Universität Münster</sa:organization><sa:organization>Institut für Kernphysik</sa:organization><sa:city>Münster</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0670">Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1350" affiliation-id="S0370269322007833-6b1ae45228f1f040c55767dc107b589d"><ce:label>135</ce:label><ce:textfn>Wigner Research Centre for Physics, Budapest, Hungary</ce:textfn><sa:affiliation><sa:organization>Wigner Research Centre for Physics</sa:organization><sa:city>Budapest</sa:city><sa:country>Hungary</sa:country></sa:affiliation><ce:source-text id="srct0675">Wigner Research Centre for Physics, Budapest, Hungary</ce:source-text></ce:affiliation><ce:affiliation id="aff1360" affiliation-id="S0370269322007833-96e79e7381eab77664732c3553e247e8"><ce:label>136</ce:label><ce:textfn>Yale University, New Haven, CT, United States</ce:textfn><sa:affiliation><sa:organization>Yale University</sa:organization><sa:city>New Haven</sa:city><sa:state>CT</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0680">Yale University, New Haven, Connecticut, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1370" affiliation-id="S0370269322007833-f3db17ceaf6ea4190e2176a6a129c96b"><ce:label>137</ce:label><ce:textfn>Yonsei University, Seoul, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Yonsei University</sa:organization><sa:city>Seoul</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0685">Yonsei University, Seoul, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff1380" affiliation-id="S0370269322007833-2e935afd39a812eb118b95cc53ef2ac3"><ce:label>138</ce:label><ce:textfn>Zentrum für Technologie und Transfer (ZTT), Worms, Germany</ce:textfn><sa:affiliation><sa:organization>Zentrum für Technologie und Transfer (ZTT)</sa:organization><sa:city>Worms</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0690">Zentrum für Technologie und Transfer (ZTT), Worms, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1390" affiliation-id="S0370269322007833-7972cea8d4a6e5b7150baf4a74e03970"><ce:label>139</ce:label><ce:textfn>Affiliated with an institute covered by a cooperation agreement with CERN</ce:textfn><sa:affiliation><sa:address-line>Affiliated with an institute covered by a cooperation agreement with CERN</sa:address-line></sa:affiliation><ce:source-text id="srct0695">Affiliated with an institute covered by a cooperation agreement with CERN</ce:source-text></ce:affiliation><ce:affiliation id="aff1400" affiliation-id="S0370269322007833-3c2eaa2a494293a99583fccbf5d95e9f"><ce:label>140</ce:label><ce:textfn>Affiliated with an international laboratory covered by a cooperation agreement with CERN</ce:textfn><sa:affiliation><sa:address-line>Affiliated with an international laboratory covered by a cooperation agreement with CERN</sa:address-line></sa:affiliation><ce:source-text id="srct0700">Affiliated with an international laboratory covered by a cooperation agreement with CERN</ce:source-text></ce:affiliation><ce:footnote id="fn0010"><ce:label>I</ce:label><ce:note-para id="np0010">Deceased.</ce:note-para></ce:footnote><ce:footnote id="fn0020"><ce:label>II</ce:label><ce:note-para id="np0020">Also at: Max-Planck-Institut für Physik, Munich, Germany.</ce:note-para></ce:footnote><ce:footnote id="fn0030"><ce:label>III</ce:label><ce:note-para id="np0030">Also at: Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA), Bologna, Italy.</ce:note-para></ce:footnote><ce:footnote id="fn0040"><ce:label>IV</ce:label><ce:note-para id="np0040">Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy.</ce:note-para></ce:footnote><ce:footnote id="fn0050"><ce:label>V</ce:label><ce:note-para id="np0050">Also at: Department of Applied Physics, Aligarh Muslim University, Aligarh, India.</ce:note-para></ce:footnote><ce:footnote id="fn0060"><ce:label>VI</ce:label><ce:note-para id="np0060">Also at: Institute of Theoretical Physics, University of Wroclaw, Poland.</ce:note-para></ce:footnote><ce:footnote id="fn0070"><ce:label>VII</ce:label><ce:note-para id="np0070">Also at: An institution covered by a cooperation agreement with CERN.</ce:note-para></ce:footnote></ce:author-group></ce:collaboration><ce:footnote id="fn0080"><ce:label>⋆</ce:label><ce:note-para id="np0080"><ce:italic>E-mail address:</ce:italic> <ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/text/html" xlink:href="mailto:alice-publications@cern.ch" id="inf0020">alice-publications@cern.ch</ce:inter-ref>.</ce:note-para></ce:footnote></ce:author-group><ce:date-received day="5" month="5" year="2022"/><ce:date-revised day="2" month="12" year="2022"/><ce:date-accepted day="23" month="12" year="2022"/><ce:miscellaneous id="ms0010">Editor: M. Pierini</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0070">This letter reports measurements which characterize the underlying event associated with hard scatterings at mid-pseudorapidity (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math>) in pp, p–Pb and Pb–Pb collisions at centre-of-mass energy per nucleon pair, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The measurements are performed with ALICE at the LHC. Different multiplicity classes are defined based on the event activity measured at forward rapidities. The hard scatterings are identified by the leading particle defined as the charged particle with the largest transverse momentum (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) in the collision and having 8 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mo linebreak="badbreak" linebreakstyle="after"><</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra of associated particles (0.5 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) are measured in different azimuthal regions defined with respect to the leading particle direction: toward, transverse, and away. The associated charged particle yields in the transverse region are subtracted from those of the away and toward regions. The remaining jet-like yields are reported as a function of the multiplicity measured in the transverse region. The measurements show a suppression of the jet-like yield in the away region and an enhancement of high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> associated particles in the toward region in central Pb–Pb collisions, as compared to minimum-bias pp collisions. These observations are consistent with previous measurements that used two-particle correlations, and with an interpretation in terms of parton energy loss in a high-density quark gluon plasma. These yield modifications vanish in peripheral Pb–Pb collisions and are not observed in either high-multiplicity pp or p–Pb collisions.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0080">Data availability</ce:section-title><ce:para id="pr0200">This manuscript has associated data in a HEPData repository at <ce:inter-ref xlink:href="https://www.hepdata.net/" xlink:role="http://www.elsevier.com/xml/linking-roles/research-data" id="inf0550">https://www.hepdata.net/</ce:inter-ref>.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">In proton-proton (pp) collisions, jets, originating from partonic scatterings with large momentum transfer, are accompanied by particles produced by initial- and final-state radiation (ISR and FSR, respectively), as well as, by a plethora of other mechanisms. These include proton break-up, and, in a scenario incorporating multi-parton interactions (MPI) <ce:cross-refs refid="br0010 br0020" id="crs0010">[1,2]</ce:cross-refs>, several semi-hard parton-parton scatterings in a single pp collision. These jet-accompanying particles experimentally make up the underlying event (UE) and are commonly studied via azimuthal separations from the jets to minimise the influence of hard scatterings. The present study follows the strategy originally introduced by the CDF collaboration <ce:cross-ref refid="br0030" id="crf10910">[3]</ce:cross-ref>. First, the leading charged particle in the event is found, i.e., the charged particle with the highest transverse momentum in the collision (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math>). Secondly, the associated particles (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math>) are measured in three topological regions depending on their azimuthal angle relative to the leading particle, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>φ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msup><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">assoc</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">|</mml:mo></mml:math>, see <ce:cross-ref refid="fg0010" id="crf10920">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/>.</ce:para><ce:para id="pr0020">The toward region contains the primary jet within the acceptance of the detector, while the away region contains the back-scattered particles of the recoil jet <ce:cross-ref refid="br0040" id="crf10930">[4]</ce:cross-ref>. In contrast, the transverse region is dominated by the underlying-event dynamics, but it also includes contributions from ISR and FSR <ce:cross-ref refid="br0050" id="crf10940">[5]</ce:cross-ref>.</ce:para><ce:para id="pr0030">The measurements performed at RHIC and LHC in small systems (pp, p–A, and d–A collisions) have shown for high particle multiplicities similar phenomena as were originally observed only in A–A collisions and have been attributed there to the formation of the strongly interacting quark gluon plasma <ce:cross-refs refid="br0060 br0070" id="crs0020">[6,7]</ce:cross-refs>, namely, long range angular correlations and collectivity <ce:cross-ref refid="br0080" id="crf10950">[8]</ce:cross-ref>. The origin of these effects in small systems is still an open question; on one hand, hydrodynamical calculations describe some aspects of the data <ce:cross-ref refid="br0090" id="crf10960">[9]</ce:cross-ref>; on the other hand, mechanisms like colour reconnection <ce:cross-ref refid="br0100" id="crf10970">[10]</ce:cross-ref>, rope hadronisation <ce:cross-ref refid="br0110" id="crf10980">[11]</ce:cross-ref>, and string shoving <ce:cross-ref refid="br0120" id="crf10990">[12]</ce:cross-ref> can produce collective-like effects in Monte Carlo event generators such as <ce:small-caps>PYTHIA</ce:small-caps> 8 <ce:cross-ref refid="br0130" id="crf11000">[13]</ce:cross-ref>. Thus, investigating pp collisions as a function of the charged particle multiplicity has become ever more pertinent <ce:cross-refs refid="br0090 br0140 br0150 br0160 br0170 br0180" id="crs0030">[9,14–18]</ce:cross-refs>. The interpretation of the results from the analysis of high-multiplicity pp collisions is challenging due to the selection biases of the sample towards events in which partonic scatterings with large momentum transfer (hard scatterings) occurred. To mitigate this inherent bias, Martin et al. <ce:cross-ref refid="br0190" id="crf11010">[19]</ce:cross-ref> suggested to use the charged-particle multiplicity in the transverse region (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>) as a classifier of the activity in the collisions, since the correlation between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math> and the hardest scattering in the collision is small. The ALICE collaboration has reported the first <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math> spectra measured in pp collisions at centre-of-mass energy, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:msqrt><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>13</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV <ce:cross-ref refid="br0200" id="crf11020">[20]</ce:cross-ref>. Event generators, such as <ce:small-caps>PYTHIA</ce:small-caps> 8 <ce:cross-ref refid="br0130" id="crf11030">[13]</ce:cross-ref> and EPOS-LHC <ce:cross-ref refid="br0210" id="crf11040">[21]</ce:cross-ref>, do not provide a good description of the measured distribution of the ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>/<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> is the event-averaged charged-particle multiplicity in the transverse region, underestimating in particular the number of collisions with large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:math>. In the framework of MPI-based models, like those implemented in <ce:small-caps>PYTHIA</ce:small-caps> 8 and <ce:small-caps>HERWIG</ce:small-caps> 7 <ce:cross-ref refid="br0220" id="crf11050">[22]</ce:cross-ref>, the probability for a hard scattering in the collision increases with decreasing impact parameter<ce:cross-ref refid="fn0090" id="crf11060"><ce:sup>VIII</ce:sup></ce:cross-ref><ce:footnote id="fn0090"><ce:label>VIII</ce:label><ce:note-para id="np0090">In event generators like <ce:small-caps>PYTHIA</ce:small-caps> 8 the impact parameter profile is described by an overlap matter distribution of the two incoming hadrons.</ce:note-para></ce:footnote> between the colliding protons. Thus, requiring a high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> particle (e.g., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>8</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) in a given pp collision biases the selection of collisions towards those with a smaller impact parameter <ce:cross-ref refid="br0230" id="crf11070">[23]</ce:cross-ref>, which in turn biases the selection towards pp collisions with more MPI <ce:cross-ref refid="br0200" id="crf11080">[20]</ce:cross-ref>. This feature of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>-based analysis is important for the isolation of potential MPI and colour reconnection effects, which according to <ce:small-caps>PYTHIA</ce:small-caps> 8, produce effects resembling collective behaviour <ce:cross-ref refid="br0100" id="crf11090">[10]</ce:cross-ref>. By construction, MPI and colour reconnection effects are expected to be more relevant in the transverse region than in the away and toward regions <ce:cross-ref refid="br0240" id="crf11100">[24]</ce:cross-ref>. It is worth mentioning that the MPI picture has been used to explain the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in p–Pb collisions and peripheral Pb–Pb collisions <ce:cross-refs refid="br0250 br0260 br0270" id="crs0040">[25–27]</ce:cross-refs>. Studies, as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>, are therefore important to the understanding of the effects observed in high-multiplicity pp collisions. Last but not least, measurements of UE observables are also important to tune event generators <ce:cross-ref refid="br0280" id="crf11110">[28]</ce:cross-ref> that include hard partonic scatterings and MPI.</ce:para><ce:para id="pr0040">This letter reports the inclusive charged-particle transverse momentum spectra in pp, p–Pb and Pb–Pb collisions at centre-of-mass energy per nucleon pair <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV containing a high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> leading particle within the kinematic intervals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>8</mml:mn><mml:mo>≤</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math>. This guarantees the selection of collisions in which the average activity in the transverse region is roughly flat as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math> <ce:cross-ref refid="br0200" id="crf11120">[20]</ce:cross-ref>, and therefore, any additional selection on the charged particle multiplicity will only modulate the UE activity. The measurements are performed considering different event classes defined in terms of the multiplicity registered in the forward detectors. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra of associated charged particles (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mn>0.5</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math>) are measured in the toward, away, and transverse regions as a function of the average charged particle multiplicity in the transverse region. To further investigate the possible modification of the particles produced in the hard scattering in pp, p–Pb, and Pb–Pb collisions, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> distributions in the toward (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) and away (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) regions obtained after the subtraction of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in the transverse region (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) are also reported. The subtracted yields (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) are further normalised to those measured in minimum-bias (MB) pp collisions,<ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pp</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">MB</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pp</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">MB</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>X</ce:italic> indicates the collision system and the event multiplicity class. In this way, the hard process <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in the toward and away regions are isolated, and thus allowing us to study possible modifications to the produced particles due to medium effects in high-multiplicity pp, p–Pb, and Pb–Pb collisions. In heavy-ion collisions, this ratio is sensitive to the same effects which were studied using the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math> quantity <ce:cross-refs refid="br0290 br0300 br0310" id="crs0050">[29–31]</ce:cross-refs>, where jets produced in the early stage of the collision propagate through the hot and dense quark–gluon plasma. Their interaction with the coloured medium lead to parton-energy loss (jet quenching) <ce:cross-ref refid="br0320" id="crf11130">[32]</ce:cross-ref> which, for example, results in the suppression of the charged-particle yield at high <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> <ce:cross-ref refid="br0330" id="crf11140">[33]</ce:cross-ref>, and the suppression of the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> yield in the away region <ce:cross-refs refid="br0290 br0300" id="crs0060">[29,30]</ce:cross-refs>. It is worth mentioning that jet quenching effects have not been observed so far in small systems <ce:cross-refs refid="br0330 br0340" id="crs0070">[33,34]</ce:cross-refs>.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">Experiment and data analysis</ce:section-title><ce:para id="pr0050">This analysis is based on the data recorded by the ALICE apparatus during the pp and Pb–Pb runs at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV in 2015, and the p–Pb run at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV in 2016. The present study uses the V0 detector, and the Silicon Pixel Detector (SPD) for triggering and background rejection. The V0 consists of two arrays of scintillating tiles placed on each side of the interaction point covering the full azimuthal acceptance and the pseudorapidity intervals of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:mn>2.8</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5.1</mml:mn></mml:math> (V0A) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.7</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1.7</mml:mn></mml:math> (V0C). The SPD is the innermost part of the Inner Tracking System (ITS) and it is the closest detector to the interaction point. It consists of two cylindrical silicon pixel layers at radial distances of 3.9 and 7.6 cm from the beam line and the pseudorapidity coverages of the two layers are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1.4</mml:mn></mml:math>, respectively. The data were collected using a minimum-bias trigger, which required a signal in both V0A and V0C detectors. The offline event selection was optimised to reject beam-induced background in all collision systems by utilising the timing signals in the two V0 detectors. In Pb–Pb collisions, the beam-induced background is further suppressed by correlating the timing signals of the neutron zero degree calorimeters, which are positioned on both sides of the interaction point at 112.5<ce:hsp sp="0.20"/>m distance along the beam axis <ce:cross-ref refid="br0350" id="crf11150">[35]</ce:cross-ref>. The signals from the zero degree calorimeters are also used to suppress the contamination from electromagnetic interactions. This is performed by requesting the coincidence of the signals coming from both side zero degree calorimeters by which the background due to single nucleus electromagnetic dissociation processes is excluded. A criterion based on the offline reconstruction of multiple primary vertices in the SPD is applied to reduce the pileup caused by multiple interactions in the same bunch crossing <ce:cross-ref refid="br0360" id="crf11160">[36]</ce:cross-ref>. The results presented in this letter are for minimum-bias triggered pp collisions having at least one charged particle in the pseudorapidity interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1</mml:mn></mml:math> (INEL>0). The INEL>0 event class corresponds to about 75% of the total inelastic cross section <ce:cross-ref refid="br0370" id="crf11170">[37]</ce:cross-ref>. For pp and Pb–Pb collisions, the sample is subdivided into different multiplicity classes based on the total charge deposited in both V0 sub-detectors, which is termed as V0M amplitude <ce:cross-ref refid="br0380" id="crf11180">[38]</ce:cross-ref>. For p–Pb collisions, the sample is subdivided based on the total charge deposited in V0A sub-detector (V0A amplitude) <ce:cross-ref refid="br0390" id="crf11190">[39]</ce:cross-ref>, which is located in the Pb-going direction. The V0A estimator has been implemented in previous measurements that used p–Pb data (see e.g. <ce:cross-ref refid="br0400" id="crf11200">[40]</ce:cross-ref>). This allows for comparisons with other observables for similar V0A multiplicity classes. To ensure that a hard scattering took place in the collision, events are required to have a trigger particle within <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>8</mml:mn><mml:mo>≤</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math> GeV/<ce:italic>c</ce:italic>. In this <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math> interval, the momentum resolution effects are negligible on the extracted yields, and therefore, no <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math> resolution correction is applied. The total number of analysed collisions before the trigger particle selection are about 10<ce:sup>8</ce:sup>, 10<ce:sup>8</ce:sup>, and 10<ce:sup>7</ce:sup> for pp, p–Pb, and Pb–Pb collisions, respectively.</ce:para><ce:para id="pr0060">The transverse momentum of particles is determined from measurements in the central barrel with the ITS and the Time Projection Chamber (TPC). The ITS is a tracking detector which consists of six cylindrical layers of silicon detectors. The TPC is a cylindrical drift detector which covers a radial distance of 85-247<ce:hsp sp="0.20"/>cm from the beam axis and it has longitudinal dimension extending from about -250<ce:hsp sp="0.20"/>cm to +250<ce:hsp sp="0.20"/>cm around the nominal interaction point. Primary charged particles are measured in the pseudorapidity range of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math> and with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0.5</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, where <ce:italic>η</ce:italic> is measured in the laboratory frame for the three collision systems. The configuration for p–Pb collisions with protons at 4<ce:hsp sp="0.20"/>TeV energy colliding with Pb ions that have per-nucleon energies of (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mi>Z</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>A</mml:mi></mml:math>) × 4<ce:hsp sp="0.20"/>TeV ∼ 1.58<ce:hsp sp="0.20"/>TeV results in a shift in the rapidity of the nucleon–nucleon centre-of-mass system by 0.465 in the direction of the proton beam (negative z-direction). Here <ce:italic>Z</ce:italic> and <ce:italic>A</ce:italic> are the atomic and mass numbers of the Pb ion, respectively. Therefore, the detector coverage <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math> corresponds to roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>0.3</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cms</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1.3</mml:mn></mml:math> for p–Pb collisions. The particles with mean proper lifetime larger than 1<ce:hsp sp="0.20"/>cm/<ce:italic>c</ce:italic>, which are either produced directly in the interaction or from decays of particles with mean proper lifetime smaller than 1<ce:hsp sp="0.20"/>cm/<ce:italic>c</ce:italic> are termed as primary particles <ce:cross-ref refid="br0410" id="crf11210">[41]</ce:cross-ref>. The track selection follows a procedure similar to the one described in Ref. <ce:cross-ref refid="br0420" id="crf11220">[42]</ce:cross-ref> and only few specific details are reported here. Tracks (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">tracks</mml:mi></mml:mrow></mml:msub></mml:math>) are required to have two hits in the ITS, out of which at least one should be in either of the two innermost layers. The geometrical track length <ce:italic>L</ce:italic> is calculated in the TPC readout plane, excluding the information from the pads at the sector boundaries (≈3<ce:hsp sp="0.20"/>cm from the sector edges). The trajectory lengths built from radial segments, i.e. the crossed TPC pad rows, traversed in the TPC by a particle are required to be larger than 85% of the geometrical track length. The pad rows are made of at least 3 neighbouring individual observations (clusters), and their height varies from 7.5<ce:hsp sp="0.20"/>mm to 15<ce:hsp sp="0.20"/>mm <ce:cross-ref refid="br0430" id="crf11230">[43]</ce:cross-ref>. The trajectory lengths built from clusters (one cluster per pad row) is required to be larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mn>0.7</mml:mn><mml:mo>×</mml:mo><mml:mi>L</mml:mi></mml:math>. The fraction of TPC clusters shared with another track is required to be lower than 0.4. The fit quality for the ITS and TPC track points must satisfy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">hits</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>36</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">clusters</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>4</mml:mn></mml:math>, respectively, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">hits</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">clusters</mml:mi></mml:mrow></mml:msub></mml:math> are the numbers of hits in the ITS and the number of clusters in the TPC, respectively. Only tracks with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>36</mml:mn></mml:math> are included in the analysis, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> is calculated comparing the track parameters from the combined ITS and TPC track reconstruction to that derived only from the TPC and constrained to the interaction point. The definition of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> can be found in Ref. <ce:cross-ref refid="br0440" id="crf11240">[44]</ce:cross-ref>. To reduce the contamination from secondary particles, tracks are accepted if their distance-of-closest-approach (DCA) to the reconstructed primary interaction vertex satisfies in the longitudinal (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow></mml:msub></mml:math>) and transverse (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">xy</mml:mi></mml:mrow></mml:msub></mml:math>) directions the conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math><ce:hsp sp="0.20"/>cm and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">xy</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.018</mml:mn></mml:math><ce:hsp sp="0.20"/>cm + 0.035<ce:hsp sp="0.20"/>(cm×GeV/<ce:italic>c</ce:italic>)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>.</ce:para><ce:para id="pr0070">The measurement of the transverse momentum spectra of charged particles follows the standard procedure of the ALICE collaboration <ce:cross-refs refid="br0420 br0450" id="crs0080">[42,45]</ce:cross-refs>. The raw yields are corrected for efficiency and contamination from secondary particles. The efficiency correction is calculated from Monte Carlo simulations with GEANT3 <ce:cross-ref refid="br0460" id="crf11250">[46]</ce:cross-ref> transport code, which made use of PYTHIA 8 (Monash) <ce:cross-ref refid="br0280" id="crf11260">[28]</ce:cross-ref>, EPOS-LHC <ce:cross-ref refid="br0210" id="crf11270">[21]</ce:cross-ref> and HIJING <ce:cross-ref refid="br0470" id="crf11280">[47]</ce:cross-ref> event generators for pp, p–Pb and Pb–Pb collisions, respectively and incorporated a detailed description of the detector material, geometry and response. Since the event generators do not reproduce the relative abundances of different particle species in the real data, the efficiency obtained from the simulations is re-weighted considering the particle composition from data as outlined in <ce:cross-ref refid="br0420" id="crf11290">[42]</ce:cross-ref>. A multi-component template fit based on the DCA distributions from the simulation is used for the estimation of secondary contamination <ce:cross-ref refid="br0420" id="crf11300">[42]</ce:cross-ref>.</ce:para><ce:para id="pr0080">The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra for the toward and away regions include contributions from the jet fragmentation, ISR, and FSR, as well as, the contribution from the underlying event. In order to increase the sensitivity to the hardest process of the event, the particle yields measured in the transverse region are subtracted from the corresponding yields in both the toward and away regions: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>. This approach assumes that the background (UE, ISR, and FSR) in the toward and away regions is similar to the activity in the transverse region. However, one has to keep in mind that in Pb–Pb collisions two-particle correlations are affected by anisotropic transverse flow. In particular, the main contribution is due to the elliptic flow, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>, which is the second order coefficient in the Fourier expansion of the azimuthal distribution of the particle momenta <ce:cross-ref refid="br0480" id="crf11310">[48]</ce:cross-ref>. This elliptic azimuthal anisotropy modulates the background according to:<ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi>B</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>φ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant="normal">cos</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>φ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> is approximately given by the product of anisotropic flow coefficients for trigger and associated particles at their respective momenta i.e. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant="normal">assoc</mml:mi></mml:mrow></mml:msubsup></mml:math>. The existing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> measurements over a broad transverse momentum range <ce:cross-ref refid="br0490" id="crf11320">[49]</ce:cross-ref> suggest that the effect of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulation of background should be more relevant in semi-central Pb–Pb collisions. The effect is expected to be important at low and intermediate transverse momenta and decreases for high transverse momentum particles <ce:cross-ref refid="br0300" id="crf11330">[30]</ce:cross-ref>. In the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> region of interest for the jet quenching studies, namely <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>4</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, the effect of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulation is estimated to be small (about 5%) for Pb–Pb collisions. Given that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> effect is larger in Pb–Pb collisions than in pp and p–Pb collisions, no correction for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulation is applied for pp and p–Pb collisions since its effect is smaller than the other sources of systematic uncertainty.</ce:para><ce:para id="pr0090">The results are shown as a function of the average number of charged particles in the transverse region <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math>. The values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> are extracted in each multiplicity class from the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">tracks</mml:mi></mml:mrow></mml:msub></mml:math> distributions in the transverse region that are corrected for detector effects using a Bayesian unfolding <ce:cross-ref refid="br0500" id="crf11340">[50]</ce:cross-ref>. The Bayesian unfolding requires the multiplicity response matrix, which is built from the correlation between the measured multiplicity and the multiplicity at generator level (without detector effects) in the transverse region. This has been obtained from MC simulations which include the propagation of particles through the detector using GEANT 3. As a crosscheck, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> values are also calculated by integrating the transverse momentum distributions in the interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mn>0.5</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>8</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>. The difference between the results from the two strategies is assigned as the systematic uncertainty on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math>, where the effects related to the discrepancy between data and MC in the particle composition and secondary contamination are considered. This uncertainty amounts up to 3.5%, 4% and 6.5% for pp, p–Pb and Pb–Pb collisions, respectively.</ce:para><ce:para id="pr0100">The systematic uncertainties related to the track selection criteria were studied by repeating the analysis varying one-by-one the track selection criteria <ce:cross-refs refid="br0420 br0450" id="crs0090">[42,45]</ce:cross-refs>. In particular, the upper limits of the track fit quality parameters in the ITS (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">hits</mml:mi></mml:mrow></mml:msub></mml:math>) and in the TPC (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">clusters</mml:mi></mml:mrow></mml:msub></mml:math>) were varied in the ranges of 25–49 and 3–5, respectively. The maximum fraction of shared TPC clusters was varied between 0.2 to 1 and the maximum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow></mml:msub></mml:math> was varied between 1 and 5<ce:hsp sp="0.20"/>cm <ce:cross-ref refid="br0420" id="crf11350">[42]</ce:cross-ref>. We have also quantified the impact of not including the ITS hit requirement in the track selection. The systematic uncertainty on the primary particle composition was estimated using a procedure similar to the one described in <ce:cross-ref refid="br0420" id="crf11360">[42]</ce:cross-ref>. To quantify the uncertainty due to the imperfect simulation of the detector response, the track matching between the TPC and the ITS information in the data and in the simulation were compared. To achieve this, the fraction of secondary particles was rescaled according to fits to the measured DCA distributions. After this rescaling, the agreement between data and model was found to be within 3% for all collision systems. This value was assigned as an additional systematic uncertainty <ce:cross-ref refid="br0420" id="crf11370">[42]</ce:cross-ref>. The systematic uncertainty on the secondary particle contamination considers the imperfection of the method (multi-component template fit) used to extract the correction. The fit ranges were varied and the fit was repeated using templates with two (primaries, secondaries) or three (primaries, secondaries from material, secondaries from weak decays) components. The maximum spread among these variations was assigned as the systematic uncertainty on the secondary contamination. This contribution dominates at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>. The density of materials used in simulations of the experimental setup was varied by ± 4.5% <ce:cross-ref refid="br0350" id="crf11380">[35]</ce:cross-ref>, resulting in a negligible systematic uncertainty in the considered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> range of 0.5 to 6.0 GeV/<ce:italic>c</ce:italic>. For the estimation of total systematic uncertainty, all the above listed contributions were summed in quadrature. The systematic uncertainties are independent of the difference between the azimuthal angle of the associated particle and that of the trigger particle. The estimated systematic uncertainties on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra significantly depend on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>, while the dependence on the multiplicity classes is mild. The ranges of systematic uncertainties in the three considered collision systems are reported in <ce:cross-ref refid="tbl0010" id="crf11390">Table 1</ce:cross-ref><ce:float-anchor refid="tbl0010"/> for the various sources described above.</ce:para></ce:section><ce:section id="se0030" role="results"><ce:label>3</ce:label><ce:section-title id="st0040">Results and discussion</ce:section-title><ce:para id="pr0110">The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra measured in the transverse region for pp, p–Pb, and Pb–Pb collisions are shown in <ce:cross-ref refid="fg0020" id="crf11400">Fig. 2</ce:cross-ref><ce:float-anchor refid="fg0020"/> (top panel). Results are presented for different multiplicity classes. The ratios between the spectra in the individual multiplicity classes and the MB (0−100%) one are shown in the bottom panel. In the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:mn>0.5</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, the ratios for the highest multiplicity class (0−5%) are larger than unity and show an increasing trend with increasing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:mo linebreak="badbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) followed at higher <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> by a slow decrease. Instead, for the lowest multiplicity classes (40−60% and 60−90%) the ratios are lower than unity and follow an opposite trend with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>, decreasing at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> and increasing for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>3</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>. The behaviour of the ratios as a function of the event activity is reminiscent of analogous ratios as a function of the number of MPI in pp collisions simulated with <ce:small-caps>PYTHIA</ce:small-caps> 8, including colour reconnection <ce:cross-ref refid="br0510" id="crf11410">[51]</ce:cross-ref>. In particular, at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:math> GeV/<ce:italic>c</ce:italic> the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectrum of pp collisions with large MPI activity exhibits an enhancement with respect to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectrum of MB pp collisions. The effect was not observed before in data because, in contrast to the present analysis, the jet contribution was included in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra <ce:cross-ref refid="br0450" id="crf11420">[45]</ce:cross-ref>.</ce:para><ce:para id="pr0120">The top (bottom) panel of <ce:cross-ref refid="fg0030" id="crf11430">Fig. 3</ce:cross-ref><ce:float-anchor refid="fg0030"/> shows the charged particle yields for the toward (away) region after the subtraction of the yields measured in the transverse region in pp, p–Pb and Pb–Pb collisions. Results are compared with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra measured for MB pp collisions (0−100% V0M pp event class) quantified with the ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math>, as defined in Eq. <ce:cross-ref refid="fm0010" id="crf11440">(1)</ce:cross-ref>. At low transverse momenta, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>4</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is close to unity in pp and p–Pb collisions. In contrast, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> in Pb–Pb collisions exhibits a strong multiplicity dependence over the whole measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> interval. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> magnitude is larger for semi-peripheral Pb–Pb collisions, the maximum is observed for 20−40% Pb–Pb collisions, and is smaller for the most central and most peripheral classes. Given that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> contribution is not subtracted from the jet-like yields reported in <ce:cross-ref refid="fg0030" id="crf11450">Fig. 3</ce:cross-ref>, the centrality dependence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> follows the behaviour of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> as a function of collision centrality and particle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> in Pb–Pb collisions at LHC energies <ce:cross-ref refid="br0520" id="crf11460">[52]</ce:cross-ref>.</ce:para><ce:para id="pr0130"><ce:cross-ref refid="fg0040" id="crf11660">Fig. 4</ce:cross-ref><ce:float-anchor refid="fg0040"/> shows the measured values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> in the transverse momentum interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mn>4</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> as a function of the average multiplicity in the transverse region for all the multiplicity classes considered in pp, p–Pb and Pb–Pb collisions. The figure shows that, within uncertainties, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are close to unity for all the multiplicity classes measured in pp and p–Pb collisions. This indicates that effects induced by possible energy loss in these systems are not observed within uncertainties. This result is consistent with previous studies of nuclear modification factor <ce:cross-ref refid="br0330" id="crf11480">[33]</ce:cross-ref> and hadron-jet recoil measurements <ce:cross-ref refid="br0340" id="crf11490">[34]</ce:cross-ref>. By contrast, for Pb–Pb collisions the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are compatible to unity for peripheral collisions, and show a gradual enhancement (reduction) with the increase in multiplicity for the toward (away) region. The behaviour is the same for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values measured either assuming a flat background or a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>-modulated background. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>-modulated background was estimated following the approach depicted in Eq. <ce:cross-ref refid="fm0020" id="crf11500">(2)</ce:cross-ref> and using the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> data reported in <ce:cross-ref refid="br0490" id="crf11510">[49]</ce:cross-ref>. This behaviour is qualitatively similar to the di-hadron correlation results reported by the STAR and ALICE collaborations <ce:cross-refs refid="br0290 br0300" id="crs0100">[29,30]</ce:cross-refs>. In Pb–Pb collisions, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> provides information about the fragmenting jet leaving the medium, while on the away side, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> reflects the survival probability of the recoiling parton during passage through the medium. Thus a suppression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> would indicate that fewer partons survive the passage through the medium and is expected from the strong in-medium energy loss. On the other hand, the enhancement observed in the toward region is also subject to medium effects. The ratio is sensitive to a) a possible change of the fragmentation functions, b) a possible modification of the quark to gluon jet ratio in the final state due to different coupling with medium, and c) a possible bias on the parton spectrum due to trigger particle selection. Moreover, given that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is sensitive to the same effects as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math>, the interpretation of the results is similar to that reported in <ce:cross-ref refid="br0300" id="crf11520">[30]</ce:cross-ref>. It is likely that all three effects play a role <ce:cross-ref refid="br0300" id="crf11530">[30]</ce:cross-ref>. A detailed quantification of the contribution of each effect is beyond the scope of the present paper.</ce:para><ce:para id="pr0140">In order to get further insight into the effect, the measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are compared in <ce:cross-ref refid="fg0050" id="crf11540">Fig. 5</ce:cross-ref><ce:float-anchor refid="fg0050"/> with model predictions. Following the similar treatment of the experimental data, for the models, the total sample is subdivided into different V0M classes and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> is calculated for each class. For high-multiplicity pp collisions, although <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is close to unity, a small trend with multiplicity is visible, which is not seen at similar multiplicities (20−90% V0A) in p–Pb data. To understand the source of these slight deviations from unity, the data are compared with the predictions from the <ce:small-caps>PYTHIA</ce:small-caps> 8 (Monash tune <ce:cross-ref refid="br0280" id="crf11550">[28]</ce:cross-ref>) and EPOS-LHC <ce:cross-ref refid="br0210" id="crf11560">[21]</ce:cross-ref> event generators. In PYTHIA, the hadronization of quarks is simulated using the Lund string fragmentation model <ce:cross-ref refid="br0530" id="crf11570">[53]</ce:cross-ref>. Various PYTHIA tunes have been developed through extensive comparison of Monte Carlo distributions with the minimum-bias data from different experiments. The Monash tune of <ce:small-caps>PYTHIA</ce:small-caps> 8 is tuned to LHC data and uses an updated set of hadronization parameters compared to the previous tunes <ce:cross-ref refid="br0280" id="crf11580">[28]</ce:cross-ref>. EPOS-LHC is built on the Parton-Based Gribov Regge Theory. Utilising the colour exchange mechanism of string excitation, the model is tuned to LHC data <ce:cross-ref refid="br0210" id="crf11590">[21]</ce:cross-ref>. In this model, a part of the collision system which has high parton densities becomes a “core” region that evolves hydrodynamically as a quark–gluon plasma and it is surrounded by a more dilute “corona” for which fragmentation occurs in the vacuum. The upper panel of <ce:cross-ref refid="fg0050" id="crf11600">Fig. 5</ce:cross-ref> shows <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> for different multiplicity classes. The observed deviations from unity are reproduced by <ce:small-caps>PYTHIA</ce:small-caps> 8 for both the toward and away regions. Given that <ce:small-caps>PYTHIA</ce:small-caps> 8 does not incorporate any jet quenching mechanism, the origin of the effect in high <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> collisions is related to a remaining bias towards harder fragmentation and more activity from initial and final state radiation <ce:cross-ref refid="br0540" id="crf11610">[54]</ce:cross-ref>. These effects enhance the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> yield in the toward region, and produce a broadening in the away region <ce:cross-ref refid="br0550" id="crf11620">[55]</ce:cross-ref>. The EPOS-LHC results in the away region are similar to both data and <ce:small-caps>PYTHIA</ce:small-caps> 8. However, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> EPOS-LHC exhibits a trend with a maximum at intermediate multiplicity and a reduction toward low and high multiplicities, which is not consistent with the measurements.</ce:para><ce:para id="pr0150">The middle and bottom panels of <ce:cross-ref refid="fg0050" id="crf11630">Fig. 5</ce:cross-ref> show <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> measured for p–Pb and Pb–Pb collisions, respectively. The data are compared to <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr <ce:cross-ref refid="br0560" id="crf11640">[56]</ce:cross-ref> and EPOS-LHC predictions. The Angantyr model in <ce:small-caps>PYTHIA</ce:small-caps> 8 extrapolates the dynamics from pp collisions to p–Pb and Pb–Pb collisions, generalising the formalism adopted for pp collisions by including a description of the nucleon positions within the colliding nuclei and utilising the Glauber model to calculate the number of interacting nucleons and binary nucleon–nucleon collisions. <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr, which does not include jet quenching effects, predicts <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values consistent with unity for all the multiplicity classes in Pb–Pb collisions. Whereas for p–Pb collisions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is consistent with unity, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> is slightly below unity. In EPOS-LHC, a certain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> cutoff is defined in such a way that, above this cutoff, a particle loses part of its momentum in the core but survives as an independent particle produced by a flux tube. Soft particles, which are below the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> cutoff, get completely absorbed and form the core. This sort of energy loss mechanism implemented in EPOS-LHC depends on the system size <ce:cross-refs refid="br0210 br0570 br0580" id="crs0110">[21,57,58]</ce:cross-refs>. <ce:cross-ref refid="fg0050" id="crf11650">Fig. 5</ce:cross-ref> (middle) shows that for p–Pb collisions, EPOS-LHC does not describe either the magnitude or the trend of the multiplicity dependence of the measured ratio in the toward region, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math>. However, the model is in reasonable agreement with data in the away region. For Pb–Pb collisions, EPOS-LHC predicts a significant enhancement of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> for low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> ranges and deviates significantly from the experimental results.</ce:para><ce:para id="pr0160">In summary, while the data from Pb–Pb collisions are in qualitative agreement with expectations from parton energy loss due to the presence of a hot and dense medium, pp and p–Pb data do not show any hint of medium effects in the multiplicity range which is reported.</ce:para></ce:section><ce:section id="se0040"><ce:label>4</ce:label><ce:section-title id="st0050">Summary</ce:section-title><ce:para id="pr0170">The transverse momentum spectra (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mn>0.5</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) of primary charged particles in three azimuthal regions (toward, away and transverse) defined with respect to the direction of the particle with the highest transverse momentum in the event (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>8</mml:mn><mml:mo>≤</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) are reported. The spectra are studied in intervals of the multiplicity measured at forward pseudorapidities for pp, p–Pb, and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in the transverse region are subtracted from those of the away and toward regions. This is based on the assumption that the transverse side provides a good estimation of the underlying event contribution in both the toward and away regions. However, for the interpretation of the results one has to keep in mind that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulates the background and this effect is important for semi-central Pb–Pb collisions and for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>4</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> the effect is less than 5% in central and peripheral Pb–Pb collisions. Ratios to MB pp (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math>), i.e., the multiplicity dependent yields normalised to the yield measured in MB pp collisions, are reported. At low transverse momentum (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>), within 20%, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are multiplicity independent for both the toward and away regions in pp and p–Pb collisions. In contrast, in Pb–Pb collisions for both toward and away regions the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values exhibit a centrality dependence which is expected given the residual presence of elliptic flow. In the highest transverse momentum interval (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mn>4</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>), the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values in pp collisions are closer to unity but they exhibit a small reduction (increase) towards high V0 activity in pp collisions. This trend is well reproduced by <ce:small-caps>PYTHIA</ce:small-caps> 8. In the model, it is due to a selection bias towards pp collisions with harder fragmentation and larger activity from initial and final state radiation. For p–Pb collisions, within uncertainties, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are consistent with unity and do not show a multiplicity dependence. <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr fairly describes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math>, but it underestimates by about 10% the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> values in the low multiplicity classes (40−90% V0A event class). For Pb–Pb collisions, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are close to unity for peripheral collisions, and show a gradual increase (reduction) in the toward (away) region with increasing multiplicity. A similar observable, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math>, based on the per-trigger yield of associated particles in di-hadron correlation has been studied for central and peripheral Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The behaviour of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> exhibits the same features as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math>: in central collisions, on the away-side, a suppression is observed as expected from strong in-medium energy loss. In the toward region, an enhancement is observed. <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr predicts <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:math> for all multiplicity intervals, and it does not reproduce the observed away-side suppression or toward-side enhancement. Generally, EPOS-LHC does not describe the measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> ratios.</ce:para><ce:para id="pr0180">In summary, within the multiplicity reach reported in this paper, no jet quenching effects are observed in pp and p–Pb collisions within uncertainties. Further studies are required to extend the present analysis to higher multiplicities, which are currently limited by the event selection based on the forward V0 detector. The analysis of future pp and p–Pb collisions with much larger integrated luminosity may remove this limitation.</ce:para> </ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0090">Declaration of Competing Interest</ce:section-title><ce:para id="pr0210">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0060">Acknowledgements</ce:section-title><ce:para id="pr0190">The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: <ce:grant-sponsor id="gsp0010">A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation (ANSL)</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0020" sponsor-id="https://doi.org/10.13039/501100007029">State Committee of Science</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0030">World Federation of Scientists (WFS)</ce:grant-sponsor>, Armenia; <ce:grant-sponsor id="gsp0040" sponsor-id="https://doi.org/10.13039/501100001822">Austrian Academy of Sciences</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0050" sponsor-id="https://doi.org/10.13039/501100002428">Austrian Science Fund</ce:grant-sponsor> (FWF): [<ce:grant-number refid="gsp0050">M 2467-N36</ce:grant-number>] and <ce:grant-sponsor id="gsp0060">Nationalstiftung für Forschung, Technologie und Entwicklung</ce:grant-sponsor>, Austria; <ce:grant-sponsor id="gsp0070">Ministry of Communications and High Technologies, National Nuclear Research Center</ce:grant-sponsor>, Azerbaijan; Conselho Nacional de Desenvolvimento Científico e Tecnológico (<ce:grant-sponsor id="gsp0080" sponsor-id="https://doi.org/10.13039/501100003593">CNPq</ce:grant-sponsor>), <ce:grant-sponsor id="gsp0090" sponsor-id="https://doi.org/10.13039/501100004809">Financiadora de Estudos e Projetos</ce:grant-sponsor> (Finep), <ce:grant-sponsor id="gsp0100" sponsor-id="https://doi.org/10.13039/501100001807">Fundação de Amparo à Pesquisa do Estado de São Paulo</ce:grant-sponsor> (<ce:grant-sponsor id="gsp0110" sponsor-id="https://doi.org/10.13039/501100001807">FAPESP</ce:grant-sponsor>) and <ce:grant-sponsor id="gsp0120" sponsor-id="https://doi.org/10.13039/501100004909">Universidade Federal do Rio Grande do Sul</ce:grant-sponsor> (<ce:grant-sponsor id="gsp0130" sponsor-id="https://doi.org/10.13039/501100004909">UFRGS</ce:grant-sponsor>), Brazil; Bulgarian <ce:grant-sponsor id="gsp0140" sponsor-id="https://doi.org/10.13039/501100005992">Ministry of Education and Science</ce:grant-sponsor>, within the National Roadmap for Research Infrastructures 2020–2027 (object CERN), Bulgaria; <ce:grant-sponsor id="gsp0150" sponsor-id="https://doi.org/10.13039/501100002338">Ministry of Education of China</ce:grant-sponsor> (MOEC), <ce:grant-sponsor id="gsp0160">Ministry of Science & Technology of China</ce:grant-sponsor> (MSTC) and <ce:grant-sponsor id="gsp0170" sponsor-id="https://doi.org/10.13039/501100001809">National Natural Science Foundation of China</ce:grant-sponsor> (NSFC), China; <ce:grant-sponsor id="gsp0180" sponsor-id="https://doi.org/10.13039/100015526">Ministry of Science and Education</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0190" sponsor-id="https://doi.org/10.13039/501100004488">Croatian Science Foundation</ce:grant-sponsor>, Croatia; <ce:grant-sponsor id="gsp0200" sponsor-id="https://doi.org/10.13039/501100019929">Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear</ce:grant-sponsor> (CEADEN), <ce:grant-sponsor id="gsp0210">Cubaenergía</ce:grant-sponsor>, Cuba; <ce:grant-sponsor id="gsp0220">Ministry of Education, Youth and Sports of the Czech Republic</ce:grant-sponsor>, Czech Republic; The <ce:grant-sponsor id="gsp0230">Danish Council for Independent Research | Natural Sciences</ce:grant-sponsor>, the <ce:grant-sponsor id="gsp0240" sponsor-id="https://doi.org/10.13039/100008398">Villum Fonden</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0250" sponsor-id="https://doi.org/10.13039/501100001732">Danish National Research Foundation</ce:grant-sponsor> (DNRF), Denmark; <ce:grant-sponsor id="gsp0260">Helsinki Institute of Physics</ce:grant-sponsor> (HIP), Finland; Commissariat à l'Energie Atomique (<ce:grant-sponsor id="gsp0270" sponsor-id="https://doi.org/10.13039/501100006489">CEA</ce:grant-sponsor>) and <ce:grant-sponsor id="gsp0280" sponsor-id="https://doi.org/10.13039/501100012441">Institut National de Physique Nucléaire et de Physique des Particules</ce:grant-sponsor> (IN2P3) and <ce:grant-sponsor id="gsp0290" sponsor-id="https://doi.org/10.13039/501100004794">Centre National de la Recherche Scientifique</ce:grant-sponsor> (CNRS), France; Bundesministerium für Bildung und Forschung (<ce:grant-sponsor id="gsp0300" sponsor-id="https://doi.org/10.13039/501100002347">BMBF</ce:grant-sponsor>) and <ce:grant-sponsor id="gsp0310" sponsor-id="https://doi.org/10.13039/501100010958">GSI Helmholtzzentrum für Schwerionenforschung GmbH</ce:grant-sponsor>, Germany; <ce:grant-sponsor id="gsp0320" sponsor-id="https://doi.org/10.13039/501100003448">General Secretariat for Research and Technology</ce:grant-sponsor>, Ministry of Education, Research and Religions, Greece; <ce:grant-sponsor id="gsp0330" sponsor-id="https://doi.org/10.13039/501100018818">National Research, Development and Innovation Office</ce:grant-sponsor>, Hungary; Department of Atomic Energy Government of India (<ce:grant-sponsor id="gsp0340" sponsor-id="https://doi.org/10.13039/501100001502">DAE</ce:grant-sponsor>), Department of Science and Technology, Government of India (<ce:grant-sponsor id="gsp0350" sponsor-id="https://doi.org/10.13039/501100001409">DST</ce:grant-sponsor>), <ce:grant-sponsor id="gsp0360" sponsor-id="https://doi.org/10.13039/501100001501">University Grants Commission</ce:grant-sponsor>, Government of India (UGC) and <ce:grant-sponsor id="gsp0370" sponsor-id="https://doi.org/10.13039/501100001412">Council of Scientific and Industrial Research</ce:grant-sponsor> (CSIR), India; National Research and Innovation Agency - <ce:grant-sponsor id="gsp0380" sponsor-id="https://doi.org/10.13039/100020473">BRIN</ce:grant-sponsor>, Indonesia; Istituto Nazionale di Fisica Nucleare (<ce:grant-sponsor id="gsp0390" sponsor-id="https://doi.org/10.13039/501100004007">INFN</ce:grant-sponsor>), Italy; Japanese <ce:grant-sponsor id="gsp0400" sponsor-id="https://doi.org/10.13039/501100001700">Ministry of Education, Culture, Sports, Science and Technology</ce:grant-sponsor> (MEXT) and <ce:grant-sponsor id="gsp0410" sponsor-id="https://doi.org/10.13039/501100001691">Japan Society for the Promotion of Science</ce:grant-sponsor> (JSPS) KAKENHI, Japan; Consejo Nacional de Ciencia (<ce:grant-sponsor id="gsp0420" sponsor-id="https://doi.org/10.13039/501100003141">CONACYT</ce:grant-sponsor>) y Tecnología, through <ce:grant-sponsor id="gsp0430" sponsor-id="https://doi.org/10.13039/501100007709">Fondo de Cooperación Internacional en Ciencia y Tecnología</ce:grant-sponsor> (FONCICYT) and <ce:grant-sponsor id="gsp0440" sponsor-id="https://doi.org/10.13039/501100006087">Dirección General de Asuntos del Personal Académico</ce:grant-sponsor> (DGAPA), Mexico; <ce:grant-sponsor id="gsp0450" sponsor-id="https://doi.org/10.13039/501100003246">Nederlandse Organisatie voor Wetenschappelijk Onderzoek</ce:grant-sponsor> (NWO), Netherlands; The <ce:grant-sponsor id="gsp0460" sponsor-id="https://doi.org/10.13039/501100005416">Research Council of Norway</ce:grant-sponsor>, Norway; <ce:grant-sponsor id="gsp0470">Commission on Science and Technology for Sustainable Development in the South</ce:grant-sponsor> (COMSATS), Pakistan; <ce:grant-sponsor id="gsp0480" sponsor-id="https://doi.org/10.13039/501100011871">Pontificia Universidad Católica del Perú</ce:grant-sponsor>, Peru; <ce:grant-sponsor id="gsp0490">Ministry of Education and Science</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0500" sponsor-id="https://doi.org/10.13039/501100004281">National Science Centre</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0510">WUT ID-UB</ce:grant-sponsor>, Poland; <ce:grant-sponsor id="gsp0520" sponsor-id="https://doi.org/10.13039/501100003708">Korea Institute of Science and Technology Information</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0530" sponsor-id="https://doi.org/10.13039/501100003725">National Research Foundation of Korea</ce:grant-sponsor> (NRF), Republic of Korea; <ce:grant-sponsor id="gsp0540">Ministry of Education and Scientific Research</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0550" sponsor-id="https://doi.org/10.13039/501100019278">Institute of Atomic Physics</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0560" sponsor-id="https://doi.org/10.13039/501100015622">Ministry of Research and Innovation</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0570" sponsor-id="https://doi.org/10.13039/501100019278">Institute of Atomic Physics</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0580">University Politehnica of Bucharest</ce:grant-sponsor>, Romania; <ce:grant-sponsor id="gsp0590" sponsor-id="https://doi.org/10.13039/501100003193">Ministry of Education, Science, Research and Sport of the Slovak Republic</ce:grant-sponsor>, Slovakia; <ce:grant-sponsor id="gsp0600">National Research Foundation of South Africa</ce:grant-sponsor>, South Africa; <ce:grant-sponsor id="gsp0610" sponsor-id="https://doi.org/10.13039/501100004359">Swedish Research Council</ce:grant-sponsor> (VR) and <ce:grant-sponsor id="gsp0620">Knut & Alice Wallenberg Foundation</ce:grant-sponsor> (KAW), Sweden; <ce:grant-sponsor id="gsp0630" sponsor-id="https://doi.org/10.13039/100012470">European Organization for Nuclear Research</ce:grant-sponsor>, Switzerland; <ce:grant-sponsor id="gsp0640" sponsor-id="https://doi.org/10.13039/501100004352">Suranaree University of Technology</ce:grant-sponsor> (SUT), <ce:grant-sponsor id="gsp0650" sponsor-id="https://doi.org/10.13039/501100004192">National Science and Technology Development Agency</ce:grant-sponsor> (NSTDA), <ce:grant-sponsor id="gsp0660" sponsor-id="https://doi.org/10.13039/501100017170">Thailand Science Research and Innovation</ce:grant-sponsor> (TSRI) and <ce:grant-sponsor id="gsp0670">National Science, Research and Innovation Fund</ce:grant-sponsor> (NSRF), Thailand; <ce:grant-sponsor id="gsp0680" sponsor-id="https://doi.org/10.13039/100020381">Turkish Energy, Nuclear and Mineral Research Agency</ce:grant-sponsor> (TENMAK), Turkey; <ce:grant-sponsor id="gsp0690" sponsor-id="https://doi.org/10.13039/501100004742">National Academy of Sciences of Ukraine</ce:grant-sponsor>, Ukraine; <ce:grant-sponsor id="gsp0700" sponsor-id="https://doi.org/10.13039/501100000271">Science and Technology Facilities Council</ce:grant-sponsor> (STFC), United Kingdom; National Science Foundation of the United States of America (<ce:grant-sponsor id="gsp0710" sponsor-id="https://doi.org/10.13039/100000001">NSF</ce:grant-sponsor>) and <ce:grant-sponsor id="gsp0720" sponsor-id="https://doi.org/10.13039/100000015">United States Department of Energy</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0730" sponsor-id="https://doi.org/10.13039/100006209">Office of Nuclear Physics</ce:grant-sponsor> (DOE NP), United States of America. In addition, individual groups or members have received support from: Marie Skłodowska Curie, Strong 2020 - <ce:grant-sponsor id="gsp0740" sponsor-id="https://doi.org/10.13039/100010661">Horizon 2020</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0750" sponsor-id="https://doi.org/10.13039/501100000781">European Research Council</ce:grant-sponsor> (grant nos. <ce:grant-number refid="gsp0750">824093</ce:grant-number>, <ce:grant-number refid="gsp0750">896850</ce:grant-number>, <ce:grant-number refid="gsp0750">950692</ce:grant-number>), <ce:grant-sponsor id="gsp0760" sponsor-id="https://doi.org/10.13039/501100000780">European Union</ce:grant-sponsor>; <ce:grant-sponsor id="gsp0770" sponsor-id="https://doi.org/10.13039/501100002341">Academy of Finland</ce:grant-sponsor> (Center of Excellence in Quark Matter) (grant nos. <ce:grant-number refid="gsp0770">346327</ce:grant-number>, <ce:grant-number refid="gsp0770">346328</ce:grant-number>), Finland; <ce:grant-sponsor id="gsp0780">Programa de Apoyos para la Superación del Personal Académico</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0790" sponsor-id="https://doi.org/10.13039/501100005739">UNAM</ce:grant-sponsor>, Mexico.</ce:para></ce:acknowledgment></body><tail><ce:bibliography id="bl0010"><ce:section-title id="st0070">References</ce:section-title><ce:bibliography-sec id="bs0010"><ce:bib-reference id="br0010"><ce:label>[1]</ce:label><sb:reference id="bibD17CEA27EFFA6FC144ED8C3DC8A6A4C2s1"><sb:contribution><sb:authors><sb:author><ce:given-name>T.</ce:given-name><ce:surname>Sjöstrand</ce:surname></sb:author><sb:author><ce:given-name>M.</ce:given-name><ce:surname>van Zijl</ce:surname></sb:author></sb:authors><sb:title><sb:maintitle>A multiple interaction model for the event structure in hadron collisions</sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Phys. 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The horizontal line in brown indicates the bounce action in the absence of gravity, <ce:italic>S</ce:italic><ce:inf>0</ce:inf> = 8<ce:italic>π</ce:italic><ce:sup>2</ce:sup>/(3|<ce:italic>λ</ce:italic>(<ce:italic>μ</ce:italic>)|). The metric and Palatini cases are represented by the black-dashed and gray-dot-dashed curves, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0010">Fig. 1</ce:alt-text><ce:link locator="gr001" xlink:type="simple" xlink:href="pii:S0370269323004434/gr001" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0010"/></ce:figure></ce:floats><head><ce:title id="ti0010">Electroweak vacuum decay in metric-affine gravity</ce:title><ce:author-group id="ag0010"><ce:author orcid="0000-0002-2957-5276" id="au0010" author-id="S0370269323004434-957dd146f6e243df4b6f309334983b99"><ce:given-name>Ioannis D.</ce:given-name><ce:surname>Gialamas</ce:surname><ce:cross-ref refid="cr0010" id="crf0450"><ce:sup>⁎</ce:sup></ce:cross-ref><ce:e-address type="email" xlink:href="mailto:ioannis.gialamas@kbfi.ee" id="ea0010">ioannis.gialamas@kbfi.ee</ce:e-address></ce:author><ce:author id="au0020" author-id="S0370269323004434-11ce32aef4ff6906bb946b4904582471"><ce:given-name>Hardi</ce:given-name><ce:surname>Veermäe</ce:surname><ce:e-address type="email" xlink:href="mailto:hardi.veermae@cern.ch" id="ea0020">hardi.veermae@cern.ch</ce:e-address></ce:author><ce:affiliation id="aff0010" affiliation-id="S0370269323004434-e0efc38af768f3818fce829f251dffc5"><ce:textfn>Laboratory of High Energy and Computational Physics, National Institute of Chemical Physics and Biophysics, Rävala pst. 10, 10143, Tallinn, Estonia</ce:textfn><sa:affiliation><sa:organization>Laboratory of High Energy and Computational Physics</sa:organization><sa:organization>National Institute of Chemical Physics and Biophysics</sa:organization><sa:address-line>Rävala pst. 10</sa:address-line><sa:city>Tallinn</sa:city><sa:postal-code>10143</sa:postal-code><sa:country>Estonia</sa:country></sa:affiliation><ce:source-text id="srct0005">Laboratory of High Energy and Computational Physics, National Institute of Chemical Physics and Biophysics, Rävala pst. 10, 10143, Tallinn, Estonia</ce:source-text></ce:affiliation><ce:correspondence id="cr0010"><ce:label>⁎</ce:label><ce:text>Corresponding author.</ce:text></ce:correspondence></ce:author-group><ce:date-received day="1" month="6" year="2023"/><ce:date-revised day="12" month="7" year="2023"/><ce:date-accepted day="27" month="7" year="2023"/><ce:miscellaneous id="ms0010">Editor: R. Gregory</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0020">We investigate the stability of the electroweak vacuum in metric-affine gravity in which the Standard Model Higgs boson can be non-minimally coupled to both the Ricci scalar and the Holst invariant. We find that vacuum stability is improved in this framework across a wide range of model parameters.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0080">Data availability</ce:section-title><ce:para id="pr0220">No data was used for the research described in the article.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">It is well known that the potential of the Higgs boson in the Standard Model (SM) is deeper at high energies than in the electroweak vacuum permitting its decay through quantum tunneling <ce:cross-refs refid="br0010 br0020 br0030 br0040" id="crs0010">[1–4]</ce:cross-refs>. Although this does not invalidate the SM, the electroweak vacuum is predicted to be metastable in the absence of contributions from UV physics <ce:cross-refs refid="br0050 br0060 br0070 br0080 br0090 br0100 br0110 br0120 br0130" id="crs0020">[5–13]</ce:cross-refs>.</ce:para><ce:para id="pr0020">Coleman and De Luccia <ce:cross-ref refid="br0140" id="crf0020">[14]</ce:cross-ref> were the first to delve into the matter of gravitational effects on vacuum decay. Subsequently, multiple studies of gravitational corrections have been performed <ce:cross-refs refid="br0150 br0160 br0170 br0180 br0190 br0200 br0210 br0220 br0230" id="crs0030">[15–23]</ce:cross-refs>, along with discussions about the impact of black holes on the amplification or reduction of the vacuum decay rate <ce:cross-refs refid="br0240 br0250 br0260 br0270 br0280 br0290 br0300 br0310 br0320 br0330 br0340 br0350 br0360 br0370 br0380 br0390 br0400" id="crs0040">[24–40]</ce:cross-refs>.</ce:para><ce:para id="pr0030">In this letter, we extend the calculation of gravitational corrections to vacuum decay in the context of metric-affine gravity. There, in contrast to general relativity, the connection is taken to be an independent variable without the usual symmetries of the Levi-Civita one. As a result, the Riemann tensor does not possess the symmetries it has in the metric case, and thus the gravitational action should be extended by including an additional scalar curvature invariant - the Holst invariant. This term commonly appears in Loop Quantum Gravity <ce:cross-ref refid="br0410" id="crf0030">[41]</ce:cross-ref> and has been studied in various branches of high energy physics, such as black hole thermodynamics <ce:cross-refs refid="br0420 br0430" id="crs0050">[42,43]</ce:cross-refs>. Lately, its significance in inflationary cosmology <ce:cross-refs refid="br0440 br0450 br0460 br0470 br0480 br0490" id="crs0060">[44–49]</ce:cross-refs>, and high energy physics phenomenology <ce:cross-refs refid="br0500 br0510 br0520 br0530 br0540" id="crs0070">[50–54]</ce:cross-refs>, has garnered a great deal of attention.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">The gravitational action</ce:section-title><ce:para id="pr0040">Metric-affine theories of gravity treat the metric tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and the connection <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts></mml:math> as independent variables. This should be contrasted with the usual metric gravity that uses the Levi-Civita connection <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mo stretchy="true">{</mml:mo><mml:mmultiscripts><mml:mrow/><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/><mml:mprescripts/><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow></mml:mmultiscripts><mml:mo stretchy="true">}</mml:mo></mml:math>, which is completely determined by the metric. It is useful to decompose the connection <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts></mml:math> as<ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mo>≡</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mmultiscripts><mml:mrow/><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/><mml:mprescripts/><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow></mml:mmultiscripts><mml:mo stretchy="true">}</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts></mml:math> is dubbed the distortion tensor. The Riemann tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts></mml:math> is constructed from the connection Γ in the usual way and one can form two scalars that are linear on it. These are the Ricci scalar and the Holst invariant <ce:cross-refs refid="br0550 br0560" id="crs0080">[55,56]</ce:cross-refs><ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:mi mathvariant="script">R</mml:mi><mml:mo>≡</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>g</ce:italic> is the determinant of the metric tensor and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:msup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msup></mml:math> is the totally antisymmetric tensor. In metric gravity, the Holst invariant vanishes identically due to the symmetries of the Riemann tensor.</ce:para><ce:para id="pr0050">The most general action linear in the Riemann tensor and containing terms of at most dimension 4 has the form<ce:cross-ref refid="fn0010" id="crf0040"><ce:sup>1</ce:sup></ce:cross-ref><ce:footnote id="fn0010"><ce:label>1</ce:label><ce:note-para id="np0010">Natural units with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:mi>c</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>ħ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:math> are used throughout this paper.</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0030"><ce:label>(3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:mi>S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is the Higgs potential,<ce:display><ce:formula id="fm0040"><ce:label>(4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mi>κ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>ξ</mml:mi><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>κ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> are non-minimal couplings, <ce:italic>β</ce:italic>, <ce:italic>ξ</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math> are constant couplings, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>κ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>. The function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> can be thought of as a field-dependent Barbero-Immirzi parameter <ce:cross-refs refid="br0560 br0570" id="crs0090">[56,57]</ce:cross-refs>. We will first work out the formalism without assuming a specific functional form of <ce:italic>f</ce:italic>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>, and <ce:italic>V</ce:italic> and suppress their arguments for notational brevity.</ce:para><ce:para id="pr0060">In addition to the Ricci and Holst terms, metric-affine gravity permits the construction of 20 additional scalars with mass dimension 2 from torsion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msub></mml:math> and non-metricity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> <ce:cross-refs refid="br0580 br0590" id="crs0100">[58,59]</ce:cross-refs>. Although such terms are not considered in this work, our perturbative results can be straightforwardly extended to include them as will be explained below. We will also neglect couplings between the connection and fermions as they will only generate Planck-suppressed four-fermion and higher-order scalar-fermion interactions <ce:cross-refs refid="br0470 br0510 br0530 br0600" id="crs0110">[47,51,53,60]</ce:cross-refs> and do not affect the leading order corrections to vacuum stability.</ce:para><ce:para id="pr0070">We remark that the action <ce:cross-ref refid="fm0030" id="crf0050">(3)</ce:cross-ref> also appears in Einstein-Cartan gravity. A crucial distinction with the current case is that the Einstein-Cartan connection is decomposed using the Levi-Civita connection and torsion and, unlike in metric-affine gravity, the non-metricity is taken to be zero. Nevertheless, as we will show below, the metric-affine framework contains both the metric and Palatini formulations as limiting cases and the non-metricity may be taken to zero without loss of generality.</ce:para><ce:para id="pr0080">In order to study bounce solutions, we construct the Euclidean action <ce:cross-ref refid="fm0030" id="crf0060">(3)</ce:cross-ref> by analytically continuing the Lorentzian signature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mo stretchy="false">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> to the Euclidean one <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mo stretchy="false">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>. Then, to bring the action to a more conventional form, we will first integrate out the connection. To this aim, we will express the Ricci scalar and the Holst invariant in terms of the metric Ricci scalar <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mi>R</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:math> and the distortion tensor,<ce:display><ce:formula id="fm0050"><ce:label>(5a)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>R</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>λ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>λ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0060"><ce:label>(5b)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>ν</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msub><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>σ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>D</ce:italic> denotes the covariant derivative of the Levi-Civita connection. Substituting Eq. <ce:cross-ref refid="fm0050" id="crf0070">(5a)</ce:cross-ref> and <ce:cross-ref refid="fm0060" id="crf0080">(5b)</ce:cross-ref> into the action <ce:cross-ref refid="fm0030" id="crf0090">(3)</ce:cross-ref>, yields<ce:cross-ref refid="fn0020" id="crf0100"><ce:sup>2</ce:sup></ce:cross-ref><ce:footnote id="fn0020"><ce:label>2</ce:label><ce:note-para id="np0020">Although the Holst term picks up an imaginary unit when continuing to Euclidean space, analogously to the CP violating topological term in Yang-Mills theory (e.g., see Ref. <ce:cross-ref refid="br0610" id="crf0110">[61]</ce:cross-ref>), its effect is negated due to the dependence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mi mathvariant="normal">sign</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mn>4</mml:mn><mml:mo>!</mml:mo></mml:math> on the sign of the metric determinant.</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0070"><ce:label>(6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo id="mmlbr0001">∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>f</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>V</mml:mi><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="before">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>λ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>λ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="before">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>i</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>ν</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msub><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>σ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The distortion tensor obeys an algebraic non-homogeneous linear equation of motion. Thus, in order to integrate it out in full generality, it is sufficient to find a particular solution to this equation <ce:cross-ref refid="br0470" id="crf0120">[47]</ce:cross-ref>. Such a solution is given by <ce:cross-ref refid="br0490" id="crf0130">[49]</ce:cross-ref><ce:display><ce:formula id="fm0080"><ce:label>(7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mspace width="0.2em"/><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:msub><mml:mi>X</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>X</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>i</mml:mi><mml:msub><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:mi>Y</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where<ce:display><ce:formula id="fm0090"><ce:label>(8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:mi>f</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">cos</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">sin</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> This solution is metric compatible, i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>ρ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, and has torsion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>ν</mml:mi><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>ν</mml:mi><mml:mi>μ</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msub></mml:math>. So, the theory is dynamically equivalent to the Einstein-Cartan theory. On the other hand, the particular solution <ce:cross-ref refid="fm0080" id="crf0140">(7)</ce:cross-ref> is not the general one because of the projective symmetry of the action, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>ν</mml:mi><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>ν</mml:mi><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>, which can be used to induce the non-metricity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:msub></mml:math>. In particular, the Palatini limit with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> is obtained by choosing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>X</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math>.</ce:para><ce:para id="pr0090">Substituting <ce:cross-ref refid="fm0080" id="crf0150">(7)</ce:cross-ref> in the action <ce:cross-ref refid="fm0070" id="crf0160">(6)</ce:cross-ref> gives<ce:display><ce:formula id="fm0100"><ce:label>(9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>f</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>K</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where the contribution of the independent connection is now fully captured by the kinetic function<ce:cross-ref refid="fn0030" id="crf0170"><ce:sup>3</ce:sup></ce:cross-ref><ce:footnote id="fn0030"><ce:label>3</ce:label><ce:note-para id="np0030">The primes denote differentiation with respect to the argument of the function, that is, depending on the context, with respect to <ce:italic>h</ce:italic> or <ce:italic>r</ce:italic>.</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0110"><ce:label>(10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mi>K</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mi>f</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>f</mml:mi><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Note that the action <ce:cross-ref refid="fm0100" id="crf0180">(9)</ce:cross-ref> does not depend on the sign of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>.</ce:para><ce:para id="pr0100">For specific combinations of the involved functions, the general metric-affine theory interpolates continuously between the metric (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:mi>K</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:math>) and the Palatini (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:mi>K</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mi>f</mml:mi></mml:math>) theories. In accordance with Ref. <ce:cross-ref refid="br0440" id="crf0190">[44]</ce:cross-ref>, we find that these scenarios correspond to<ce:display><ce:formula id="fm0120"><ce:label>(11a)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>c</mml:mi><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mi>c</mml:mi></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mtext>(metric)</mml:mtext></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0130"><ce:label>(11b)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>c</mml:mi><mml:mi>f</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mspace width="2em"/><mml:mspace width="2em"/><mml:mtext>(Palatini)</mml:mtext></mml:mrow></mml:math></ce:formula></ce:display> with <ce:italic>c</ce:italic> a constant. As an important case, the metric formulation can be obtained in the limit in which the constant part of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math> is large. More specifically, without loss of generality we can consider <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>κ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>, <ce:italic>f</ce:italic> are arbitrary functions of <ce:italic>h</ce:italic>. If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mi>κ</mml:mi><mml:mo>≫</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>f</mml:mi></mml:math>, then<ce:display><ce:formula id="fm0140"><ce:label>(12)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mi>K</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> and thus the metric-affine theory approaches the purely metric theory when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mo>∞</mml:mo></mml:math>. With <ce:italic>f</ce:italic>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math> given by <ce:cross-ref refid="fm0040" id="crf0200">(4)</ce:cross-ref>, the Palatini formulation corresponds to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">/</mml:mo><mml:mi>ξ</mml:mi></mml:math>, of which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> is only a special case. Consequently, as <ce:italic>β</ce:italic> ranges from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo>∞</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">/</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mo>∞</mml:mo></mml:math>, the metric formulation is continuously deformed to the Palatini one and back.</ce:para></ce:section><ce:section id="se0030"><ce:label>3</ce:label><ce:section-title id="st0040">Corrections to vacuum decay in metric-affine gravity</ce:section-title><ce:para id="pr0110">To compute the minimal bounce action, we will look for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> symmetric solutions <ce:cross-ref refid="br0620" id="crf0210">[62]</ce:cross-ref>, with the line element <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Ω</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Ω</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> denotes the line element of the unit 3-sphere and the Higgs field depends only on the radial coordinate <ce:italic>r</ce:italic>, <ce:italic>i.e.</ce:italic> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:mi>h</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. In this background, the metric Ricci scalar reads <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:mi>R</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>6</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>ρ</mml:mi><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>″</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> and the Euclidean action can be recast as<ce:display><ce:formula id="fm0150"><ce:label>(13)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>r</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true" maxsize="6.6ex" minsize="6.6ex">[</mml:mo><mml:mn>3</mml:mn><mml:mi>f</mml:mi><mml:mspace width="0.2em"/><mml:mfrac><mml:mrow><mml:mi>ρ</mml:mi><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>″</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>K</mml:mi><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="true" maxsize="6.6ex" minsize="6.6ex">]</mml:mo><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The bounce solution is determined by the following equations of motion<ce:cross-ref refid="fn0040" id="crf0220"><ce:sup>4</ce:sup></ce:cross-ref><ce:footnote id="fn0040"><ce:label>4</ce:label><ce:note-para id="np0040">The <ce:italic>ρ</ce:italic> equations of motion follow from the <ce:italic>rr</ce:italic> component of the Einstein equations, but can also be derived by varying the action <ce:cross-ref refid="fm0150" id="crf0230">(13)</ce:cross-ref>.</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0160"><ce:label>(14a)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>f</mml:mi></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>K</mml:mi><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>V</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0170"><ce:label>(14b)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>″</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> To obtain the bounce action, we will adopt the perturbative method proposed in Ref. <ce:cross-ref refid="br0150" id="crf0240">[15]</ce:cross-ref> and look for solutions as a series in <ce:italic>κ</ce:italic>, <ce:italic>i.e.</ce:italic><ce:display><ce:formula id="fm0180"><ce:label>(15a)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>κ</mml:mi><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0190"><ce:label>(15b)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>κ</mml:mi><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> This approach is suitable when the gravitational corrections are relatively small, and, as we will demonstrate, this technique is adequate for elucidating the differences that emerge due to the inclusion of the Holst invariant. In a similar vein, the bounce action <ce:cross-ref refid="fm0150" id="crf0250">(13)</ce:cross-ref> can be expanded as<ce:display><ce:formula id="fm0200"><ce:label>(16)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>κ</mml:mi><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0120">The leading order solution<ce:cross-ref refid="fn0050" id="crf0260"><ce:sup>5</ce:sup></ce:cross-ref><ce:footnote id="fn0050"><ce:label>5</ce:label><ce:note-para id="np0050">See also <ce:cross-ref refid="br0630" id="crf0270">[63]</ce:cross-ref> for exact solutions for vacuum decay in Higgs-like unbounded potentials.</ce:note-para></ce:footnote> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> of <ce:cross-refs refid="fm0160 fm0170" id="crs0120">(14)</ce:cross-refs> is the so-called Fubini instanton <ce:cross-refs refid="br0640 br0650" id="crs0130">[64,65]</ce:cross-refs><ce:display><ce:formula id="fm0210"><ce:label>(17)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:mfrac><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> with <ce:italic>μ</ce:italic> being an arbitrary scale of the bounce. It solves the equation of motion in the absence of gravity (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>r</mml:mi></mml:math>) and with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>λ</mml:mi><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mn>4</mml:mn></mml:math> assuming that the Higgs quartic coupling <ce:italic>λ</ce:italic> is constant and negative. The leading order contribution to the action <ce:cross-ref refid="fm0150" id="crf0280">(13)</ce:cross-ref> is<ce:display><ce:formula id="fm0220"><ce:label>(18)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>r</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>8</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> and gives the bounce action in the absence of gravity. To obtain the gravitationally corrected action one must account for the running of <ce:italic>λ</ce:italic> <ce:cross-refs refid="br0170 br0190" id="crs0140">[17,19]</ce:cross-refs>. We will evaluate <ce:italic>λ</ce:italic> at the scale of the bounce <ce:italic>μ</ce:italic> and then minimize the action with respect to <ce:italic>μ</ce:italic>. The running of <ce:italic>λ</ce:italic> is computed at a 3-loop level <ce:cross-ref refid="br0110" id="crf0290">[11]</ce:cross-ref> with the relevant parameters taken from <ce:cross-ref refid="br0660" id="crf0300">[66]</ce:cross-ref>.</ce:para><ce:para id="pr0130">Evaluating the gravitational correction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> relies on the specific form of <ce:italic>f</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>, for which we will assume the form <ce:cross-ref refid="fm0040" id="crf0310">(4)</ce:cross-ref> when needed. We will assume that the leading order gravitational corrections to the kinetic function <ce:cross-ref refid="fm0110" id="crf0320">(10)</ce:cross-ref> can be expressed as<ce:display><ce:formula id="fm0230"><ce:label>(19)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:mi>K</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>κ</mml:mi><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> is a dimensionless constant. Indeed, with <ce:italic>f</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math> given by <ce:cross-ref refid="fm0040" id="crf0330">(4)</ce:cross-ref>, we have that<ce:display><ce:formula id="fm0240"><ce:label>(20)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:mi>β</mml:mi><mml:mi>ξ</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Note that for large <ce:italic>β</ce:italic> we recover Eq. <ce:cross-ref refid="fm0140" id="crf0340">(12)</ce:cross-ref>. At order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>κ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, the equation of motion <ce:cross-ref refid="fm0160" id="crf0350">(14a)</ce:cross-ref> is<ce:display><ce:formula id="fm0250"><ce:label>(21)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msubsup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> independently of the shape of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>. For the Fubini bounce <ce:cross-ref refid="fm0210" id="crf0360">(17)</ce:cross-ref>, it is solved by<ce:display><ce:formula id="fm0260"><ce:label>(22)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>r</mml:mi><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mi mathvariant="normal">arctan</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Knowing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> is sufficient to compute the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>κ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> correction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> to the action. We checked explicitly that the dependence on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> can be eliminated by the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> equations of motion and partial integration. This is a general result, however, because <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> minimizes the action in the absence of gravity and thus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>κ</mml:mi><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> <ce:cross-ref refid="br0190" id="crf0370">[19]</ce:cross-ref>. In all, we obtain that<ce:display><ce:formula id="fm0270"><ce:label>(23)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo id="mmlbr0002" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>32</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>45</mml:mn><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:mi>ξ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="before">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>32</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>45</mml:mn><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:mi>ξ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>36</mml:mn><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:mi>β</mml:mi><mml:mi>ξ</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> encodes the modifications resulting from an independent connection, <ce:italic>i.e.</ce:italic>, setting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> recovers the metric case.</ce:para><ce:para id="pr0140">The gravitational correction <ce:cross-ref refid="fm0270" id="crf0380">(23)</ce:cross-ref> can be negative for certain values of model parameters. If this happens, then the action cannot be minimized with respect to <ce:italic>μ</ce:italic> and the adopted perturbative approach is not applicable. However, by minimizing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> with respect to <ce:italic>ξ</ce:italic>, it is straightforward to show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> is always positive when<ce:display><ce:formula id="fm0280"><ce:label>(24)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:mi>β</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo>≥</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>12</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Otherwise, the positivity of the gravitational correction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> can be achieved only in certain regions of the parameter space. Two special cases warrant being considered more closely:<ce:list id="ls0010"><ce:list-item id="li0010"><ce:label>1.</ce:label><ce:para id="pr0150"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>: A minimally coupled Holst term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si79.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>κ</mml:mi></mml:math> is probably the simplest scenario. The region allowed by the positivity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> is<ce:display><ce:formula id="fm0290"><ce:label>(25)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mi>ξ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>≥</mml:mo><mml:mfrac><mml:mrow><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt></mml:mrow><mml:mrow><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si81.svg"><mml:mi>β</mml:mi><mml:mo>≪</mml:mo><mml:mn>1</mml:mn></mml:math>, this gives <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si82.svg"><mml:mi>ξ</mml:mi><mml:mo>≤</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi>ξ</mml:mi><mml:mo>≥</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>12</mml:mn></mml:math>, while, when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:mi>β</mml:mi><mml:mo>≫</mml:mo><mml:mn>1</mml:mn></mml:math>, only a narrow region around the conformal coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si85.svg"><mml:mi>ξ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>6</mml:mn></mml:math> is forbidden, that is, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si86.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>ξ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>6</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mo>≥</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>6</mml:mn><mml:mi>β</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>.</ce:para></ce:list-item><ce:list-item id="li0020"><ce:label>2.</ce:label><ce:para id="pr0160"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>: In this case, the contribution from the Holst term<ce:display><ce:formula id="fm0300"><ce:label>(26)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>32</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>45</mml:mn><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>12</mml:mn><mml:mi>ξ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>36</mml:mn><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> is always positive and will thus always improve the stability of the SM vacuum when compared to the Palatini case. This special case is depicted in the middle panel of <ce:cross-ref refid="fg0010" id="crf0390">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/> However, as in the Palatini limit, the positivity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> implies a strict lower bound<ce:display><ce:formula id="fm0310"><ce:label>(27)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si89.svg"><mml:mi>ξ</mml:mi><mml:mo>≥</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>12</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display></ce:para></ce:list-item></ce:list></ce:para><ce:para id="pr0170">The minimal bounce action is shown in <ce:cross-ref refid="fg0010" id="crf0400">Fig. 1</ce:cross-ref> with the running of <ce:italic>λ</ce:italic> computed at the 3-loop level <ce:cross-ref refid="br0110" id="crf0410">[11]</ce:cross-ref>. It shows that the metric limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si90.svg"><mml:mi>β</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mo>∞</mml:mo></mml:math> is realized quite well already for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>6</mml:mn></mml:math> when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si92.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo>≈</mml:mo><mml:mn>0</mml:mn></mml:math>. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si93.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo>≫</mml:mo><mml:mn>1</mml:mn></mml:math>, we see that the bounce action is typically enhanced when <ce:italic>β</ce:italic> and <ce:italic>ξ</ce:italic> have the opposite signs, thus improving the stability of the vacuum. Additionally, in comparison to the metric case, the stability is improved when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si94.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:math>. In all depicted cases, the regions in which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0</mml:mn></mml:math> can be observed: When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>6</mml:mn></mml:math>, this region exists only for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> line and is contained in a narrow range around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si97.svg"><mml:mi>ξ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>37</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>216</mml:mn></mml:math>. In the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn></mml:math> case, a parameter region with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0</mml:mn></mml:math> can be observed for every <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>. The disallowed <ce:italic>ξ</ce:italic> range varies with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>. Since the theory is independent of the sign of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>, then the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>6</mml:mn></mml:math> panel covers the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn></mml:math> case with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si99.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>100</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math> and vice versa.</ce:para><ce:para id="pr0180">Finally, it is important to point out that, as in the action <ce:cross-ref refid="fm0100" id="crf0420">(9)</ce:cross-ref>, the contributions from mass dimension 2 terms constructed from torsion and non-metricity that can appear in metric-affine gravity can be reduced to a non-canonical kinetic term in a metric theory <ce:cross-ref refid="br0590" id="crf0430">[59]</ce:cross-ref>. This implies that our results can be straightforwardly extended to include corrections to vacuum stability from such terms by computing their contribution to the small <ce:italic>κ</ce:italic> expansion <ce:cross-ref refid="fm0230" id="crf0440">(19)</ce:cross-ref>.</ce:para></ce:section><ce:section id="se0040" role="conclusion"><ce:label>4</ce:label><ce:section-title id="st0050">Conclusions</ce:section-title><ce:para id="pr0190">We analyzed the stability of the electroweak vacuum in metric-affine gravity, where the Higgs boson is expected to have an additional non-minimal coupling to the Holst invariant. This scenario can be reformulated in terms of an equivalent metric theory with a non-canonical kinetic term, where the gravitational corrections to the bounce action can be studied with established perturbative methods. Our results show that the stability of the electroweak vacuum in metric-affine gravity is improved across a wide range of model parameters.</ce:para><ce:para id="pr0200">A non-minimally coupled Holst term provides a class of models that continuously connects metric and Palatini gravity. We find that the limiting case of Palatini gravity displays the mildest improvement to vacuum stability.</ce:para> </ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0090">Declaration of Competing Interest</ce:section-title><ce:para id="pr0230">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0060">Acknowledgements</ce:section-title><ce:para id="pr0210">We thank Tomi Koivisto for useful comments. 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Workman et al., Review of Particle Physics, PTEP 2022 (2022) 083C01.</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> \ No newline at end of file +<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd" [<!ENTITY gr001 SYSTEM "gr001" NDATA IMAGE>]><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="sco" xml:lang="en"><item-info><jid>PLB</jid><aid>138109</aid><ce:article-number>138109</ce:article-number><ce:pii>S0370-2693(23)00443-4</ce:pii><ce:doi>10.1016/j.physletb.2023.138109</ce:doi><ce:copyright year="2023" type="other">The Author(s)</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>Theory</ce:text></ce:doctopic></ce:doctopics></item-info><ce:floats><ce:figure id="fg0010"><ce:label>Fig. 1</ce:label><ce:caption id="cp0010"><ce:simple-para id="sp0010">The minimal bounce action <ce:cross-ref refid="fm0200" id="crf0010">(16)</ce:cross-ref> for <ce:italic>β</ce:italic> = 0,±6 and various values of <ce:italic>ξ</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>. The horizontal line in brown indicates the bounce action in the absence of gravity, <ce:italic>S</ce:italic><ce:inf>0</ce:inf> = 8<ce:italic>π</ce:italic><ce:sup>2</ce:sup>/(3|<ce:italic>λ</ce:italic>(<ce:italic>μ</ce:italic>)|). The metric and Palatini cases are represented by the black-dashed and gray-dot-dashed curves, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0010">Fig. 1</ce:alt-text><ce:link locator="gr001" xlink:type="simple" xlink:href="pii:S0370269323004434/gr001" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0010"/></ce:figure></ce:floats><head><ce:title id="ti0010">Electroweak vacuum decay in metric-affine gravity</ce:title><ce:author-group id="ag0010"><ce:author orcid="0000-0002-2957-5276" id="au0010" author-id="S0370269323004434-957dd146f6e243df4b6f309334983b99"><ce:given-name>Ioannis D.</ce:given-name><ce:surname>Gialamas</ce:surname><ce:cross-ref refid="cr0010" id="crf0450"><ce:sup>⁎</ce:sup></ce:cross-ref><ce:e-address type="email" xlink:href="mailto:ioannis.gialamas@kbfi.ee" id="ea0010">ioannis.gialamas@kbfi.ee</ce:e-address></ce:author><ce:author id="au0020" author-id="S0370269323004434-11ce32aef4ff6906bb946b4904582471"><ce:given-name>Hardi</ce:given-name><ce:surname>Veermäe</ce:surname><ce:e-address type="email" xlink:href="mailto:hardi.veermae@cern.ch" id="ea0020">hardi.veermae@cern.ch</ce:e-address></ce:author><ce:affiliation id="aff0010" affiliation-id="S0370269323004434-e0efc38af768f3818fce829f251dffc5"><ce:textfn>Laboratory of High Energy and Computational Physics, National Institute of Chemical Physics and Biophysics, Rävala pst. 10, 10143, Tallinn, Estonia</ce:textfn><sa:affiliation><sa:organization>Laboratory of High Energy and Computational Physics</sa:organization><sa:organization>National Institute of Chemical Physics and Biophysics</sa:organization><sa:address-line>Rävala pst. 10</sa:address-line><sa:city>Tallinn</sa:city><sa:postal-code>10143</sa:postal-code><sa:country>Estonia</sa:country></sa:affiliation><ce:source-text id="srct0005">Laboratory of High Energy and Computational Physics, National Institute of Chemical Physics and Biophysics, Rävala pst. 10, 10143, Tallinn, Estonia</ce:source-text></ce:affiliation><ce:correspondence id="cr0010"><ce:label>⁎</ce:label><ce:text>Corresponding author.</ce:text></ce:correspondence></ce:author-group><ce:date-received day="1" month="6" year="2023"/><ce:date-revised day="12" month="7" year="2023"/><ce:date-accepted day="27" month="7" year="2023"/><ce:miscellaneous id="ms0010">Editor: R. Gregory</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0020">We investigate the stability of the electroweak vacuum in metric-affine gravity in which the Standard Model Higgs boson can be non-minimally coupled to both the Ricci scalar and the Holst invariant. We find that vacuum stability is improved in this framework across a wide range of model parameters.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0080">Data availability</ce:section-title><ce:para id="pr0220">No data was used for the research described in the article.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">It is well known that the potential of the Higgs boson in the Standard Model (SM) is deeper at high energies than in the electroweak vacuum permitting its decay through quantum tunneling <ce:cross-refs refid="br0010 br0020 br0030 br0040" id="crs0010">[1–4]</ce:cross-refs>. Although this does not invalidate the SM, the electroweak vacuum is predicted to be metastable in the absence of contributions from UV physics <ce:cross-refs refid="br0050 br0060 br0070 br0080 br0090 br0100 br0110 br0120 br0130" id="crs0020">[5–13]</ce:cross-refs>.</ce:para><ce:para id="pr0020">Coleman and De Luccia <ce:cross-ref refid="br0140" id="crf0020">[14]</ce:cross-ref> were the first to delve into the matter of gravitational effects on vacuum decay. Subsequently, multiple studies of gravitational corrections have been performed <ce:cross-refs refid="br0150 br0160 br0170 br0180 br0190 br0200 br0210 br0220 br0230" id="crs0030">[15–23]</ce:cross-refs>, along with discussions about the impact of black holes on the amplification or reduction of the vacuum decay rate <ce:cross-refs refid="br0240 br0250 br0260 br0270 br0280 br0290 br0300 br0310 br0320 br0330 br0340 br0350 br0360 br0370 br0380 br0390 br0400" id="crs0040">[24–40]</ce:cross-refs>.</ce:para><ce:para id="pr0030">In this letter, we extend the calculation of gravitational corrections to vacuum decay in the context of metric-affine gravity. There, in contrast to general relativity, the connection is taken to be an independent variable without the usual symmetries of the Levi-Civita one. As a result, the Riemann tensor does not possess the symmetries it has in the metric case, and thus the gravitational action should be extended by including an additional scalar curvature invariant - the Holst invariant. This term commonly appears in Loop Quantum Gravity <ce:cross-ref refid="br0410" id="crf0030">[41]</ce:cross-ref> and has been studied in various branches of high energy physics, such as black hole thermodynamics <ce:cross-refs refid="br0420 br0430" id="crs0050">[42,43]</ce:cross-refs>. Lately, its significance in inflationary cosmology <ce:cross-refs refid="br0440 br0450 br0460 br0470 br0480 br0490" id="crs0060">[44–49]</ce:cross-refs>, and high energy physics phenomenology <ce:cross-refs refid="br0500 br0510 br0520 br0530 br0540" id="crs0070">[50–54]</ce:cross-refs>, has garnered a great deal of attention.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">The gravitational action</ce:section-title><ce:para id="pr0040">Metric-affine theories of gravity treat the metric tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> and the connection <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts></mml:math> as independent variables. This should be contrasted with the usual metric gravity that uses the Levi-Civita connection <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mo stretchy="true">{</mml:mo><mml:mmultiscripts><mml:mrow/><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/><mml:mprescripts/><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow></mml:mmultiscripts><mml:mo stretchy="true">}</mml:mo></mml:math>, which is completely determined by the metric. It is useful to decompose the connection <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts></mml:math> as<ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mo>≡</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mmultiscripts><mml:mrow/><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/><mml:mprescripts/><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow></mml:mmultiscripts><mml:mo stretchy="true">}</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts></mml:math> is dubbed the distortion tensor. The Riemann tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts></mml:math> is constructed from the connection Γ in the usual way and one can form two scalars that are linear on it. These are the Ricci scalar and the Holst invariant <ce:cross-refs refid="br0550 br0560" id="crs0080">[55,56]</ce:cross-refs><ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:mi mathvariant="script">R</mml:mi><mml:mo>≡</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>g</ce:italic> is the determinant of the metric tensor and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:msup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msup></mml:math> is the totally antisymmetric tensor. In metric gravity, the Holst invariant vanishes identically due to the symmetries of the Riemann tensor.</ce:para><ce:para id="pr0050">The most general action linear in the Riemann tensor and containing terms of at most dimension 4 has the form<ce:cross-ref refid="fn0010" id="crf0040"><ce:sup>1</ce:sup></ce:cross-ref><ce:footnote id="fn0010"><ce:label>1</ce:label><ce:note-para id="np0010">Natural units with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:mi>c</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>ħ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:math> are used throughout this paper.</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0030"><ce:label>(3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:mi>S</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is the Higgs potential,<ce:display><ce:formula id="fm0040"><ce:label>(4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mi>κ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>ξ</mml:mi><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>κ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> are non-minimal couplings, <ce:italic>β</ce:italic>, <ce:italic>ξ</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math> are constant couplings, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mi>κ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Pl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>. The function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> can be thought of as a field-dependent Barbero-Immirzi parameter <ce:cross-refs refid="br0560 br0570" id="crs0090">[56,57]</ce:cross-refs>. We will first work out the formalism without assuming a specific functional form of <ce:italic>f</ce:italic>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>, and <ce:italic>V</ce:italic> and suppress their arguments for notational brevity.</ce:para><ce:para id="pr0060">In addition to the Ricci and Holst terms, metric-affine gravity permits the construction of 20 additional scalars with mass dimension 2 from torsion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msub></mml:math> and non-metricity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">∇</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math> <ce:cross-refs refid="br0580 br0590" id="crs0100">[58,59]</ce:cross-refs>. Although such terms are not considered in this work, our perturbative results can be straightforwardly extended to include them as will be explained below. We will also neglect couplings between the connection and fermions as they will only generate Planck-suppressed four-fermion and higher-order scalar-fermion interactions <ce:cross-refs refid="br0470 br0510 br0530 br0600" id="crs0110">[47,51,53,60]</ce:cross-refs> and do not affect the leading order corrections to vacuum stability.</ce:para><ce:para id="pr0070">We remark that the action <ce:cross-ref refid="fm0030" id="crf0050">(3)</ce:cross-ref> also appears in Einstein-Cartan gravity. A crucial distinction with the current case is that the Einstein-Cartan connection is decomposed using the Levi-Civita connection and torsion and, unlike in metric-affine gravity, the non-metricity is taken to be zero. Nevertheless, as we will show below, the metric-affine framework contains both the metric and Palatini formulations as limiting cases and the non-metricity may be taken to zero without loss of generality.</ce:para><ce:para id="pr0080">In order to study bounce solutions, we construct the Euclidean action <ce:cross-ref refid="fm0030" id="crf0060">(3)</ce:cross-ref> by analytically continuing the Lorentzian signature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mo stretchy="false">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> to the Euclidean one <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mo stretchy="false">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo>,</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>. Then, to bring the action to a more conventional form, we will first integrate out the connection. To this aim, we will express the Ricci scalar and the Holst invariant in terms of the metric Ricci scalar <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mi>R</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:math> and the distortion tensor,<ce:display><ce:formula id="fm0050"><ce:label>(5a)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>R</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>λ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>λ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0060"><ce:label>(5b)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mo>˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>ν</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msub><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>σ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>D</ce:italic> denotes the covariant derivative of the Levi-Civita connection. Substituting Eq. <ce:cross-ref refid="fm0050" id="crf0070">(5a)</ce:cross-ref> and <ce:cross-ref refid="fm0060" id="crf0080">(5b)</ce:cross-ref> into the action <ce:cross-ref refid="fm0030" id="crf0090">(3)</ce:cross-ref>, yields<ce:cross-ref refid="fn0020" id="crf0100"><ce:sup>2</ce:sup></ce:cross-ref><ce:footnote id="fn0020"><ce:label>2</ce:label><ce:note-para id="np0020">Although the Holst term picks up an imaginary unit when continuing to Euclidean space, analogously to the CP violating topological term in Yang-Mills theory (e.g., see Ref. <ce:cross-ref refid="br0610" id="crf0110">[61]</ce:cross-ref>), its effect is negated due to the dependence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:msup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mi mathvariant="normal">sign</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mn>4</mml:mn><mml:mo>!</mml:mo></mml:math> on the sign of the metric determinant.</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0070"><ce:label>(6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo id="mmlbr0001">∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">[</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>f</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>V</mml:mi><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="before">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>λ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>λ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="before">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>i</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>ν</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msub><mml:mmultiscripts><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:none/><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>σ</mml:mi></mml:mrow><mml:none/></mml:mmultiscripts><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">]</mml:mo><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The distortion tensor obeys an algebraic non-homogeneous linear equation of motion. Thus, in order to integrate it out in full generality, it is sufficient to find a particular solution to this equation <ce:cross-ref refid="br0470" id="crf0120">[47]</ce:cross-ref>. Such a solution is given by <ce:cross-ref refid="br0490" id="crf0130">[49]</ce:cross-ref><ce:display><ce:formula id="fm0080"><ce:label>(7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.2em"/><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mspace width="0.2em"/><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:msub><mml:mi>X</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>X</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>i</mml:mi><mml:msub><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:mi>Y</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where<ce:display><ce:formula id="fm0090"><ce:label>(8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:mi>f</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">cos</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">sin</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> This solution is metric compatible, i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ν</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>ρ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>, and has torsion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>ν</mml:mi><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>ν</mml:mi><mml:mi>μ</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msub></mml:math>. So, the theory is dynamically equivalent to the Einstein-Cartan theory. On the other hand, the particular solution <ce:cross-ref refid="fm0080" id="crf0140">(7)</ce:cross-ref> is not the general one because of the projective symmetry of the action, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>ν</mml:mi><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>ν</mml:mi><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>, which can be used to induce the non-metricity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:msub></mml:math>. In particular, the Palatini limit with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> is obtained by choosing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>X</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math>.</ce:para><ce:para id="pr0090">Substituting <ce:cross-ref refid="fm0080" id="crf0150">(7)</ce:cross-ref> in the action <ce:cross-ref refid="fm0070" id="crf0160">(6)</ce:cross-ref> gives<ce:display><ce:formula id="fm0100"><ce:label>(9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:msqrt><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>f</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>K</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∂</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where the contribution of the independent connection is now fully captured by the kinetic function<ce:cross-ref refid="fn0030" id="crf0170"><ce:sup>3</ce:sup></ce:cross-ref><ce:footnote id="fn0030"><ce:label>3</ce:label><ce:note-para id="np0030">The primes denote differentiation with respect to the argument of the function, that is, depending on the context, with respect to <ce:italic>h</ce:italic> or <ce:italic>r</ce:italic>.</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0110"><ce:label>(10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mi>K</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mi>f</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>f</mml:mi><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Note that the action <ce:cross-ref refid="fm0100" id="crf0180">(9)</ce:cross-ref> does not depend on the sign of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>.</ce:para><ce:para id="pr0100">For specific combinations of the involved functions, the general metric-affine theory interpolates continuously between the metric (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:mi>K</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn></mml:math>) and the Palatini (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:mi>K</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mi>f</mml:mi></mml:math>) theories. In accordance with Ref. <ce:cross-ref refid="br0440" id="crf0190">[44]</ce:cross-ref>, we find that these scenarios correspond to<ce:display><ce:formula id="fm0120"><ce:label>(11a)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>c</mml:mi><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mi>c</mml:mi></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="2em"/><mml:mtext>(metric)</mml:mtext></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0130"><ce:label>(11b)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>c</mml:mi><mml:mi>f</mml:mi><mml:mspace width="0.2em"/><mml:mo>,</mml:mo><mml:mspace width="0.2em"/><mml:mspace width="2em"/><mml:mspace width="2em"/><mml:mtext>(Palatini)</mml:mtext></mml:mrow></mml:math></ce:formula></ce:display> with <ce:italic>c</ce:italic> a constant. As an important case, the metric formulation can be obtained in the limit in which the constant part of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math> is large. More specifically, without loss of generality we can consider <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>κ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>, <ce:italic>f</ce:italic> are arbitrary functions of <ce:italic>h</ce:italic>. If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mi>κ</mml:mi><mml:mo>≫</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>f</mml:mi></mml:math>, then<ce:display><ce:formula id="fm0140"><ce:label>(12)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mi>K</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> and thus the metric-affine theory approaches the purely metric theory when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mo>∞</mml:mo></mml:math>. With <ce:italic>f</ce:italic>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math> given by <ce:cross-ref refid="fm0040" id="crf0200">(4)</ce:cross-ref>, the Palatini formulation corresponds to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">/</mml:mo><mml:mi>ξ</mml:mi></mml:math>, of which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> is only a special case. Consequently, as <ce:italic>β</ce:italic> ranges from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo>∞</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">/</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mo>∞</mml:mo></mml:math>, the metric formulation is continuously deformed to the Palatini one and back.</ce:para></ce:section><ce:section id="se0030"><ce:label>3</ce:label><ce:section-title id="st0040">Corrections to vacuum decay in metric-affine gravity</ce:section-title><ce:para id="pr0110">To compute the minimal bounce action, we will look for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> symmetric solutions <ce:cross-ref refid="br0620" id="crf0210">[62]</ce:cross-ref>, with the line element <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Ω</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Ω</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> denotes the line element of the unit 3-sphere and the Higgs field depends only on the radial coordinate <ce:italic>r</ce:italic>, <ce:italic>i.e.</ce:italic> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:mi>h</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. In this background, the metric Ricci scalar reads <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:mi>R</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>6</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>ρ</mml:mi><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>″</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> and the Euclidean action can be recast as<ce:display><ce:formula id="fm0150"><ce:label>(13)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>r</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true" maxsize="6.6ex" minsize="6.6ex">[</mml:mo><mml:mn>3</mml:mn><mml:mi>f</mml:mi><mml:mspace width="0.2em"/><mml:mfrac><mml:mrow><mml:mi>ρ</mml:mi><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>″</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>K</mml:mi><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="true" maxsize="6.6ex" minsize="6.6ex">]</mml:mo><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The bounce solution is determined by the following equations of motion<ce:cross-ref refid="fn0040" id="crf0220"><ce:sup>4</ce:sup></ce:cross-ref><ce:footnote id="fn0040"><ce:label>4</ce:label><ce:note-para id="np0040">The <ce:italic>ρ</ce:italic> equations of motion follow from the <ce:italic>rr</ce:italic> component of the Einstein equations, but can also be derived by varying the action <ce:cross-ref refid="fm0150" id="crf0230">(13)</ce:cross-ref>.</ce:note-para></ce:footnote><ce:display><ce:formula id="fm0160"><ce:label>(14a)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>f</mml:mi></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>K</mml:mi><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>V</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0170"><ce:label>(14b)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>″</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> To obtain the bounce action, we will adopt the perturbative method proposed in Ref. <ce:cross-ref refid="br0150" id="crf0240">[15]</ce:cross-ref> and look for solutions as a series in <ce:italic>κ</ce:italic>, <ce:italic>i.e.</ce:italic><ce:display><ce:formula id="fm0180"><ce:label>(15a)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>κ</mml:mi><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0190"><ce:label>(15b)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>κ</mml:mi><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> This approach is suitable when the gravitational corrections are relatively small, and, as we will demonstrate, this technique is adequate for elucidating the differences that emerge due to the inclusion of the Holst invariant. In a similar vein, the bounce action <ce:cross-ref refid="fm0150" id="crf0250">(13)</ce:cross-ref> can be expanded as<ce:display><ce:formula id="fm0200"><ce:label>(16)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>κ</mml:mi><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0120">The leading order solution<ce:cross-ref refid="fn0050" id="crf0260"><ce:sup>5</ce:sup></ce:cross-ref><ce:footnote id="fn0050"><ce:label>5</ce:label><ce:note-para id="np0050">See also <ce:cross-ref refid="br0630" id="crf0270">[63]</ce:cross-ref> for exact solutions for vacuum decay in Higgs-like unbounded potentials.</ce:note-para></ce:footnote> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> of <ce:cross-refs refid="fm0160 fm0170" id="crs0120">(14)</ce:cross-refs> is the so-called Fubini instanton <ce:cross-refs refid="br0640 br0650" id="crs0130">[64,65]</ce:cross-refs><ce:display><ce:formula id="fm0210"><ce:label>(17)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msqrt><mml:mrow><mml:mfrac><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> with <ce:italic>μ</ce:italic> being an arbitrary scale of the bounce. It solves the equation of motion in the absence of gravity (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>r</mml:mi></mml:math>) and with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>λ</mml:mi><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mn>4</mml:mn></mml:math> assuming that the Higgs quartic coupling <ce:italic>λ</ce:italic> is constant and negative. The leading order contribution to the action <ce:cross-ref refid="fm0150" id="crf0280">(13)</ce:cross-ref> is<ce:display><ce:formula id="fm0220"><ce:label>(18)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>r</mml:mi><mml:mspace width="0.2em"/><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>8</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> and gives the bounce action in the absence of gravity. To obtain the gravitationally corrected action one must account for the running of <ce:italic>λ</ce:italic> <ce:cross-refs refid="br0170 br0190" id="crs0140">[17,19]</ce:cross-refs>. We will evaluate <ce:italic>λ</ce:italic> at the scale of the bounce <ce:italic>μ</ce:italic> and then minimize the action with respect to <ce:italic>μ</ce:italic>. The running of <ce:italic>λ</ce:italic> is computed at a 3-loop level <ce:cross-ref refid="br0110" id="crf0290">[11]</ce:cross-ref> with the relevant parameters taken from <ce:cross-ref refid="br0660" id="crf0300">[66]</ce:cross-ref>.</ce:para><ce:para id="pr0130">Evaluating the gravitational correction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> relies on the specific form of <ce:italic>f</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>, for which we will assume the form <ce:cross-ref refid="fm0040" id="crf0310">(4)</ce:cross-ref> when needed. We will assume that the leading order gravitational corrections to the kinetic function <ce:cross-ref refid="fm0110" id="crf0320">(10)</ce:cross-ref> can be expressed as<ce:display><ce:formula id="fm0230"><ce:label>(19)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:mi>K</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>κ</mml:mi><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> is a dimensionless constant. Indeed, with <ce:italic>f</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math> given by <ce:cross-ref refid="fm0040" id="crf0330">(4)</ce:cross-ref>, we have that<ce:display><ce:formula id="fm0240"><ce:label>(20)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:mi>β</mml:mi><mml:mi>ξ</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Note that for large <ce:italic>β</ce:italic> we recover Eq. <ce:cross-ref refid="fm0140" id="crf0340">(12)</ce:cross-ref>. At order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>κ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, the equation of motion <ce:cross-ref refid="fm0160" id="crf0350">(14a)</ce:cross-ref> is<ce:display><ce:formula id="fm0250"><ce:label>(21)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msubsup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mspace width="0.2em"/><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> independently of the shape of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>. For the Fubini bounce <ce:cross-ref refid="fm0210" id="crf0360">(17)</ce:cross-ref>, it is solved by<ce:display><ce:formula id="fm0260"><ce:label>(22)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>r</mml:mi><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mi mathvariant="normal">arctan</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Knowing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> is sufficient to compute the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>κ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> correction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> to the action. We checked explicitly that the dependence on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> can be eliminated by the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> equations of motion and partial integration. This is a general result, however, because <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> minimizes the action in the absence of gravity and thus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>κ</mml:mi><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>κ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> <ce:cross-ref refid="br0190" id="crf0370">[19]</ce:cross-ref>. In all, we obtain that<ce:display><ce:formula id="fm0270"><ce:label>(23)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo id="mmlbr0002" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>32</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>45</mml:mn><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:mi>ξ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="before">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>32</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>45</mml:mn><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>6</mml:mn><mml:mi>ξ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>36</mml:mn><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:mi>β</mml:mi><mml:mi>ξ</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> encodes the modifications resulting from an independent connection, <ce:italic>i.e.</ce:italic>, setting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> recovers the metric case.</ce:para><ce:para id="pr0140">The gravitational correction <ce:cross-ref refid="fm0270" id="crf0380">(23)</ce:cross-ref> can be negative for certain values of model parameters. If this happens, then the action cannot be minimized with respect to <ce:italic>μ</ce:italic> and the adopted perturbative approach is not applicable. However, by minimizing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> with respect to <ce:italic>ξ</ce:italic>, it is straightforward to show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> is always positive when<ce:display><ce:formula id="fm0280"><ce:label>(24)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:mi>β</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo>≥</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>12</mml:mn><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Otherwise, the positivity of the gravitational correction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> can be achieved only in certain regions of the parameter space. Two special cases warrant being considered more closely:<ce:list id="ls0010"><ce:list-item id="li0010"><ce:label>1.</ce:label><ce:para id="pr0150"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>: A minimally coupled Holst term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si79.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>κ</mml:mi></mml:math> is probably the simplest scenario. The region allowed by the positivity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> is<ce:display><ce:formula id="fm0290"><ce:label>(25)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mi>ξ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>≥</mml:mo><mml:mfrac><mml:mrow><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt></mml:mrow><mml:mrow><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si81.svg"><mml:mi>β</mml:mi><mml:mo>≪</mml:mo><mml:mn>1</mml:mn></mml:math>, this gives <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si82.svg"><mml:mi>ξ</mml:mi><mml:mo>≤</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi>ξ</mml:mi><mml:mo>≥</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>12</mml:mn></mml:math>, while, when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:mi>β</mml:mi><mml:mo>≫</mml:mo><mml:mn>1</mml:mn></mml:math>, only a narrow region around the conformal coupling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si85.svg"><mml:mi>ξ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>6</mml:mn></mml:math> is forbidden, that is, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si86.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>ξ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>6</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mo>≥</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>6</mml:mn><mml:mi>β</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>.</ce:para></ce:list-item><ce:list-item id="li0020"><ce:label>2.</ce:label><ce:para id="pr0160"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>: In this case, the contribution from the Holst term<ce:display><ce:formula id="fm0300"><ce:label>(26)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>32</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>45</mml:mn><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>12</mml:mn><mml:mi>ξ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>36</mml:mn><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> is always positive and will thus always improve the stability of the SM vacuum when compared to the Palatini case. This special case is depicted in the middle panel of <ce:cross-ref refid="fg0010" id="crf0390">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/> However, as in the Palatini limit, the positivity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> implies a strict lower bound<ce:display><ce:formula id="fm0310"><ce:label>(27)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si89.svg"><mml:mi>ξ</mml:mi><mml:mo>≥</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>12</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mspace width="0.2em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display></ce:para></ce:list-item></ce:list></ce:para><ce:para id="pr0170">The minimal bounce action is shown in <ce:cross-ref refid="fg0010" id="crf0400">Fig. 1</ce:cross-ref> with the running of <ce:italic>λ</ce:italic> computed at the 3-loop level <ce:cross-ref refid="br0110" id="crf0410">[11]</ce:cross-ref>. It shows that the metric limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si90.svg"><mml:mi>β</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mo>∞</mml:mo></mml:math> is realized quite well already for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si91.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>6</mml:mn></mml:math> when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si92.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo>≈</mml:mo><mml:mn>0</mml:mn></mml:math>. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si93.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo>≫</mml:mo><mml:mn>1</mml:mn></mml:math>, we see that the bounce action is typically enhanced when <ce:italic>β</ce:italic> and <ce:italic>ξ</ce:italic> have the opposite signs, thus improving the stability of the vacuum. Additionally, in comparison to the metric case, the stability is improved when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si94.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:math>. In all depicted cases, the regions in which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0</mml:mn></mml:math> can be observed: When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>6</mml:mn></mml:math>, this region exists only for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> line and is contained in a narrow range around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si97.svg"><mml:mi>ξ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>37</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>216</mml:mn></mml:math>. In the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn></mml:math> case, a parameter region with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si95.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0</mml:mn></mml:math> can be observed for every <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>. The disallowed <ce:italic>ξ</ce:italic> range varies with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>. Since the theory is independent of the sign of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mover accent="true"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:math>, then the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si96.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>6</mml:mn></mml:math> panel covers the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si98.svg"><mml:mi>β</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn></mml:math> case with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si99.svg"><mml:mover accent="true"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>100</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math> and vice versa.</ce:para><ce:para id="pr0180">Finally, it is important to point out that, as in the action <ce:cross-ref refid="fm0100" id="crf0420">(9)</ce:cross-ref>, the contributions from mass dimension 2 terms constructed from torsion and non-metricity that can appear in metric-affine gravity can be reduced to a non-canonical kinetic term in a metric theory <ce:cross-ref refid="br0590" id="crf0430">[59]</ce:cross-ref>. This implies that our results can be straightforwardly extended to include corrections to vacuum stability from such terms by computing their contribution to the small <ce:italic>κ</ce:italic> expansion <ce:cross-ref refid="fm0230" id="crf0440">(19)</ce:cross-ref>.</ce:para></ce:section><ce:section id="se0040" role="conclusion"><ce:label>4</ce:label><ce:section-title id="st0050">Conclusions</ce:section-title><ce:para id="pr0190">We analyzed the stability of the electroweak vacuum in metric-affine gravity, where the Higgs boson is expected to have an additional non-minimal coupling to the Holst invariant. This scenario can be reformulated in terms of an equivalent metric theory with a non-canonical kinetic term, where the gravitational corrections to the bounce action can be studied with established perturbative methods. Our results show that the stability of the electroweak vacuum in metric-affine gravity is improved across a wide range of model parameters.</ce:para><ce:para id="pr0200">A non-minimally coupled Holst term provides a class of models that continuously connects metric and Palatini gravity. We find that the limiting case of Palatini gravity displays the mildest improvement to vacuum stability.</ce:para> </ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0090">Declaration of Competing Interest</ce:section-title><ce:para id="pr0230">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0060">Acknowledgements</ce:section-title><ce:para id="pr0210">We thank Tomi Koivisto for useful comments. 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Gregory</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0010">One of the leading issues in quantum field theory and cosmology is the mismatch between the observed and calculated values for the cosmological constant in Einstein's field equations of up to 120 orders of magnitude. In this paper, we discuss new methods to potentially bridge this chasm using the generalized uncertainty principle (GUP). We find that if quantum gravity GUP models are the solution to this puzzle, then it may require the gravitationally modified position operator undergoes a parity transformation at high energies.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0090">Data availability</ce:section-title><ce:para id="pr0280">No data was used for the research described in the article.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010"><ce:label>1</ce:label><ce:section-title id="st0020">Cosmological constant puzzle</ce:section-title><ce:para id="pr0010">The cosmological constant problem is the naive mismatch by ∼ 120 orders of magnitude between the observed and a simple theoretical value for the cosmological constant in Einstein's field equations; we will clarify in what sense this 120 order of magnitude mismatch is misleading/naive below. Quantum gravity has long been advertised as a potential solution to various puzzles like this cosmological constant problem. In this paper, we lay out the reasoning that led us to the conclusion that if GUP models are to resolve the cosmological constant problem, they would require the gravitationally modified position operator to undergo a parity transformation at high energy scales or short distance scales.</ce:para><ce:para id="pr0020">GUP models modify the Heisenberg uncertainty principle to obtain a minimum length scale; that is, a positive lower bound on the uncertainty in position. GUP models are not elegant, top-down theories of quantum gravity like string theory <ce:cross-ref refid="br0010" id="crf0050">[1]</ce:cross-ref> or loop quantum gravity <ce:cross-ref refid="br0020" id="crf0060">[2]</ce:cross-ref>, but they have the advantage of being easy to work with and provide a phenomenological window into quantum gravity. We will begin by briefly reviewing the salient features of the cosmological constant problem; a more complete discussion can be found in the excellent review by Weinberg <ce:cross-ref refid="br0030" id="crf0070">[3]</ce:cross-ref>. We will adopt the notation and units of <ce:cross-ref refid="br0030" id="crf0080">[3]</ce:cross-ref>.</ce:para><ce:para id="pr0030">The cosmological constant, <ce:italic>λ</ce:italic>, in Einstein's field equations (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mi>G</mml:mi><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>λ</mml:mi><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>) is equivalent to space-time having a constant energy density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mi>G</mml:mi></mml:mrow></mml:mfrac></mml:math>, and is representative of the expansion of space. The approach for the calculation of <ce:italic>λ</ce:italic> in QFT is to add up all the energies of the zero modes (vacuum modes) of quantum fields. The vacuum modes of quantum fields are given by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>ħ</mml:mi><mml:msub><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt></mml:math>. Integrating over all possible momenta up to some cut-off, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, yields the vacuum energy density<ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo id="mmlbr0001" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:munderover><mml:mo movablelimits="false">∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mi>d</mml:mi><mml:mi>p</mml:mi><mml:mspace width="0.25em"/><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="before">≈</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>16</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Here, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:mi>p</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:math> is the magnitude of the momentum. We use <ce:italic>p</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:math> interchangeably throughout this work.</ce:para><ce:para id="pr0040">The integral in <ce:cross-ref refid="fm0010" id="crf0090">(1)</ce:cross-ref> is divergent, so it needs to be cut off at some momentum/energy scale, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, which is usually taken to be the Planck scale, <ce:italic>i.e.</ce:italic> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. Inserting this value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> into <ce:cross-ref refid="fm0010" id="crf0100">(1)</ce:cross-ref> gives <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mn>71</mml:mn></mml:mrow></mml:msup></mml:math> GeV<ce:sup>4</ce:sup>. The observed vacuum energy density <ce:cross-ref refid="br0040" id="crf0110">[4]</ce:cross-ref> is about <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mo>≈</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>47</mml:mn></mml:mrow></mml:msup></mml:math> GeV<ce:sup>4</ce:sup>, which is a difference of 118 orders of magnitude - a terrible disagreement between theory and observation. As mentioned in the opening paragraph this approximately 120 orders of magnitude mismatch is somewhat naive. First, the hard cut-off of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> in <ce:cross-ref refid="fm0010" id="crf0120">(1)</ce:cross-ref>, leading to a quartic dependence on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, violates relativistic invariance which requires <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>p</mml:mi></mml:math> <ce:italic>i.e.</ce:italic> energy density equals the negative pressure. This was pointed in <ce:cross-refs refid="br0050 br0060" id="crs0010">[5,6]</ce:cross-refs>, and these two works also show that if one uses dimensional regularization of integrals like <ce:cross-ref refid="fm0010" id="crf0130">(1)</ce:cross-ref>, one can restore relativistic invariance, and the energy density is no longer quartically dependent on cut-off scale, but has a logarithmic dependence. Finally, in <ce:cross-ref refid="br0060" id="crf0140">[6]</ce:cross-ref> it was shown that by using dimensional regularization of integrals like those in <ce:cross-ref refid="fm0010" id="crf0150">(1)</ce:cross-ref> and using known Standard Model fields (specifically the top quark, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> bosons and Higgs boson with an estimated mass of 150 GeV) gave a disagreement between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mo>≈</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>47</mml:mn></mml:mrow></mml:msup></mml:math> GeV<ce:sup>4</ce:sup> and the theoretical value of “only” 56 orders of magnitude. This is still terrible, but essentially cuts in half the order of magnitude disagreement, from ∼ 120 to ∼ 56. This estimate of <ce:cross-ref refid="br0060" id="crf0160">[6]</ce:cross-ref> is more realistic since the only scale probed experimentally is the electroweak scale of ∼ 100 GeV. Using the Planck scale gives an overestimate of the disagreement. Nevertheless using either the electroweak scale or the Planck scale gives a huge discrepancy between the theoretical and observed value of the vacuum energy density.</ce:para><ce:para id="pr0050">One of the standard ideas for addressing the cosmological constant problem is via <ce:italic>unbroken supersymmetry</ce:italic><ce:cross-ref refid="fn0010" id="crf0170"><ce:sup>1</ce:sup></ce:cross-ref><ce:footnote id="fn0010"><ce:label>1</ce:label><ce:note-para id="np0010">In unbroken supersymmetry particles and their superpartner particles have the same mass.</ce:note-para></ce:footnote> (SUSY) models <ce:cross-refs refid="br0030 br0070" id="crs0020">[3,7]</ce:cross-refs>. SUSY models have equal numbers of bosonic and fermionic fields/degrees of freedom. Since bosonic and fermionic vacuum modes have opposite signs, their vacuum energies cancel one another exactly in unbroken SUSY and one would have a natural explanation for a cosmological constant that is exactly zero. However, one does not want an exactly zero cosmological constant and SUSY <ce:italic>is</ce:italic> broken at least up to some high energy scale greater than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:msub><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mi>S</mml:mi><mml:mi>Y</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> GeV. But if one used the lower limit of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msub><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mi>S</mml:mi><mml:mi>Y</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> GeV as the cut-off <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, then the disagreement between theory and experiment is ∼ 60 orders of magnitude. Therefore, simple SUSY models do not provide an answer to the cosmological constant puzzle.</ce:para><ce:para id="pr0060">However, despite SUSY (either broken or unbroken) not providing a solution to the cosmological constant puzzle, it nevertheless has the feature that one obtains a small/zero cosmological constant by a cancellation of large positive and negative contributions. In the GUP approach to the cosmological constant problem presented here, we find that we can similarly get a small cosmological constant by having a modified position operator which flips parity at some large energy/momentum scale. This changing of the parity of the modified position operator leads to a cancellation of large positive and large negative contributions to the cosmological constant. Such violations of parity in the gravitational interaction were studied theoretically in <ce:cross-ref refid="br0080" id="crf0180">[8]</ce:cross-ref> and recently experimental bounds have been placed on such parity violations in gravity <ce:cross-ref refid="br0090" id="crf0190">[9]</ce:cross-ref>.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">Quantum gravity via generalized uncertainty principle</ce:section-title><ce:para id="pr0070">In this section, we will lay out the GUP approach to quantum gravity <ce:cross-refs refid="br0100 br0110 br0120 br0130 br0140 br0150 br0160 br0170 br0180 br0190" id="crs0030">[10–19]</ce:cross-refs>. GUP models are phenomenological methods to quantize gravity which modify the canonical position and momentum operators and by extension their commutator. This leads to a modified Heisenberg uncertainty principle which gives another avenue to analyze how quantum gravity works at short distances and high energies.</ce:para><ce:para id="pr0080">Inspired by <ce:cross-ref refid="br0160" id="crf0200">[16]</ce:cross-ref>, we will only modify the position operator and keep the standard momentum operator:<ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>i</mml:mi><mml:mi>ħ</mml:mi><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mrow><mml:mi mathvariant="normal">and</mml:mi></mml:mrow><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>;</mml:mo></mml:math></ce:formula></ce:display> the capitalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> indicates modification. These operators have the modified commutator<ce:display><ce:formula id="fm0030"><ce:label>(3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>i</mml:mi><mml:mi>ħ</mml:mi><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> In <ce:cross-ref refid="br0160" id="crf0210">[16]</ce:cross-ref>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> where <ce:italic>β</ce:italic> is a phenomenological parameter which sets the scale for quantum gravity. Generally, <ce:italic>β</ce:italic> is taken to be the Planck scale <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mi>β</mml:mi><mml:mo>∼</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>P</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>ħ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:math>. The modified commutator in <ce:cross-ref refid="fm0030" id="crf0220">(3)</ce:cross-ref> implies an uncertainty relationship <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>X</mml:mi><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>p</mml:mi><mml:mo>∼</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>β</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> which leads to a minimum length of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mi mathvariant="normal">Δ</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>ħ</mml:mi><mml:msqrt><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msqrt></mml:math> at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:mi mathvariant="normal">Δ</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac></mml:math>. Additionally, in order for position and momentum operators to be symmetric, <ce:italic>i.e.</ce:italic> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>, the scalar product of this model must have the form<ce:display><ce:formula id="fm0040"><ce:label>(4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munderover><mml:mo movablelimits="false">∫</mml:mo><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo>∞</mml:mo></mml:mrow><mml:mrow><mml:mo>∞</mml:mo></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The modification of the scalar product as given by <ce:cross-ref refid="fm0040" id="crf0230">(4)</ce:cross-ref> is for three dimensions; even in <ce:italic>n</ce:italic> dimensions one still has the same modifying factor for the momentum integration, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac></mml:math>.</ce:para></ce:section><ce:section id="se0030"><ce:label>3</ce:label><ce:section-title id="st0040">Vacuum energy calculations with GUP</ce:section-title><ce:para id="pr0090">With these modified operators and scalar products, one can modify the vacuum energy integration in <ce:cross-ref refid="fm0010" id="crf0240">(1)</ce:cross-ref> following <ce:cross-ref refid="br0200" id="crf0250">[20]</ce:cross-ref>. In <ce:cross-ref refid="br0200" id="crf0260">[20]</ce:cross-ref>, the authors calculate how the GUP modifies Liouville's theorem and the phase space volume, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mspace width="0.25em"/><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:math>. The modified phase space associated with the model from <ce:cross-ref refid="fm0020" id="crf0270">(2)</ce:cross-ref> and <ce:cross-ref refid="fm0030" id="crf0280">(3)</ce:cross-ref> is<ce:display><ce:formula id="fm0050"><ce:label>(5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mspace width="0.25em"/><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Upon integrating out the volume, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>V</mml:mi></mml:math>, and after quantization, the phase space volume is given by<ce:display><ce:formula id="fm0060"><ce:label>(6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mfrac><mml:mrow><mml:mi>V</mml:mi><mml:mspace width="0.25em"/><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>ħ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="false">→</mml:mo><mml:mfrac><mml:mrow><mml:mi>V</mml:mi><mml:mspace width="0.25em"/><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> In the last step in <ce:cross-ref refid="fm0060" id="crf0290">(6)</ce:cross-ref>, we set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:mi>ħ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mn>1</mml:mn></mml:math> to match the units of reference <ce:cross-ref refid="br0030" id="crf0300">[3]</ce:cross-ref>. Using the modified phase space volume in <ce:cross-ref refid="fm0060" id="crf0310">(6)</ce:cross-ref> for the calculation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> in <ce:cross-ref refid="fm0010" id="crf0320">(1)</ce:cross-ref> gives<ce:display><ce:formula id="fm0070"><ce:label>(7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> This is beneficial because the factor of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> makes the integrand integrable and makes the vacuum energy density finite without using a ‘by hand’ cut-off.</ce:para><ce:para id="pr0100">Substituting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> from <ce:cross-refs refid="br0160 br0200" id="crs0040">[16,20]</ce:cross-refs> into <ce:cross-ref refid="fm0070" id="crf0330">(7)</ce:cross-ref>, one finds that the GUP vacuum energy density is finite with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>∝</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. However, <ce:italic>β</ce:italic> is typically taken to be of the Planck scale and this leads to the GUP vacuum energy density from <ce:cross-ref refid="fm0070" id="crf0340">(7)</ce:cross-ref> yielding the same value as the by hand cut-off value obtained from <ce:cross-ref refid="fm0010" id="crf0350">(1)</ce:cross-ref>. Unfortunately, nothing has been gained by exchanging the by hand cut-off for a GUP-inspired functional cut-off, as already noted in <ce:cross-ref refid="br0200" id="crf0360">[20]</ce:cross-ref>. Moreover, the inner product used for the vacuum energy density in <ce:cross-ref refid="fm0070" id="crf0370">(7)</ce:cross-ref> does not preserve the symmetry of the modified position operator. Equations <ce:cross-ref refid="fm0050" id="crf0380">(5)</ce:cross-ref>, <ce:cross-ref refid="fm0060" id="crf0390">(6)</ce:cross-ref>, and <ce:cross-ref refid="fm0070" id="crf0400">(7)</ce:cross-ref> imply that integration over momentum should come with the factor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msup><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>. This violates the symmetry requirement for the position operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>. In order for the position operator to be symmetric, one needs the factor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msup><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> as in equation <ce:cross-ref refid="fm0040" id="crf0410">(4)</ce:cross-ref>.</ce:para><ce:para id="pr0110">To resolve this conflict between <ce:cross-ref refid="fm0040" id="crf0420">(4)</ce:cross-ref> and <ce:cross-ref refid="fm0060" id="crf0430">(6)</ce:cross-ref>, we need to reconsider the spatial/volume calculation. In the transition from <ce:cross-ref refid="fm0050" id="crf0440">(5)</ce:cross-ref> to <ce:cross-ref refid="fm0060" id="crf0450">(6)</ce:cross-ref>, it is assumed that the spatial volume does not change, <ce:italic>i.e.</ce:italic> the spatial volume is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>V</mml:mi></mml:math>; but since this approach utilizes the GUP, one would imagine that the introduction of a minimal length should change the calculation of volumes.</ce:para><ce:para id="pr0120">We pair a single factor of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> to go with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:math> integration, which would be consistent with the symmetry of the position operator required by <ce:cross-ref refid="fm0040" id="crf0460">(4)</ce:cross-ref>. The remaining two factors are paired with the volume integration to account for the introduction of a minimal length. The phase space integration with the spatial volume in spherical coordinates becomes <ce:cross-ref refid="br0210" id="crf0470">[21]</ce:cross-ref><ce:display><ce:formula id="fm0080"><ce:label>(8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>d</mml:mi><mml:mi>r</mml:mi><mml:mi>d</mml:mi><mml:mi mathvariant="normal">Ω</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The finite length factor in <ce:cross-ref refid="fm0080" id="crf0480">(8)</ce:cross-ref> is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, so that the spatial integration that takes into account the GUP minimal length is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:math>. Now, the modified momentum integration in <ce:cross-ref refid="fm0080" id="crf0490">(8)</ce:cross-ref> agrees with the requirement of symmetry of the position and momentum operators as given in <ce:cross-ref refid="fm0040" id="crf0500">(4)</ce:cross-ref>. The corrected GUP-modified vacuum energy is then<ce:display><ce:formula id="fm0090"><ce:label>(9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Whether we use <ce:cross-ref refid="fm0010" id="crf0510">(1)</ce:cross-ref>, <ce:cross-ref refid="fm0070" id="crf0520">(7)</ce:cross-ref>, or <ce:cross-ref refid="fm0090" id="crf0530">(9)</ce:cross-ref> to calculated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, the discrepancy between the calculated and observed vacuum energy density is still enormous. This discrepancy is due to the fact that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> is always positive.</ce:para><ce:para id="pr0130">One possible resolution is to allow <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> to become negative at large momentum. With this modification, the vacuum energy density integral has a negative contribution which can bring the calculated vacuum energy density closer to the observed value. Generally, GUP models do not consider an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> which can be negative, because this results in a parity flip of the position operator, <ce:italic>i.e.</ce:italic> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:math>, at large momentum. Admittedly having a parity flip like this is very unusual and may lead to difficulties coming up with a good physical interpretation. However, the weak interaction is known to violate parity, and additionally there are works that have examined parity violation in gravity. For example, reference <ce:cross-ref refid="br0080" id="crf0540">[8]</ce:cross-ref> examined parity violation in gravity, and recently reference <ce:cross-ref refid="br0090" id="crf0550">[9]</ce:cross-ref> placed experimental bounds on parity violation and time-reversal symmetry violation in gravity using spin-gravity interactions. Further in <ce:cross-ref refid="br0220" id="crf0560">[22]</ce:cross-ref> a model of parity violation is constructed with potential signatures appearing in the cosmic microwave background (CMB). As another example references <ce:cross-refs refid="br0230 br0240" id="crs0050">[23,24]</ce:cross-refs> attempt to solve the cosmological constant problem in a loop quantum gravity model with degenerate geometry, with an implied parity violation. Thus while there are certainly questions as to what a parity flip like <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:math> means physically, there is work examining parity violation in gravity both theoretically and experimentally.</ce:para><ce:para id="pr0140">In the next section we investigate the possibility that quantum gravity may lead to a modified position operator which changes sign/violates parity at some high energy/momentum scale leading to a small cosmological constant consistent with observations.</ce:para></ce:section><ce:section id="se0040"><ce:label>4</ce:label><ce:section-title id="st0050">GUP cosmological constant and parity transformation</ce:section-title><ce:para id="pr0150">In order to recover the standard position operator at low energy scales, the function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> which modifies the position operator must satisfy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:math> when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo>≪</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> designates the momentum where the parity flip occurs. As mentioned previously, one expects <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:math> up to, for example, the electroweak scale with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo>≈</mml:mo><mml:mn>100</mml:mn></mml:math> GeV. At this point the calculated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> is already ∼56 orders of magnitude larger than the observed vacuum energy density.</ce:para><ce:para id="pr0160">As discussed in the previous section, if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> is positive definite, then the integration from the electroweak scale upward will only make further positive contributions to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, increasing the discrepancy between the calculated and observed vacuum energy density. The only way we can see to counter this large reserve of positive vacuum energy density is to have the vacuum density integrand in <ce:cross-ref refid="fm0090" id="crf0570">(9)</ce:cross-ref> become negative as the energy/momentum scale increases. This is conceptually similar to unbroken supersymmetry, where the positive contribution of bosonic zero modes balances the negative contribution of fermionic zero modes.</ce:para><ce:para id="pr0170">An example of a GUP model which satisfies the above requirement is<ce:display><ce:formula id="fm0100"><ce:label>(10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>i</mml:mi><mml:mi>ħ</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mo>;</mml:mo><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where we have introduced a second momentum scale, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>. We will see the need for this later, but <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> should be of the order of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math>.</ce:para><ce:para id="pr0180">If we do an expansion to second order in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> and, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> we find that <ce:cross-ref refid="fm0100" id="crf0580">(10)</ce:cross-ref> is equivalent to the GUP model of reference <ce:cross-ref refid="br0160" id="crf0590">[16]</ce:cross-ref> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mi>i</mml:mi><mml:mi>ħ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math>, and <ce:italic>β</ce:italic> depending on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>. The specific form of the modified position operator in <ce:cross-ref refid="fm0100" id="crf0600">(10)</ce:cross-ref> is driven by two constraints: (i) we want the modified operators in <ce:cross-ref refid="fm0100" id="crf0610">(10)</ce:cross-ref> to give a minimum length; (ii) we want the position operator to flip sign at some large momentum scale so that the vacuum energy density integral will have positive (at low momentum) and negative (at high momentum) contributions. This flipping of the sign of the position operator can be seen as a form of parity violation; which proposes that the gravitational interaction may violate parity, as is also the case for the weak interaction.</ce:para><ce:para id="pr0190">Taking the modified operators from <ce:cross-ref refid="fm0100" id="crf0620">(10)</ce:cross-ref> and using them to calculate the vacuum energy density <ce:cross-ref refid="fm0090" id="crf0630">(9)</ce:cross-ref>, taking into account that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:math>, we get<ce:display><ce:formula id="fm0110"><ce:label>(11)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo id="mmlbr0002" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>∫</mml:mo><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="before">≈</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:munderover><mml:mo movablelimits="false">∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>∞</mml:mo></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>p</mml:mi><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="before">=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>32</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> From the first line to the second, we have written <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:math> simply as <ce:italic>p</ce:italic>, we have done the integration over the solid angle giving 4<ce:italic>π</ce:italic>, and we have assumed that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> is small compared to both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> allowing us to use <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mo>≈</mml:mo><mml:mi>p</mml:mi></mml:math>.</ce:para><ce:para id="pr0200">From the last expression in <ce:cross-ref refid="fm0110" id="crf0640">(11)</ce:cross-ref>, we can see a balancing between the positive contribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:math> coming from the integration over low <ce:italic>p</ce:italic>, and the negative contribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac></mml:math> coming from the integration over high <ce:italic>p</ce:italic>. This is reminiscent of the balancing of positive and negative contributions to the vacuum energy density in unbroken supersymmetric models.</ce:para><ce:para id="pr0210">One can use <ce:cross-ref refid="fm0110" id="crf0650">(11)</ce:cross-ref> to “solve” the cosmological constant problem. Setting the calculated vacuum energy density in <ce:cross-ref refid="fm0110" id="crf0660">(11)</ce:cross-ref> to the observed vacuum energy density, one can solve for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> and obtain<ce:display><ce:formula id="fm0120"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>32</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> is at the Planck scale this implies <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo>∼</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>118</mml:mn></mml:mrow></mml:msup></mml:math>, which in turn leads to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> are both of the Planck scale.</ce:para><ce:para id="pr0220">Furthermore <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> do not necessarily need to be at the Planck scale to resolve the cosmological constant problem – one just needs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mo>≪</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:math>, which can be obtained even with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> at a much lower scale than the Planck scale. One still has the fine tuning problem of why the two scales should deviate from the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> by such an incredibly small amount compared to either <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>. Nonetheless, this example illustrates how a negative contribution to the vacuum energy density, from high momentum, helps the GUP approach to the cosmological constant problem.</ce:para><ce:para id="pr0230">One could ask if this model would be able to connect the present small value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup></mml:math> with a much larger value of the vacuum energy density required for an inflationary epoch in the very early Universe. Looking at the last line of <ce:cross-ref refid="fm0110" id="crf0670">(11)</ce:cross-ref> one can have a large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> if, in the early Universe, the two momentum scales satisfy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>, but not <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>. Then from this initial state the momentum scales would need to evolve toward <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> to give the small observed vacuum energy density of the present Universe.</ce:para></ce:section><ce:section id="se0050"><ce:label>5</ce:label><ce:section-title id="st0060">Summary and conclusions</ce:section-title><ce:para id="pr0240">In this paper we examined how GUP models might address the cosmological constant problem. For GUP models given by modified position and momentum operators of the form <ce:cross-ref refid="fm0020" id="crf0680">(2)</ce:cross-ref> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>i</mml:mi><mml:mi>ħ</mml:mi><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math>, there were two variants of the associated modified vacuum energy density given in <ce:cross-ref refid="fm0070" id="crf0690">(7)</ce:cross-ref> and <ce:cross-ref refid="fm0090" id="crf0700">(9)</ce:cross-ref>. These two expressions differed by the number of factors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.svg"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac></mml:math>. We argued for the validity of the expression in <ce:cross-ref refid="fm0090" id="crf0710">(9)</ce:cross-ref> over <ce:cross-ref refid="fm0070" id="crf0720">(7)</ce:cross-ref>, since only the former expression satisfied the requirement of symmetry of the modified position and momentum operators discussed around equation <ce:cross-ref refid="fm0040" id="crf0730">(4)</ce:cross-ref>.</ce:para><ce:para id="pr0250">However, regardless of whether one used <ce:cross-ref refid="fm0070" id="crf0740">(7)</ce:cross-ref> or <ce:cross-ref refid="fm0090" id="crf0750">(9)</ce:cross-ref> to calculate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, one still has essentially the same problem as the by-hand cut-off of <ce:cross-ref refid="fm0010" id="crf0760">(1)</ce:cross-ref>: the vacuum energy density was proportional the momentum cut-off scale to the fourth power, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:math>, which for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> near the Planck scale (or even the electroweak scale) made <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> too large. With the GUP modified energy density of either <ce:cross-ref refid="fm0070" id="crf0770">(7)</ce:cross-ref> or <ce:cross-ref refid="fm0090" id="crf0780">(9)</ce:cross-ref> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, while formally finite, would nevertheless be proportional to the inverse square of the functional cut-off parameter <ce:italic>β i.e.</ce:italic> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, and if the scale of <ce:italic>β</ce:italic> were taken as the Planck scale, then one finds the same problem as using a by-hand cut-off of the formally infinite integral in <ce:cross-ref refid="fm0010" id="crf0790">(1)</ce:cross-ref> - the vacuum energy density from GUP models will have the same, large, order-of-magnitude disagreement compared to the observed vacuum energy density.</ce:para><ce:para id="pr0260">We propose that the modified position operator changes sign at some momentum scale, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math>, as in <ce:cross-ref refid="fm0100" id="crf0800">(10)</ce:cross-ref>. Then because of the link between the function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> from the modified position operator in <ce:cross-ref refid="fm0020" id="crf0810">(2)</ce:cross-ref>, and how it changes the vacuum energy density in <ce:cross-ref refid="fm0090" id="crf0820">(9)</ce:cross-ref>, one finds that the positive contribution to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> from the integration below <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> is balanced by negative contribution from above <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> This balancing of large positive and negative contributions to the vacuum energy density is similar to unbroken SUSY, where a large, positive bosonic contribution is balanced by a large, negative fermionic contribution. Here, however this balancing of positive and negative contributions comes from the parity flip - the change in sign of the modified position operator at some momentum scale.</ce:para> </ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0100">Declaration of Competing Interest</ce:section-title><ce:para id="pr0290">The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Douglas Singleton reports financial support was provided by <ce:grant-sponsor id="gsp0010" sponsor-id="https://doi.org/10.13039/100000001">National Science Foundation</ce:grant-sponsor>. Michael Bishop reports financial support was provided by <ce:grant-sponsor id="gsp0020" sponsor-id="https://doi.org/10.13039/100010075">California State University, Fresno</ce:grant-sponsor>. Douglas Singleton reports financial support was provided by <ce:grant-sponsor id="gsp0030" sponsor-id="https://doi.org/10.13039/100000001">National Science Foundation</ce:grant-sponsor>.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0070">Acknowledgement</ce:section-title><ce:para id="pr0270">DS is supported by a 2023-2024 <ce:grant-sponsor id="gsp0040" sponsor-id="https://doi.org/10.13039/100005956">KITP</ce:grant-sponsor> Fellows Award. This research was supported in part by the <ce:grant-sponsor id="gsp0050" sponsor-id="https://doi.org/10.13039/100000001">National Science Foundation</ce:grant-sponsor> under Grant No. <ce:grant-number refid="gsp0050">NSF PHY-1748958</ce:grant-number>. 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B 672, 386 (2009).</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> \ No newline at end of file +<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd"><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="sco" xml:lang="en"><item-info><jid>PLB</jid><aid>138173</aid><ce:article-number>138173</ce:article-number><ce:pii>S0370-2693(23)00507-5</ce:pii><ce:doi>10.1016/j.physletb.2023.138173</ce:doi><ce:copyright year="2023" type="other">The Author(s)</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>Astrophysics & Cosmology</ce:text></ce:doctopic></ce:doctopics></item-info><head><ce:title id="ti0010">Quantum gravity, the cosmological constant, and parity transformation</ce:title><ce:author-group id="ag0010"><ce:author id="au0010" author-id="S0370269323005075-fd9eaebec5bc0310ea4d5d786e4bf21c"><ce:given-name>Michael</ce:given-name><ce:surname>Bishop</ce:surname><ce:cross-ref refid="aff0010" id="crf0010"><ce:sup>a</ce:sup></ce:cross-ref><ce:e-address type="email" xlink:href="mailto:mibishop@mail.fresnostate.edu" id="ea0010">mibishop@mail.fresnostate.edu</ce:e-address></ce:author><ce:author id="au0020" author-id="S0370269323005075-53653b508b89f206bafd29356cbb841b"><ce:given-name>Peter</ce:given-name><ce:surname>Martin</ce:surname><ce:cross-ref refid="aff0020" id="crf0020"><ce:sup>b</ce:sup></ce:cross-ref><ce:e-address type="email" xlink:href="mailto:kotor2@mail.fresnostate.edu" id="ea0020">kotor2@mail.fresnostate.edu</ce:e-address></ce:author><ce:author id="au0030" author-id="S0370269323005075-4e51edfc44e12f1e7fbb2c6e553f0f1d"><ce:given-name>Douglas</ce:given-name><ce:surname>Singleton</ce:surname><ce:cross-ref refid="aff0020" id="crf0030"><ce:sup>b</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0030" id="crf0040"><ce:sup>c</ce:sup></ce:cross-ref><ce:e-address type="email" xlink:href="mailto:dougs@mail.fresnostate.edu" id="ea0030">dougs@mail.fresnostate.edu</ce:e-address></ce:author><ce:affiliation id="aff0010" affiliation-id="S0370269323005075-7290786d872411b4e6f444401cecd9ba"><ce:label>a</ce:label><ce:textfn>Mathematics Department, California State University Fresno, Fresno, CA 93740, USA</ce:textfn><sa:affiliation><sa:organization>Mathematics Department</sa:organization><sa:organization>California State University Fresno</sa:organization><sa:city>Fresno</sa:city><sa:state>CA</sa:state><sa:postal-code>93740</sa:postal-code><sa:country>USA</sa:country></sa:affiliation><ce:source-text id="srct0005">Mathematics Department, California State University Fresno, Fresno, CA 93740</ce:source-text></ce:affiliation><ce:affiliation id="aff0020" affiliation-id="S0370269323005075-9fd291008f3d5aa9316b7301a9346c31"><ce:label>b</ce:label><ce:textfn>Physics Department, California State University Fresno, Fresno, CA 93740, USA</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>California State University Fresno</sa:organization><sa:city>Fresno</sa:city><sa:state>CA</sa:state><sa:postal-code>93740</sa:postal-code><sa:country>USA</sa:country></sa:affiliation><ce:source-text id="srct0010">Physics Department, California State University Fresno, Fresno, CA 93740</ce:source-text></ce:affiliation><ce:affiliation id="aff0030" affiliation-id="S0370269323005075-6bba475b52d869ff1b2682ebf27a62e5"><ce:label>c</ce:label><ce:textfn>Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA</ce:textfn><sa:affiliation><sa:organization>Kavli Institute for Theoretical Physics</sa:organization><sa:organization>University of California Santa Barbara</sa:organization><sa:city>Santa Barbara</sa:city><sa:state>CA</sa:state><sa:postal-code>93106</sa:postal-code><sa:country>USA</sa:country></sa:affiliation><ce:source-text id="srct0015">Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA</ce:source-text></ce:affiliation></ce:author-group><ce:date-received day="13" month="7" year="2023"/><ce:date-revised day="2" month="8" year="2023"/><ce:date-accepted day="6" month="9" year="2023"/><ce:miscellaneous id="ms0010">Editor: R. Gregory</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0010">One of the leading issues in quantum field theory and cosmology is the mismatch between the observed and calculated values for the cosmological constant in Einstein's field equations of up to 120 orders of magnitude. In this paper, we discuss new methods to potentially bridge this chasm using the generalized uncertainty principle (GUP). We find that if quantum gravity GUP models are the solution to this puzzle, then it may require the gravitationally modified position operator undergoes a parity transformation at high energies.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0090">Data availability</ce:section-title><ce:para id="pr0280">No data was used for the research described in the article.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010"><ce:label>1</ce:label><ce:section-title id="st0020">Cosmological constant puzzle</ce:section-title><ce:para id="pr0010">The cosmological constant problem is the naive mismatch by ∼ 120 orders of magnitude between the observed and a simple theoretical value for the cosmological constant in Einstein's field equations; we will clarify in what sense this 120 order of magnitude mismatch is misleading/naive below. Quantum gravity has long been advertised as a potential solution to various puzzles like this cosmological constant problem. In this paper, we lay out the reasoning that led us to the conclusion that if GUP models are to resolve the cosmological constant problem, they would require the gravitationally modified position operator to undergo a parity transformation at high energy scales or short distance scales.</ce:para><ce:para id="pr0020">GUP models modify the Heisenberg uncertainty principle to obtain a minimum length scale; that is, a positive lower bound on the uncertainty in position. GUP models are not elegant, top-down theories of quantum gravity like string theory <ce:cross-ref refid="br0010" id="crf0050">[1]</ce:cross-ref> or loop quantum gravity <ce:cross-ref refid="br0020" id="crf0060">[2]</ce:cross-ref>, but they have the advantage of being easy to work with and provide a phenomenological window into quantum gravity. We will begin by briefly reviewing the salient features of the cosmological constant problem; a more complete discussion can be found in the excellent review by Weinberg <ce:cross-ref refid="br0030" id="crf0070">[3]</ce:cross-ref>. We will adopt the notation and units of <ce:cross-ref refid="br0030" id="crf0080">[3]</ce:cross-ref>.</ce:para><ce:para id="pr0030">The cosmological constant, <ce:italic>λ</ce:italic>, in Einstein's field equations (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mi>G</mml:mi><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>λ</mml:mi><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math>) is equivalent to space-time having a constant energy density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mi>G</mml:mi></mml:mrow></mml:mfrac></mml:math>, and is representative of the expansion of space. The approach for the calculation of <ce:italic>λ</ce:italic> in QFT is to add up all the energies of the zero modes (vacuum modes) of quantum fields. The vacuum modes of quantum fields are given by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mi>ħ</mml:mi><mml:msub><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt></mml:math>. Integrating over all possible momenta up to some cut-off, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, yields the vacuum energy density<ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo id="mmlbr0001" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:munderover><mml:mo movablelimits="false">∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mi>d</mml:mi><mml:mi>p</mml:mi><mml:mspace width="0.25em"/><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="before">≈</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>16</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Here, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:mi>p</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:math> is the magnitude of the momentum. We use <ce:italic>p</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:math> interchangeably throughout this work.</ce:para><ce:para id="pr0040">The integral in <ce:cross-ref refid="fm0010" id="crf0090">(1)</ce:cross-ref> is divergent, so it needs to be cut off at some momentum/energy scale, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, which is usually taken to be the Planck scale, <ce:italic>i.e.</ce:italic> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. Inserting this value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> into <ce:cross-ref refid="fm0010" id="crf0100">(1)</ce:cross-ref> gives <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mn>71</mml:mn></mml:mrow></mml:msup></mml:math> GeV<ce:sup>4</ce:sup>. The observed vacuum energy density <ce:cross-ref refid="br0040" id="crf0110">[4]</ce:cross-ref> is about <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mo>≈</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>47</mml:mn></mml:mrow></mml:msup></mml:math> GeV<ce:sup>4</ce:sup>, which is a difference of 118 orders of magnitude - a terrible disagreement between theory and observation. As mentioned in the opening paragraph this approximately 120 orders of magnitude mismatch is somewhat naive. First, the hard cut-off of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> in <ce:cross-ref refid="fm0010" id="crf0120">(1)</ce:cross-ref>, leading to a quartic dependence on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, violates relativistic invariance which requires <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mi>ρ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi>p</mml:mi></mml:math> <ce:italic>i.e.</ce:italic> energy density equals the negative pressure. This was pointed in <ce:cross-refs refid="br0050 br0060" id="crs0010">[5,6]</ce:cross-refs>, and these two works also show that if one uses dimensional regularization of integrals like <ce:cross-ref refid="fm0010" id="crf0130">(1)</ce:cross-ref>, one can restore relativistic invariance, and the energy density is no longer quartically dependent on cut-off scale, but has a logarithmic dependence. Finally, in <ce:cross-ref refid="br0060" id="crf0140">[6]</ce:cross-ref> it was shown that by using dimensional regularization of integrals like those in <ce:cross-ref refid="fm0010" id="crf0150">(1)</ce:cross-ref> and using known Standard Model fields (specifically the top quark, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> bosons and Higgs boson with an estimated mass of 150 GeV) gave a disagreement between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mo>≈</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>47</mml:mn></mml:mrow></mml:msup></mml:math> GeV<ce:sup>4</ce:sup> and the theoretical value of “only” 56 orders of magnitude. This is still terrible, but essentially cuts in half the order of magnitude disagreement, from ∼ 120 to ∼ 56. This estimate of <ce:cross-ref refid="br0060" id="crf0160">[6]</ce:cross-ref> is more realistic since the only scale probed experimentally is the electroweak scale of ∼ 100 GeV. Using the Planck scale gives an overestimate of the disagreement. Nevertheless using either the electroweak scale or the Planck scale gives a huge discrepancy between the theoretical and observed value of the vacuum energy density.</ce:para><ce:para id="pr0050">One of the standard ideas for addressing the cosmological constant problem is via <ce:italic>unbroken supersymmetry</ce:italic><ce:cross-ref refid="fn0010" id="crf0170"><ce:sup>1</ce:sup></ce:cross-ref><ce:footnote id="fn0010"><ce:label>1</ce:label><ce:note-para id="np0010">In unbroken supersymmetry particles and their superpartner particles have the same mass.</ce:note-para></ce:footnote> (SUSY) models <ce:cross-refs refid="br0030 br0070" id="crs0020">[3,7]</ce:cross-refs>. SUSY models have equal numbers of bosonic and fermionic fields/degrees of freedom. Since bosonic and fermionic vacuum modes have opposite signs, their vacuum energies cancel one another exactly in unbroken SUSY and one would have a natural explanation for a cosmological constant that is exactly zero. However, one does not want an exactly zero cosmological constant and SUSY <ce:italic>is</ce:italic> broken at least up to some high energy scale greater than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:msub><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mi>S</mml:mi><mml:mi>Y</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> GeV. But if one used the lower limit of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msub><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mi>S</mml:mi><mml:mi>Y</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> GeV as the cut-off <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, then the disagreement between theory and experiment is ∼ 60 orders of magnitude. Therefore, simple SUSY models do not provide an answer to the cosmological constant puzzle.</ce:para><ce:para id="pr0060">However, despite SUSY (either broken or unbroken) not providing a solution to the cosmological constant puzzle, it nevertheless has the feature that one obtains a small/zero cosmological constant by a cancellation of large positive and negative contributions. In the GUP approach to the cosmological constant problem presented here, we find that we can similarly get a small cosmological constant by having a modified position operator which flips parity at some large energy/momentum scale. This changing of the parity of the modified position operator leads to a cancellation of large positive and large negative contributions to the cosmological constant. Such violations of parity in the gravitational interaction were studied theoretically in <ce:cross-ref refid="br0080" id="crf0180">[8]</ce:cross-ref> and recently experimental bounds have been placed on such parity violations in gravity <ce:cross-ref refid="br0090" id="crf0190">[9]</ce:cross-ref>.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">Quantum gravity via generalized uncertainty principle</ce:section-title><ce:para id="pr0070">In this section, we will lay out the GUP approach to quantum gravity <ce:cross-refs refid="br0100 br0110 br0120 br0130 br0140 br0150 br0160 br0170 br0180 br0190" id="crs0030">[10–19]</ce:cross-refs>. GUP models are phenomenological methods to quantize gravity which modify the canonical position and momentum operators and by extension their commutator. This leads to a modified Heisenberg uncertainty principle which gives another avenue to analyze how quantum gravity works at short distances and high energies.</ce:para><ce:para id="pr0080">Inspired by <ce:cross-ref refid="br0160" id="crf0200">[16]</ce:cross-ref>, we will only modify the position operator and keep the standard momentum operator:<ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>i</mml:mi><mml:mi>ħ</mml:mi><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mrow><mml:mi mathvariant="normal">and</mml:mi></mml:mrow><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>;</mml:mo></mml:math></ce:formula></ce:display> the capitalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> indicates modification. These operators have the modified commutator<ce:display><ce:formula id="fm0030"><ce:label>(3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>i</mml:mi><mml:mi>ħ</mml:mi><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> In <ce:cross-ref refid="br0160" id="crf0210">[16]</ce:cross-ref>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> where <ce:italic>β</ce:italic> is a phenomenological parameter which sets the scale for quantum gravity. Generally, <ce:italic>β</ce:italic> is taken to be the Planck scale <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mi>β</mml:mi><mml:mo>∼</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>P</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>ħ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:math>. The modified commutator in <ce:cross-ref refid="fm0030" id="crf0220">(3)</ce:cross-ref> implies an uncertainty relationship <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>X</mml:mi><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>p</mml:mi><mml:mo>∼</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>β</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> which leads to a minimum length of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mi mathvariant="normal">Δ</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>ħ</mml:mi><mml:msqrt><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msqrt></mml:math> at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:mi mathvariant="normal">Δ</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac></mml:math>. Additionally, in order for position and momentum operators to be symmetric, <ce:italic>i.e.</ce:italic> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>, the scalar product of this model must have the form<ce:display><ce:formula id="fm0040"><ce:label>(4)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:munderover><mml:mo movablelimits="false">∫</mml:mo><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mo>∞</mml:mo></mml:mrow><mml:mrow><mml:mo>∞</mml:mo></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The modification of the scalar product as given by <ce:cross-ref refid="fm0040" id="crf0230">(4)</ce:cross-ref> is for three dimensions; even in <ce:italic>n</ce:italic> dimensions one still has the same modifying factor for the momentum integration, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac></mml:math>.</ce:para></ce:section><ce:section id="se0030"><ce:label>3</ce:label><ce:section-title id="st0040">Vacuum energy calculations with GUP</ce:section-title><ce:para id="pr0090">With these modified operators and scalar products, one can modify the vacuum energy integration in <ce:cross-ref refid="fm0010" id="crf0240">(1)</ce:cross-ref> following <ce:cross-ref refid="br0200" id="crf0250">[20]</ce:cross-ref>. In <ce:cross-ref refid="br0200" id="crf0260">[20]</ce:cross-ref>, the authors calculate how the GUP modifies Liouville's theorem and the phase space volume, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mspace width="0.25em"/><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:math>. The modified phase space associated with the model from <ce:cross-ref refid="fm0020" id="crf0270">(2)</ce:cross-ref> and <ce:cross-ref refid="fm0030" id="crf0280">(3)</ce:cross-ref> is<ce:display><ce:formula id="fm0050"><ce:label>(5)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mspace width="0.25em"/><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Upon integrating out the volume, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>V</mml:mi></mml:math>, and after quantization, the phase space volume is given by<ce:display><ce:formula id="fm0060"><ce:label>(6)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mfrac><mml:mrow><mml:mi>V</mml:mi><mml:mspace width="0.25em"/><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>ħ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="false">→</mml:mo><mml:mfrac><mml:mrow><mml:mi>V</mml:mi><mml:mspace width="0.25em"/><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> In the last step in <ce:cross-ref refid="fm0060" id="crf0290">(6)</ce:cross-ref>, we set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:mi>ħ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mn>1</mml:mn></mml:math> to match the units of reference <ce:cross-ref refid="br0030" id="crf0300">[3]</ce:cross-ref>. Using the modified phase space volume in <ce:cross-ref refid="fm0060" id="crf0310">(6)</ce:cross-ref> for the calculation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> in <ce:cross-ref refid="fm0010" id="crf0320">(1)</ce:cross-ref> gives<ce:display><ce:formula id="fm0070"><ce:label>(7)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> This is beneficial because the factor of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> makes the integrand integrable and makes the vacuum energy density finite without using a ‘by hand’ cut-off.</ce:para><ce:para id="pr0100">Substituting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> from <ce:cross-refs refid="br0160 br0200" id="crs0040">[16,20]</ce:cross-refs> into <ce:cross-ref refid="fm0070" id="crf0330">(7)</ce:cross-ref>, one finds that the GUP vacuum energy density is finite with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>∝</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. However, <ce:italic>β</ce:italic> is typically taken to be of the Planck scale and this leads to the GUP vacuum energy density from <ce:cross-ref refid="fm0070" id="crf0340">(7)</ce:cross-ref> yielding the same value as the by hand cut-off value obtained from <ce:cross-ref refid="fm0010" id="crf0350">(1)</ce:cross-ref>. Unfortunately, nothing has been gained by exchanging the by hand cut-off for a GUP-inspired functional cut-off, as already noted in <ce:cross-ref refid="br0200" id="crf0360">[20]</ce:cross-ref>. Moreover, the inner product used for the vacuum energy density in <ce:cross-ref refid="fm0070" id="crf0370">(7)</ce:cross-ref> does not preserve the symmetry of the modified position operator. Equations <ce:cross-ref refid="fm0050" id="crf0380">(5)</ce:cross-ref>, <ce:cross-ref refid="fm0060" id="crf0390">(6)</ce:cross-ref>, and <ce:cross-ref refid="fm0070" id="crf0400">(7)</ce:cross-ref> imply that integration over momentum should come with the factor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msup><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>. This violates the symmetry requirement for the position operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>. In order for the position operator to be symmetric, one needs the factor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msup><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> as in equation <ce:cross-ref refid="fm0040" id="crf0410">(4)</ce:cross-ref>.</ce:para><ce:para id="pr0110">To resolve this conflict between <ce:cross-ref refid="fm0040" id="crf0420">(4)</ce:cross-ref> and <ce:cross-ref refid="fm0060" id="crf0430">(6)</ce:cross-ref>, we need to reconsider the spatial/volume calculation. In the transition from <ce:cross-ref refid="fm0050" id="crf0440">(5)</ce:cross-ref> to <ce:cross-ref refid="fm0060" id="crf0450">(6)</ce:cross-ref>, it is assumed that the spatial volume does not change, <ce:italic>i.e.</ce:italic> the spatial volume is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:mo>∫</mml:mo><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mi>x</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>V</mml:mi></mml:math>; but since this approach utilizes the GUP, one would imagine that the introduction of a minimal length should change the calculation of volumes.</ce:para><ce:para id="pr0120">We pair a single factor of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> to go with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:math> integration, which would be consistent with the symmetry of the position operator required by <ce:cross-ref refid="fm0040" id="crf0460">(4)</ce:cross-ref>. The remaining two factors are paired with the volume integration to account for the introduction of a minimal length. The phase space integration with the spatial volume in spherical coordinates becomes <ce:cross-ref refid="br0210" id="crf0470">[21]</ce:cross-ref><ce:display><ce:formula id="fm0080"><ce:label>(8)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>d</mml:mi><mml:mi>r</mml:mi><mml:mi>d</mml:mi><mml:mi mathvariant="normal">Ω</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> The finite length factor in <ce:cross-ref refid="fm0080" id="crf0480">(8)</ce:cross-ref> is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, so that the spatial integration that takes into account the GUP minimal length is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:math>. Now, the modified momentum integration in <ce:cross-ref refid="fm0080" id="crf0490">(8)</ce:cross-ref> agrees with the requirement of symmetry of the position and momentum operators as given in <ce:cross-ref refid="fm0040" id="crf0500">(4)</ce:cross-ref>. The corrected GUP-modified vacuum energy is then<ce:display><ce:formula id="fm0090"><ce:label>(9)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo>∫</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> Whether we use <ce:cross-ref refid="fm0010" id="crf0510">(1)</ce:cross-ref>, <ce:cross-ref refid="fm0070" id="crf0520">(7)</ce:cross-ref>, or <ce:cross-ref refid="fm0090" id="crf0530">(9)</ce:cross-ref> to calculated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, the discrepancy between the calculated and observed vacuum energy density is still enormous. This discrepancy is due to the fact that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> is always positive.</ce:para><ce:para id="pr0130">One possible resolution is to allow <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> to become negative at large momentum. With this modification, the vacuum energy density integral has a negative contribution which can bring the calculated vacuum energy density closer to the observed value. Generally, GUP models do not consider an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> which can be negative, because this results in a parity flip of the position operator, <ce:italic>i.e.</ce:italic> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:math>, at large momentum. Admittedly having a parity flip like this is very unusual and may lead to difficulties coming up with a good physical interpretation. However, the weak interaction is known to violate parity, and additionally there are works that have examined parity violation in gravity. For example, reference <ce:cross-ref refid="br0080" id="crf0540">[8]</ce:cross-ref> examined parity violation in gravity, and recently reference <ce:cross-ref refid="br0090" id="crf0550">[9]</ce:cross-ref> placed experimental bounds on parity violation and time-reversal symmetry violation in gravity using spin-gravity interactions. Further in <ce:cross-ref refid="br0220" id="crf0560">[22]</ce:cross-ref> a model of parity violation is constructed with potential signatures appearing in the cosmic microwave background (CMB). As another example references <ce:cross-refs refid="br0230 br0240" id="crs0050">[23,24]</ce:cross-refs> attempt to solve the cosmological constant problem in a loop quantum gravity model with degenerate geometry, with an implied parity violation. Thus while there are certainly questions as to what a parity flip like <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:math> means physically, there is work examining parity violation in gravity both theoretically and experimentally.</ce:para><ce:para id="pr0140">In the next section we investigate the possibility that quantum gravity may lead to a modified position operator which changes sign/violates parity at some high energy/momentum scale leading to a small cosmological constant consistent with observations.</ce:para></ce:section><ce:section id="se0040"><ce:label>4</ce:label><ce:section-title id="st0050">GUP cosmological constant and parity transformation</ce:section-title><ce:para id="pr0150">In order to recover the standard position operator at low energy scales, the function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> which modifies the position operator must satisfy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:math> when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo>≪</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> designates the momentum where the parity flip occurs. As mentioned previously, one expects <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:math> up to, for example, the electroweak scale with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo>≈</mml:mo><mml:mn>100</mml:mn></mml:math> GeV. At this point the calculated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> is already ∼56 orders of magnitude larger than the observed vacuum energy density.</ce:para><ce:para id="pr0160">As discussed in the previous section, if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> is positive definite, then the integration from the electroweak scale upward will only make further positive contributions to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, increasing the discrepancy between the calculated and observed vacuum energy density. The only way we can see to counter this large reserve of positive vacuum energy density is to have the vacuum density integrand in <ce:cross-ref refid="fm0090" id="crf0570">(9)</ce:cross-ref> become negative as the energy/momentum scale increases. This is conceptually similar to unbroken supersymmetry, where the positive contribution of bosonic zero modes balances the negative contribution of fermionic zero modes.</ce:para><ce:para id="pr0170">An example of a GUP model which satisfies the above requirement is<ce:display><ce:formula id="fm0100"><ce:label>(10)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mi>i</mml:mi><mml:mi>ħ</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mo>;</mml:mo><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:mspace width="0.25em"/><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mspace width="0.25em"/><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where we have introduced a second momentum scale, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>. We will see the need for this later, but <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> should be of the order of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math>.</ce:para><ce:para id="pr0180">If we do an expansion to second order in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> and, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> we find that <ce:cross-ref refid="fm0100" id="crf0580">(10)</ce:cross-ref> is equivalent to the GUP model of reference <ce:cross-ref refid="br0160" id="crf0590">[16]</ce:cross-ref> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mi>i</mml:mi><mml:mi>ħ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math>, and <ce:italic>β</ce:italic> depending on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>. The specific form of the modified position operator in <ce:cross-ref refid="fm0100" id="crf0600">(10)</ce:cross-ref> is driven by two constraints: (i) we want the modified operators in <ce:cross-ref refid="fm0100" id="crf0610">(10)</ce:cross-ref> to give a minimum length; (ii) we want the position operator to flip sign at some large momentum scale so that the vacuum energy density integral will have positive (at low momentum) and negative (at high momentum) contributions. This flipping of the sign of the position operator can be seen as a form of parity violation; which proposes that the gravitational interaction may violate parity, as is also the case for the weak interaction.</ce:para><ce:para id="pr0190">Taking the modified operators from <ce:cross-ref refid="fm0100" id="crf0620">(10)</ce:cross-ref> and using them to calculate the vacuum energy density <ce:cross-ref refid="fm0090" id="crf0630">(9)</ce:cross-ref>, taking into account that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:math>, we get<ce:display><ce:formula id="fm0110"><ce:label>(11)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo id="mmlbr0002" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>∫</mml:mo><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="before">≈</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:munderover><mml:mo movablelimits="false">∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>∞</mml:mo></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>p</mml:mi><mml:mo linebreak="newline" indentalign="id" indenttarget="mmlbr0002" linebreakstyle="before">=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>32</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.25em"/><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> From the first line to the second, we have written <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo></mml:math> simply as <ce:italic>p</ce:italic>, we have done the integration over the solid angle giving 4<ce:italic>π</ce:italic>, and we have assumed that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> is small compared to both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> allowing us to use <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mo>≈</mml:mo><mml:mi>p</mml:mi></mml:math>.</ce:para><ce:para id="pr0200">From the last expression in <ce:cross-ref refid="fm0110" id="crf0640">(11)</ce:cross-ref>, we can see a balancing between the positive contribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:math> coming from the integration over low <ce:italic>p</ce:italic>, and the negative contribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac></mml:math> coming from the integration over high <ce:italic>p</ce:italic>. This is reminiscent of the balancing of positive and negative contributions to the vacuum energy density in unbroken supersymmetric models.</ce:para><ce:para id="pr0210">One can use <ce:cross-ref refid="fm0110" id="crf0650">(11)</ce:cross-ref> to “solve” the cosmological constant problem. Setting the calculated vacuum energy density in <ce:cross-ref refid="fm0110" id="crf0660">(11)</ce:cross-ref> to the observed vacuum energy density, one can solve for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> and obtain<ce:display><ce:formula id="fm0120"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>32</mml:mn><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display> If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> is at the Planck scale this implies <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo>∼</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>118</mml:mn></mml:mrow></mml:msup></mml:math>, which in turn leads to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> are both of the Planck scale.</ce:para><ce:para id="pr0220">Furthermore <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> do not necessarily need to be at the Planck scale to resolve the cosmological constant problem – one just needs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mo>≪</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:math>, which can be obtained even with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> at a much lower scale than the Planck scale. One still has the fine tuning problem of why the two scales should deviate from the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> by such an incredibly small amount compared to either <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>. Nonetheless, this example illustrates how a negative contribution to the vacuum energy density, from high momentum, helps the GUP approach to the cosmological constant problem.</ce:para><ce:para id="pr0230">One could ask if this model would be able to connect the present small value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:msubsup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup></mml:math> with a much larger value of the vacuum energy density required for an inflationary epoch in the very early Universe. Looking at the last line of <ce:cross-ref refid="fm0110" id="crf0670">(11)</ce:cross-ref> one can have a large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> if, in the early Universe, the two momentum scales satisfy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>, but not <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math>. Then from this initial state the momentum scales would need to evolve toward <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:math> to give the small observed vacuum energy density of the present Universe.</ce:para></ce:section><ce:section id="se0050"><ce:label>5</ce:label><ce:section-title id="st0060">Summary and conclusions</ce:section-title><ce:para id="pr0240">In this paper we examined how GUP models might address the cosmological constant problem. For GUP models given by modified position and momentum operators of the form <ce:cross-ref refid="fm0020" id="crf0680">(2)</ce:cross-ref> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mi>i</mml:mi><mml:mi>ħ</mml:mi><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math>, there were two variants of the associated modified vacuum energy density given in <ce:cross-ref refid="fm0070" id="crf0690">(7)</ce:cross-ref> and <ce:cross-ref refid="fm0090" id="crf0700">(9)</ce:cross-ref>. These two expressions differed by the number of factors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si75.svg"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac></mml:math>. We argued for the validity of the expression in <ce:cross-ref refid="fm0090" id="crf0710">(9)</ce:cross-ref> over <ce:cross-ref refid="fm0070" id="crf0720">(7)</ce:cross-ref>, since only the former expression satisfied the requirement of symmetry of the modified position and momentum operators discussed around equation <ce:cross-ref refid="fm0040" id="crf0730">(4)</ce:cross-ref>.</ce:para><ce:para id="pr0250">However, regardless of whether one used <ce:cross-ref refid="fm0070" id="crf0740">(7)</ce:cross-ref> or <ce:cross-ref refid="fm0090" id="crf0750">(9)</ce:cross-ref> to calculate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, one still has essentially the same problem as the by-hand cut-off of <ce:cross-ref refid="fm0010" id="crf0760">(1)</ce:cross-ref>: the vacuum energy density was proportional the momentum cut-off scale to the fourth power, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:math>, which for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> near the Planck scale (or even the electroweak scale) made <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> too large. With the GUP modified energy density of either <ce:cross-ref refid="fm0070" id="crf0770">(7)</ce:cross-ref> or <ce:cross-ref refid="fm0090" id="crf0780">(9)</ce:cross-ref> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math>, while formally finite, would nevertheless be proportional to the inverse square of the functional cut-off parameter <ce:italic>β i.e.</ce:italic> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, and if the scale of <ce:italic>β</ce:italic> were taken as the Planck scale, then one finds the same problem as using a by-hand cut-off of the formally infinite integral in <ce:cross-ref refid="fm0010" id="crf0790">(1)</ce:cross-ref> - the vacuum energy density from GUP models will have the same, large, order-of-magnitude disagreement compared to the observed vacuum energy density.</ce:para><ce:para id="pr0260">We propose that the modified position operator changes sign at some momentum scale, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math>, as in <ce:cross-ref refid="fm0100" id="crf0800">(10)</ce:cross-ref>. Then because of the link between the function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> from the modified position operator in <ce:cross-ref refid="fm0020" id="crf0810">(2)</ce:cross-ref>, and how it changes the vacuum energy density in <ce:cross-ref refid="fm0090" id="crf0820">(9)</ce:cross-ref>, one finds that the positive contribution to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math> from the integration below <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> is balanced by negative contribution from above <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:math> This balancing of large positive and negative contributions to the vacuum energy density is similar to unbroken SUSY, where a large, positive bosonic contribution is balanced by a large, negative fermionic contribution. Here, however this balancing of positive and negative contributions comes from the parity flip - the change in sign of the modified position operator at some momentum scale.</ce:para> </ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0100">Declaration of Competing Interest</ce:section-title><ce:para id="pr0290">The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Douglas Singleton reports financial support was provided by <ce:grant-sponsor id="gsp0010" sponsor-id="https://doi.org/10.13039/100000001">National Science Foundation</ce:grant-sponsor>. Michael Bishop reports financial support was provided by <ce:grant-sponsor id="gsp0020" sponsor-id="https://doi.org/10.13039/100010075">California State University, Fresno</ce:grant-sponsor>. Douglas Singleton reports financial support was provided by <ce:grant-sponsor id="gsp0030" sponsor-id="https://doi.org/10.13039/100000001">National Science Foundation</ce:grant-sponsor>.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0070">Acknowledgement</ce:section-title><ce:para id="pr0270">DS is supported by a 2023-2024 <ce:grant-sponsor id="gsp0040" sponsor-id="https://doi.org/10.13039/100005956">KITP</ce:grant-sponsor> Fellows Award. This research was supported in part by the <ce:grant-sponsor id="gsp0050" sponsor-id="https://doi.org/10.13039/100000001">National Science Foundation</ce:grant-sponsor> under Grant No. <ce:grant-number refid="gsp0050">NSF PHY-1748958</ce:grant-number>. 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For symmetric collision systems (Pb<ce:glyph name="sbnd"/>Pb and pp) the data has been symmetrised around <ce:italic>η</ce:italic> = 0 and points for <ce:italic>η</ce:italic> > 3.5 have been reflected around <ce:italic> η</ce:italic> = 0. The boxes around the points and to the right reflect the uncorrelated and correlated, with respect to pseudorapidity, systematic uncertainty, respectively. The relative correlated, normalisation, uncertainties are evaluated at d<ce:italic>N</ce:italic><ce:inf>ch</ce:inf>/d<ce:italic>η</ce:italic> |<ce:inf><ce:italic>η</ce:italic>=0</ce:inf>. The lines show fits of Eq. <ce:cross-ref refid="fm0030" id="crf0020">(1)</ce:cross-ref> (Pb<ce:glyph name="sbnd"/>Pb and pp) and Eq. <ce:cross-ref refid="fm0040" id="crf0030">(2)</ce:cross-ref> (p<ce:glyph name="sbnd"/>Pb) to the data (discussed in Section <ce:cross-ref refid="se0040" id="crf0040">4</ce:cross-ref>). Please note that the ordinate has been cut twice to accommodate for the very different ranges of the charged-particle pseudorapidity densities.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0010">Fig. 1</ce:alt-text><ce:link locator="gr001" xlink:type="simple" xlink:href="pii:S0370269323000643/gr001" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0010"/></ce:figure><ce:figure id="fg0020"><ce:label>Fig. 2</ce:label><ce:caption id="cp0020"><ce:simple-para id="sp0020">Charged-particle pseudorapidity density in p<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> in seven centrality classes based on the V0A and V0C estimators. The lines are obtained using a fit of a scaled, normal distribution in rapidity Eq. <ce:cross-ref refid="fm0040" id="crf0050">(2)</ce:cross-ref> to the data (discussed in Section <ce:cross-ref refid="se0040" id="crf0060">4</ce:cross-ref>).</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0020">Fig. 2</ce:alt-text><ce:link locator="gr002" xlink:type="simple" xlink:href="pii:S0370269323000643/gr002" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0020"/></ce:figure><ce:figure id="fg0030"><ce:label>Fig. 3</ce:label><ce:caption id="cp0030"><ce:simple-para id="sp0030">Ratio <ce:italic>r</ce:italic><ce:inf><ce:italic>X</ce:italic></ce:inf> of the charged-particle pseudorapidity density in Pb<ce:glyph name="sbnd"/>Pb (top) and p<ce:glyph name="sbnd"/>Pb (bottom) in different centrality classes to the charged-particle pseudorapidity density in pp in the INEL>0 event class. Note, for Pb<ce:glyph name="sbnd"/>Pb <ce:italic>η</ce:italic><ce:inf>lab</ce:inf> is the same as the centre-of-mass pseudorapidity.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0030">Fig. 3</ce:alt-text><ce:link locator="gr003" xlink:type="simple" xlink:href="pii:S0370269323000643/gr003" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0030"/></ce:figure><ce:figure id="fg0040"><ce:label>Fig. 4</ce:label><ce:caption id="cp0040"><ce:simple-para id="sp0040">The width (top) and effective <ce:italic>p</ce:italic><ce:inf> T</ce:inf>/<ce:italic>m</ce:italic> (bottom) fit parameters as a function of the mean number of participants in pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>. Vertical uncertainties are the standard error on the best-fit parameter values, while horizontal uncertainties reflect the uncertainty on 〈<ce:italic>N</ce:italic><ce:inf> part</ce:inf>〉 from the Glauber calculations. Also shown are similar fit parameters from the same parameterisation of EPOS-LHC calculations as well as direct calculations of the standard deviation of the d<ce:italic>N</ce:italic><ce:inf> ch</ce:inf>/d<ce:italic>y</ce:italic> distributions and the 〈<ce:italic>p</ce:italic><ce:inf> T</ce:inf>〉/〈<ce:italic>m</ce:italic>〉 ratio from the EPOS-LHC calculations.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0040">Fig. 4</ce:alt-text><ce:link locator="gr004" xlink:type="simple" xlink:href="pii:S0370269323000643/gr004" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0040"/></ce:figure><ce:figure id="fg0050"><ce:label>Fig. 5</ce:label><ce:caption id="cp0050"><ce:simple-para id="sp0050">The transverse area <ce:italic>S</ce:italic><ce:inf>T</ce:inf> as calculated in a numerical Glauber model for two extreme cases: a) only the exclusive overlap of nucleons is considered (∩, open markers) and b) the inclusive area of participating nucleons contribute (∪, closed markers) in both p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0050">Fig. 5</ce:alt-text><ce:link locator="gr005" xlink:type="simple" xlink:href="pii:S0370269323000643/gr005" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0050"/></ce:figure><ce:figure id="fg0060"><ce:label>Fig. 6</ce:label><ce:caption id="cp0060"><ce:simple-para id="sp0060">Estimate of the lower bound on the Bjorken transverse energy density in pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>, considering the exclusive (∩, open markers) and inclusive (∪, full markers) overlap area <ce:italic> S</ce:italic><ce:inf>T</ce:inf> of the nucleons. The expression <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:mi>C</mml:mi><mml:mmultiscripts><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow><mml:none/><mml:none/><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:mmultiscripts></mml:math> is fitted to case ∪, and we find <ce:italic>C</ce:italic> = (0.8 ± 0.3) GeV/(fm<ce:sup> 2</ce:sup><ce:italic>c</ce:italic>) and <ce:italic>p</ce:italic> = 0.44 ± 0.08. Also shown is an estimate, via d<ce:italic>E</ce:italic><ce:inf>T</ce:inf>/d<ce:italic> y</ce:italic>, of <ce:italic>ε</ce:italic><ce:inf>Bj</ce:inf> from Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> (stars with uncertainty band) <ce:cross-ref refid="br0310" id="crf0070">[31]</ce:cross-ref>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0060">Fig. 6</ce:alt-text><ce:link locator="gr006" xlink:type="simple" xlink:href="pii:S0370269323000643/gr006" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0060"/></ce:figure></ce:floats><head><ce:title id="ti0010">System-size dependence of the charged-particle pseudorapidity density at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> for pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions</ce:title><ce:author-group id="ag0010"><ce:collaboration id="co0010" collaboration-id="S0370269323000643-3bea72599603117cd9d18494a0279c47"><ce:text>ALICE Collaboration</ce:text><ce:cross-ref refid="fn0080" id="crf0080"><ce:sup>⋆</ce:sup></ce:cross-ref><ce:author-group id="ag0020"><ce:author id="au0010" author-id="S0370269323000643-e60c93a934b81cf9801254193264c6ee"><ce:given-name>S.</ce:given-name><ce:surname>Acharya</ce:surname><ce:cross-ref refid="aff1420" id="crf0090"><ce:sup>142</ce:sup></ce:cross-ref></ce:author><ce:author id="au0020" author-id="S0370269323000643-0eab85892b6d74b18661e74a7987c599"><ce:given-name>D.</ce:given-name><ce:surname>Adamová</ce:surname><ce:cross-ref refid="aff0960" id="crf0100"><ce:sup>96</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="aff1420" affiliation-id="S0370269323000643-f1ae52f852d4d7d99988b3e872f887e4"><ce:label>142</ce:label><ce:textfn>Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Variable Energy Cyclotron Centre</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0710">Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0960" affiliation-id="S0370269323000643-35f0d35e6b51b7c24d9b3bc939a45b30"><ce:label>96</ce:label><ce:textfn>Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Institute of the Czech Academy of Sciences</sa:organization><sa:city>Řež u Prahy</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0480">Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy, Czech Republic</ce:source-text></ce:affiliation></ce:author-group></ce:collaboration><ce:footnote id="fn0080"><ce:label>⋆</ce:label><ce:note-para id="np0080"><ce:italic>E-mail address:</ce:italic><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/text/html" xlink:href="mailto:alice-publications@cern.ch" id="inf0020"> alice-publications@cern.ch</ce:inter-ref>.</ce:note-para></ce:footnote></ce:author-group><ce:date-received day="20" month="5" year="2022"/><ce:date-revised day="17" month="1" year="2023"/><ce:date-accepted day="17" month="1" year="2023"/><ce:miscellaneous id="ms0010">Editor: M. Doser</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0070">We present the first systematic comparison of the charged-particle pseudorapidity densities for three widely different collision systems, pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb, at the top energy of the Large Hadron Collider (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>) measured over a wide pseudorapidity range (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.5</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5</mml:mn></mml:math>), the widest possible among the four experiments at that facility. The systematic uncertainties are minimised since the measurements are recorded by the same experimental apparatus (ALICE). The distributions for p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb collisions are determined as a function of the centrality of the collisions, while results from pp collisions are reported for inelastic events with at least one charged particle at midrapidity. The charged-particle pseudorapidity densities are, under simple and robust assumptions, transformed to charged-particle rapidity densities. This allows for the calculation and the presentation of the evolution of the width of the rapidity distributions and of a lower bound on the Bjorken energy density, as a function of the number of participants in all three collision systems. We find a decreasing width of the particle production, and roughly a smooth ten fold increase in the energy density, as the system size grows, which is consistent with a gradually higher dense phase of matter.</ce:simple-para></ce:abstract-sec></ce:abstract></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">The number of charged particles produced in energetic nuclear collisions is an important indicator for the strong interaction processes that determine the particle production at the sub-nucleonic level. In particular, the production of charged particles is expected to reflect the number of quark and gluon collisions occurring during the initial stages of the reaction. The total number of particles produced also provides information on the energy transfer available from the initial colliding beams to particle production, as a consequence of nuclear stopping <ce:cross-ref refid="br0010" id="crf10940">[1]</ce:cross-ref>. In order to help unravel this complex scenario it is important to compare the particle production amongst collision systems of different sizes over a wide kinematic range.</ce:para><ce:para id="pr0020">We present the measured charged-particle pseudorapidity density, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math>, for pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb (previously published <ce:cross-ref refid="br0020" id="crf10950">[2]</ce:cross-ref>) collisions at the same collision energy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> in the nucleon–nucleon centre-of-mass reference frame. This is, at present, the maximum available energy at CERN's Large Hadron Collider (LHC) for Pb<ce:glyph name="sbnd"/>Pb collisions. The measurements were carried out using ALICE at LHC (for earlier <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math> results see for example Refs. <ce:cross-refs refid="br0030 br0040 br0050" id="crs0010">[3–5]</ce:cross-refs>). The three studied reactions have different characteristics probing widely different particle production yields and mechanisms. In Pb<ce:glyph name="sbnd"/>Pb collisions, the total particle yield for central collisions is of the order 10<ce:sup> 4</ce:sup><ce:cross-ref refid="br0020" id="crf10960">[2]</ce:cross-ref>, and a strongly coupled plasma of quarks and gluons (sQGP) is formed <ce:cross-refs refid="br0060 br0070 br0080 br0090" id="crs0020">[6–9]</ce:cross-refs>, whose collective and transport properties are currently under intense study. On the other hand, pp collisions represent the simplest possible nuclear collision system, where the average total particle production is much smaller (≈80, by integrating the measured distributions), and is to first approximation much less subject to collective effects <ce:cross-ref refid="br0100" id="crf10970">[10]</ce:cross-ref>. The p<ce:glyph name="sbnd"/>Pb system is intermediate to the other reactions, corresponding to the situation where a single nucleon probes the nucleons in a narrow cylinder of the target nucleus. The extent to which p<ce:glyph name="sbnd"/>Pb is governed by the initial state cold nuclear matter of the lead ion or whether collective phenomena in the hot and dense medium play an important role is, at present, a matter under scrutiny by the community <ce:cross-refs refid="br0100 br0110" id="crs0030">[10,11]</ce:cross-refs>.</ce:para><ce:para id="pr0030">In this letter, we compare the three reactions and present the ratios of the charged-particle pseudorapidity density distributions (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math>) of the more complex reactions to the pp distribution. Owing to ALICE's unique large acceptance in pseudorapidity, and using simple and robust assumptions, we transform the measured charged-particle pseudorapidity density distributions into charged-particle rapidity density distributions (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math>). This allows us to calculate the width of the rapidity distributions as a function of the number of participating nucleons. The parameters of the transformation also allow us to estimate a lower bound on the energy density using the well-known formula from Bjorken <ce:cross-ref refid="br0120" id="crf10980">[12]</ce:cross-ref>. An energy density exceeding the critical energy density of roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math><ce:cross-ref refid="br0130" id="crf10990">[13]</ce:cross-ref> is a necessary condition for the formation of deconfined matter of quarks and gluons, and thus it is of the utmost interest to understand the development of these energy densities across different collision systems.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">Experimental set-up, data sample, analysis method, systematic uncertainties</ce:section-title><ce:para id="pr0040">A detailed description of the ALICE detector and its performance can be found elsewhere <ce:cross-refs refid="br0140 br0150" id="crs0040">[14,15]</ce:cross-refs>. The present analysis uses the Silicon Pixel Detector (SPD) to determine the pseudorapidity densities in the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math> and the Forward Multiplicity Detector (FMD) in the ranges <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.5</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1.8</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mn>1.8</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5</mml:mn></mml:math>. The V0, comprised of two plastic scintillator discs covering <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.7</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1.7</mml:mn></mml:math> (V0C) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mn>2.8</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5.1</mml:mn></mml:math> (V0A), and the ZDC, two zero-degree calorimeters located <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>112.5</mml:mn><mml:mspace width="0.2em"/><mml:mtext>m</mml:mtext></mml:math> from the interaction point, measurements determine the collision centrality and are used for offline event selection <ce:cross-ref refid="br0020" id="crf11000">[2]</ce:cross-ref>.</ce:para><ce:para id="pr0050">The results presented are based on data from collisions at a centre-of-mass energy per nucleon pair of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> as collected by ALICE during LHC Run 1 (2013) for p<ce:glyph name="sbnd"/>Pb, and during Run 2 (2015) for pp and Pb<ce:glyph name="sbnd"/>Pb. The FMD suffered high levels of background noise during the 2016 p<ce:glyph name="sbnd"/>Pb campaign, due to the high collision rate, and this data is therefore not used for the present analysis. About 10<ce:sup>5</ce:sup> events with a minimum bias trigger requirement <ce:cross-ref refid="br0020" id="crf11010">[2]</ce:cross-ref> were analysed in the centrality range from 0% to 90% and 0% to 100% of the visible cross section for Pb<ce:glyph name="sbnd"/>Pb and p<ce:glyph name="sbnd"/>Pb collisions, respectively. The minimum bias trigger for p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb collisions in ALICE was defined as a coincidence between the V0A and V0C sides of the V0 detector.</ce:para><ce:para id="pr0060">The data from the p<ce:glyph name="sbnd"/>Pb collisions were taken in two beam configurations: one where the lead ion travelled toward positive pseudorapidity and one where it travelled toward negative pseudorapidity. The results from the latter collisions are mirrored around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>. The centre-of-mass frame in p<ce:glyph name="sbnd"/>Pb collisions does not coincide with the laboratory frame, due to the single magnetic field in the LHC, and thus the rapidity of the centre-of-mass is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">CM</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>±</mml:mo><mml:mn>0.465</mml:mn></mml:math> for the two directions, respectively, in the laboratory frame. For this reason, pseudorapidity, calculated with respect to the laboratory frame, is denoted <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub></mml:math> whenever p<ce:glyph name="sbnd"/>Pb results are presented.</ce:para><ce:para id="pr0070">Likewise, for the pp collisions, about 10<ce:sup>5</ce:sup> events with coincidence between V0A and V0C and at least one charged particle in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1</mml:mn></mml:math> were analysed. By requiring at least one charged particle at midrapidity, the so-called INEL>0 event class, the systematic uncertainty, related to the absolute normalisation to the full inelastic cross section, is reduced, while still sampling a large fraction (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>75</mml:mn><mml:mtext>%</mml:mtext></mml:math>) of the hadronic cross section <ce:cross-refs refid="br0160 br0170" id="crs0050">[16,17]</ce:cross-refs> .</ce:para><ce:para id="pr0080">The standard ALICE event selection <ce:cross-ref refid="br0180" id="crf11020">[18]</ce:cross-ref> and centrality estimator based on the V0 amplitude <ce:cross-refs refid="br0190 br0200" id="crs0060">[19,20]</ce:cross-refs> are used in this analysis. The event selection consists of: a) exclusion of background events using the timing information from the ZDC (for Pb<ce:glyph name="sbnd"/>Pb and p<ce:glyph name="sbnd"/>Pb, e.g., beam–gas interactions) and V0 detectors, b) verification of the trigger conditions, and c) a reconstructed position of the collision (primary vertex). In Pb<ce:glyph name="sbnd"/>Pb collisions, centrality is obtained from the sum amplitude in both V0 detector arrays (V0M). For p<ce:glyph name="sbnd"/>Pb only the amplitude in the array on the lead-going side (V0A or V0C) is used. In Pb<ce:glyph name="sbnd"/>Pb collisions, the 10% most peripheral collisions have substantial contributions from electromagnetic processes and are therefore not included in the results presented here <ce:cross-ref refid="br0190" id="crf11030">[19]</ce:cross-ref>.</ce:para><ce:para id="pr0090">A primary charged particle is defined as a charged particle with a mean proper lifetime <ce:italic>τ</ce:italic> larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mtext>cm</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mtext>c</mml:mtext></mml:math>, which is either a) produced directly in the interaction, or b) from decays of particles with <ce:italic> τ</ce:italic> smaller than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mtext>cm</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mtext>c</mml:mtext></mml:math><ce:cross-ref refid="br0210" id="crf11040">[21]</ce:cross-ref>. All quantities reported here are for primary, charged particles, though “primary” is omitted in the following for brevity.</ce:para><ce:para id="pr0100">The analysis method is identical to that of previous publications <ce:cross-ref refid="br0020" id="crf11050">[2]</ce:cross-ref>: the measurement of the charged-particle pseudorapidity density at midrapidity is obtained from counting particle trajectories determined using the two layers of the SPD. The SPD has a lower transverse momentum acceptance of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:mn>50</mml:mn><mml:mspace width="0.2em"/><mml:mtext>MeV</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="italic">c</mml:mi></mml:math>, and the yield is extrapolated down to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mtext>MeV</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="italic">c</mml:mi></mml:math> via simulations. In the forward regions, the measurement is provided by the analysis of the deposited energy signal in the FMD and a statistical method is employed to calculate the inclusive number of charged particles. A data-driven correction <ce:cross-ref refid="br0220" id="crf11060">[22]</ce:cross-ref>, based on separate measurements exploiting displaced collision vertices, is applied to remove the background from secondary particles.</ce:para><ce:para id="pr0110">Systematic uncertainty estimations for the midrapidity measurements are detailed elsewhere <ce:cross-refs refid="br0020 br0160 br0200" id="crs0070">[2,16,20]</ce:cross-refs>, and are from background suppression, transverse momentum extrapolation, weak decays, and simulations. The estimates are obtained through variation of thresholds and simulation studies. For pp (p<ce:glyph name="sbnd"/>Pb), the total systematic uncertainty amounts to 1.5% (2.7%) over the whole pseudorapidity range; while for Pb<ce:glyph name="sbnd"/>Pb the total systematic uncertainty is 2.6% at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> and 2.9% at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2</mml:mn></mml:math>. The systematic uncertainty is mostly correlated over pseudorapidity for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math>, and largely independent of centrality. The uncertainty in the forward region, estimated via variations of thresholds and simulation studies, is the same for all collision systems and is uncorrelated across <ce:italic>η</ce:italic>, amounting to 6.9% for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>3.5</mml:mn></mml:math> and 6.4% elsewhere within the forward regions <ce:cross-ref refid="br0220" id="crf11070">[22]</ce:cross-ref>. In the figures of this letter, uncorrelated, local in pseudorapidity, systematic uncertainties are indicated by open boxes on the data points, while correlated systematic uncertainties, those that affect the overall scale and typically from event classification and selection, are indicated by filled boxes to the right of the data. The systematic uncertainty on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math>, due to the centrality class definition in Pb<ce:glyph name="sbnd"/>Pb, is estimated to vary from 0.6% for the most central to 9.5% for the most peripheral class <ce:cross-ref refid="br0230" id="crf11080">[23]</ce:cross-ref>. The 80% to 90% centrality class has residual contamination from electromagnetic processes as detailed elsewhere <ce:cross-ref refid="br0190" id="crf11090">[19]</ce:cross-ref>, which gives rise to an additional 4% systematic uncertainty in the measurements. No overall systematic uncertainty has been estimated for p<ce:glyph name="sbnd"/>Pb collisions, as the centrality selection in that collision system is inherently difficult to map to the underlying dynamics of the collisions <ce:cross-ref refid="br0200" id="crf11100">[20]</ce:cross-ref>.</ce:para></ce:section><ce:section id="se0030" role="results"><ce:label>3</ce:label><ce:section-title id="st0040">Results</ce:section-title><ce:para id="pr0120"><ce:cross-ref refid="fg0010" id="crf11420">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/> shows the measured pseudorapidity densities in pp, and in central p<ce:glyph name="sbnd"/>Pb, and the previously published results for Pb<ce:glyph name="sbnd"/>Pb <ce:cross-ref refid="br0020" id="crf11120">[2]</ce:cross-ref> collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> for primary particles.</ce:para><ce:para id="pr0130">For the 5% most central Pb<ce:glyph name="sbnd"/>Pb collisions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:mo>≈</mml:mo><mml:mn>2000</mml:mn></mml:math> at midrapidity (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>) <ce:cross-ref refid="br0020" id="crf11130">[2]</ce:cross-ref>, while for p<ce:glyph name="sbnd"/>Pb collisions the distribution peaks at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mn>60</mml:mn></mml:math> around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>3</mml:mn></mml:math> in the lead-going direction (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>). For pp collisions with the INEL>0 trigger condition discussed above, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.7</mml:mn><mml:mo>±</mml:mo><mml:mn>0.2</mml:mn></mml:math> at midrapidity, consistent with previous results derived from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra <ce:cross-ref refid="br0240" id="crf11140">[24]</ce:cross-ref>.</ce:para><ce:para id="pr0140"><ce:cross-ref refid="fg0020" id="crf11430">Fig. 2</ce:cross-ref><ce:float-anchor refid="fg0020"/> shows, as a function of centrality, the measured charged-particle pseudorapidity densities for p<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>. The strategy of centrality selection for proton on nucleus reactions is explained elsewhere <ce:cross-ref refid="br0200" id="crf11160">[20]</ce:cross-ref>. The ALICE Collaboration has previously presented <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math> for Pb<ce:glyph name="sbnd"/>Pb collisions at this energy <ce:cross-ref refid="br0020" id="crf11170">[2]</ce:cross-ref>.</ce:para><ce:para id="pr0150">In <ce:cross-ref refid="fg0030" id="crf11180">Fig. 3</ce:cross-ref><ce:float-anchor refid="fg0030"/>, the charged-particle pseudorapidity densities in p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb reactions are divided by the pp distributions corresponding to the INEL>0 trigger class. The ratio is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pp</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>, where <ce:italic> X</ce:italic> labels p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb collisions, in centrality classes, as a function of pseudorapidity. In the ratios, systematic uncertainties, of common origin, are partially cancelled, and, as an estimate, the magnitude of the resulting systematic uncertainties are given only by the uncertainties in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:math> measurements, since the uncertainties are independent of the collision system. In p<ce:glyph name="sbnd"/>Pb collisions the rapidity of the centre-of-mass is non-zero, which is not taken into account in the ratios. Such a correction would require prior determination of the full Jacobian of the transformation from pseudorapidity to rapidity, which is not possible to perform reliably with the ALICE apparatus.</ce:para><ce:para id="pr0160">The ratio of the p<ce:glyph name="sbnd"/>Pb relative to the pp distributions increases with pseudorapidity from the p-going to the Pb-going direction for central collisions, which Brodsky et al. and Adil et al. <ce:cross-refs refid="br0250 br0260" id="crs0080">[25,26]</ce:cross-refs> suggest is a sign of scaling of the pp distribution with the increasing number of participants as the lead nucleus is probed by the incident proton, and thus independent proton–nucleon scatterings on the lead-ion side. A similar scaling, however, does not hold for the Pb<ce:glyph name="sbnd"/>Pb reaction. The ratios cannot be obtained by simple scaling of the elementary pp distributions. Instead, the ratio of the Pb<ce:glyph name="sbnd"/>Pb relative to the pp distributions exhibits an enhancement of particle production around midrapidity for the more central collisions which is indicative of the formation of the sQGP <ce:cross-ref refid="br0070" id="crf11190">[7]</ce:cross-ref>. Likewise, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="normal">pPb</mml:mtext></mml:mrow></mml:msub></mml:math> increases for all but the two most peripheral centrality classes as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:mn>3</mml:mn></mml:math>. In Pb<ce:glyph name="sbnd"/>Pb collisions it is seen that the various mechanisms behind the pseudorapidity distributions are more transversely directed than in pp collisions by the increase of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="normal">PbPb</mml:mtext></mml:mrow></mml:msub></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mn>0</mml:mn></mml:math></ce:para></ce:section><ce:section id="se0040"><ce:label>4</ce:label><ce:section-title id="st0050">Rapidity and energy-density dependence on system size and discussion</ce:section-title><ce:para id="pr0170">It has been shown that the charged-particle <ce:italic>rapidity</ce:italic> density (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math>) in Pb<ce:glyph name="sbnd"/>Pb collisions, to a good accuracy, follows a normal distribution over the considered rapidity interval (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo>≲</mml:mo><mml:mn>5</mml:mn></mml:math>) <ce:cross-refs refid="br0020 br0270" id="crs0090">[2,27]</ce:cross-refs>. Those results relied on calculating the average Jacobian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>J</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math> using the full <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra, at midrapidity, of charged pions and kaons as well as protons and antiprotons. Here, we use the approximation<ce:display><ce:formula id="fm0010"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mrow><mml:mi>y</mml:mi><mml:mo>≈</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">cos</mml:mi><mml:mo>⁡</mml:mo><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>ϑ</ce:italic> is the polar angle of emission, and identify <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mi>a</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>m</mml:mi></mml:math> with an effective ratio of transverse momentum over mass. With this, the effective Jacobian can be written as<ce:display><ce:formula id="fm0020"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">cosh</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>⁡</mml:mo><mml:mi>η</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0180">We further make the ansatz that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> is normal distributed for symmetric collision systems (pp and Pb<ce:glyph name="sbnd"/>Pb), so that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math> can be parameterised as<ce:display><ce:formula id="fm0030"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mi>A</mml:mi><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:msqrt><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">{</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>A</ce:italic> and <ce:italic>σ</ce:italic> are the total integral and width of the distribution, respectively, and <ce:italic>y</ce:italic> the rapidity in the centre-of-mass frame. Motivated by the observed approximate linearity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pPb</mml:mi></mml:mrow></mml:msub></mml:math> (see lower panel of <ce:cross-ref refid="fg0030" id="crf11200">Fig. 3</ce:cross-ref>), we replace <ce:italic>A</ce:italic> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mi>y</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> for the asymmetric system (p<ce:glyph name="sbnd"/>Pb) and parameterise <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub></mml:math> as<ce:display><ce:formula id="fm0040"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo id="mmlbr0001" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>α</mml:mi><mml:mi>y</mml:mi><mml:mo stretchy="false">{</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="true" linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="after">)</mml:mo></mml:mrow><mml:mspace width="1em"/><mml:mo>×</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:msqrt><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">{</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">CM</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0190">The functions <ce:italic>f</ce:italic> and <ce:italic>g</ce:italic> defined in Eq. <ce:cross-ref refid="fm0030" id="crf11210">(1)</ce:cross-ref> and Eq. <ce:cross-ref refid="fm0040" id="crf11220">(2)</ce:cross-ref>, respectively, describe the measurements within the measured region with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> per degrees of freedom (<ce:italic>ν</ce:italic>) in the range of 0.1 to 0.5. The small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mi>ν</mml:mi></mml:math> values are a consequence of the relatively large uncorrelated systematic uncertainties on the measurements. That is, the charged-particle distributions for pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> follow a normal distribution in rapidity, with free parameters <ce:italic>A</ce:italic>, <ce:italic> a</ce:italic>, <ce:italic>σ</ce:italic>, and <ce:italic>α</ce:italic> in the asymmetric case.</ce:para><ce:para id="pr0200">The top panel of <ce:cross-ref refid="fg0040" id="crf11230">Fig. 4</ce:cross-ref><ce:float-anchor refid="fg0040"/> shows the best-fit parameter values of the normal width (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math>) for all three collision systems as a function of the average number of participating nucleons (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo></mml:math>) calculated using a Glauber model <ce:cross-ref refid="br0280" id="crf11240">[28]</ce:cross-ref>. The best-fit parameters are found taking statistical and uncorrelated systematic uncertainties into account. The result using the above procedure, for the most central Pb<ce:glyph name="sbnd"/>Pb collisions, is found to be compatible with previous results extracted by unfolding with the mean Jacobian estimated from transverse momentum spectra <ce:cross-ref refid="br0020" id="crf11250">[2]</ce:cross-ref>. The open points (crosses) and dashed lines on the figure are from evaluations of Eq. <ce:cross-ref refid="fm0030" id="crf11260">(1)</ce:cross-ref> and Eq. <ce:cross-ref refid="fm0040" id="crf11270">(2)</ce:cross-ref>, and direct calculations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math>, respectively, using model calculations with EPOS-LHC <ce:cross-ref refid="br0290" id="crf11280">[29]</ce:cross-ref>. EPOS-LHC was chosen as it provides predictions for all three collision systems. The parameterisation, in terms of the two functions, of this model calculation generally reproduces the widths of the charged-particle rapidity densities, except in the asymmetric case where a direct evaluation of the standard deviation is less motivated.</ce:para><ce:para id="pr0210">The general trend is that the widths decrease as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo></mml:math> increases, consistent with the behaviour of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="normal">PbPb</mml:mtext></mml:mrow></mml:msub></mml:math> ratios. Notably, the width of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> distributions in p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb, for low number of participant nucleons in the collisions, approaches the width of the pp distribution, which, presumably, is dominated by kinematic and phase space constraints.</ce:para><ce:para id="pr0220">The lower panel of <ce:cross-ref refid="fg0040" id="crf11290">Fig. 4</ce:cross-ref> shows the dependence of <ce:italic>a</ce:italic> on the average number of participants. The right-hand ordinate is the same, but multiplied by the average mass <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>0.215</mml:mn><mml:mo>±</mml:mo><mml:mn>0.001</mml:mn><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mtext>GeV</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="italic">c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> estimated from measurements of identified particles in Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math><ce:cross-ref refid="br0300" id="crf11300">[30]</ce:cross-ref>. To better understand the parameter <ce:italic>a</ce:italic>, this parameter extracted from the EPOS-LHC calculations, using the above procedure, is also shown in the figure. The dotted lines show the average <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>m</mml:mi></mml:math> predicted by EPOS-LHC <ce:cross-ref refid="br0290" id="crf11310">[29]</ce:cross-ref>. The EPOS-LHC calculations indicate that the extracted effective transverse momentum to mass ratio <ce:italic> a</ce:italic> is consistently smaller than the ratio of the average transverse momentum to the average mass. Thus <ce:italic>a</ce:italic> gives a lower bound on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math>.</ce:para><ce:para id="pr0230">We can estimate the energy density that is reached in the collisions as a function of the number of participants for the three systems. A conventional approach is to use the model originally proposed by Bjorken <ce:cross-ref refid="br0120" id="crf11320">[12]</ce:cross-ref> in which the energy density (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub></mml:math>) depends on the rapidity density of particles and the volume of a longitudinal cylinder with cross sectional area determined by the overlap between the colliding partners and length determined by a characteristic particle formation time<ce:display><ce:formula id="fm0050"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>τ</mml:mi></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">〈</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">〉</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Here, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mi>π</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>≈</mml:mo><mml:mi>π</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msubsup></mml:math> is the transverse area spanned by the participating nucleons, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> is the transverse-energy rapidity density, and <ce:italic>τ</ce:italic> is the formation time. While a formation time of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:mi>τ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">c</mml:mi></mml:math> is often assumed, it is left as a free parameter here. With <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt></mml:math>, the transverse-energy rapidity density can be approximated by<ce:display><ce:formula id="fm0060"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:mrow><mml:mrow><mml:mo stretchy="true">〈</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">〉</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0.55</mml:mn><mml:mo>±</mml:mo><mml:mn>0.01</mml:mn></mml:math>, the ratio of charged particles to all particles <ce:cross-ref refid="br0310" id="crf11330">[31]</ce:cross-ref>, accounts for neutral particles not measured in the experiment, and is assumed the same for all collision systems. Substituting the derived <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> and the effective <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mi>a</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>m</mml:mi><mml:mo>≲</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math> results in a lower bound estimate for the Bjorken energy density (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">LB</mml:mi></mml:mrow></mml:msub></mml:math> )<ce:display><ce:formula id="fm0070"><ce:label>(3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi><mml:mo>≥</mml:mo><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">LB</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">cosh</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>⁡</mml:mo><mml:mi>η</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>a</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math> are as in the top panel of <ce:cross-ref refid="fg0040" id="crf11340">Fig. 4</ce:cross-ref> .</ce:para><ce:para id="pr0240">The transverse area <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> is estimated in a numerical Glauber model <ce:cross-refs refid="br0320 br0330" id="crs0100">[32,33]</ce:cross-refs> as shown in <ce:cross-ref refid="fg0050" id="crf11350">Fig. 5</ce:cross-ref><ce:float-anchor refid="fg0050"/>. We consider two extremes for the transverse area spanned by the participating nucleons: a) the <ce:italic>exclusive</ce:italic> (or direct) overlap between participating nucleons, ∩ and open markers in <ce:cross-ref refid="fg0050" id="crf11360">Fig. 5</ce:cross-ref>, and b) the <ce:italic> inclusive</ce:italic> (or full) area of all participating nucleons, ∪ and full markers in <ce:cross-ref refid="fg0050" id="crf11370">Fig. 5</ce:cross-ref>.</ce:para><ce:para id="pr0250"><ce:cross-ref refid="fg0060" id="crf11440">Fig. 6</ce:cross-ref><ce:float-anchor refid="fg0060"/> shows the lower-bound energy density estimate, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">LB</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi></mml:math>, as a function of the number of participants, which reaches values between 10 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:mn>20</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> in the most central Pb<ce:glyph name="sbnd"/>Pb collisions. The uncertainties are from standard error propagation of Eq. <ce:cross-ref refid="fm0070" id="crf11390">(3)</ce:cross-ref> of uncertainties on the best-fit parameter values, the number of participants, mean mass, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:math>. A rise from roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si79.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> to over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:mn>10</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is observed if the transverse area is assumed to be the inclusive area of participating nucleons. This trend is illustrated by a power-law (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si81.svg"><mml:mi>C</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msubsup></mml:math>) fit to the data in the figure, with the parameter values <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si82.svg"><mml:mi>C</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>0.8</mml:mn><mml:mo>±</mml:mo><mml:mn>0.3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi>p</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0.44</mml:mn><mml:mo>±</mml:mo><mml:mn>0.08</mml:mn></mml:math>. On the other hand, if the transverse area is assumed to be the smaller exclusive overlap area, we observe a substantially larger lower bound on the energy density, but a less dramatic increase with increasing number of participating nucleons. Also shown in the figure are estimates of the Bjorken energy density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi></mml:math> for Pb<ce:glyph name="sbnd"/>Pb reactions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math><ce:cross-ref refid="br0310" id="crf11400">[31]</ce:cross-ref>. These results where obtained from measurements of the transverse energy in the collisions and using the inclusive estimate of the transverse area <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>. The trend of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> results is similar to these earlier results. Bearing in mind that for the largest LHC collision energy we show a lower bound estimate of the energy density in <ce:cross-ref refid="fg0060" id="crf11410">Fig. 6</ce:cross-ref>, we find a likely overall increase in the energy density from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>.</ce:para></ce:section><ce:section id="se0050"><ce:label>5</ce:label><ce:section-title id="st0060">Summary and conclusions</ce:section-title><ce:para id="pr0260">We have measured the charged particle pseudorapidity density in pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> over the widest possible pseudorapidity range available at the LHC. The distributions where determined using the same experimental apparatus and methods, and systematic uncertainties have been minimised to within the capabilities of the set-up. While the particle production in central Pb<ce:glyph name="sbnd"/>Pb collisions clearly exhibits an enhancement as compared to pp collisions, particle production in p<ce:glyph name="sbnd"/>Pb collisions is consistent with dominantly incoherent nucleon–nucleon collisions. By transforming the measured pseudorapidity distributions to rapidity distributions we have obtained systematic trends for the width of the rapidity distributions and a lower bound on the energy density, which shows a clear scaling behaviour as a function of the average number of participant nucleons. The decreasing width of the deduced rapidity distributions with increasing participant number suggests that the kinematic spread of particles, including longitudinal degrees of freedom, is reduced due to interactions in the early stages of the collisions. This is also reflected in the accompanying growth of the energy density. Both observations are consistent with the gradual establishment of a high-density phase of matter with increasing size of the collision domain.</ce:para></ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0080">Declaration of Competing Interest</ce:section-title><ce:para id="pr0270">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0090">Acknowledgements</ce:section-title><ce:para id="pr0280">The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: <ce:grant-sponsor id="gsp0010">A. I. 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<ce:grant-sponsor id="gsp0690" sponsor-id="https://doi.org/10.13039/100020381">Turkish Energy, Nuclear and Mineral Research Agency</ce:grant-sponsor> (TENMAK), Turkey; <ce:grant-sponsor id="gsp0700" sponsor-id="https://doi.org/10.13039/501100004742">National Academy of Sciences of Ukraine</ce:grant-sponsor>, Ukraine; <ce:grant-sponsor id="gsp0710" sponsor-id="https://doi.org/10.13039/501100000271">Science and Technology Facilities Council</ce:grant-sponsor> (STFC), United Kingdom; National Science Foundation of the United States of America (<ce:grant-sponsor id="gsp0720" sponsor-id="https://doi.org/10.13039/100000001">NSF</ce:grant-sponsor>) and United States Department of Energy, Office of Nuclear Physics (<ce:grant-sponsor id="gsp0730" sponsor-id="https://doi.org/10.13039/100006209">DOE NP</ce:grant-sponsor>), United States of America.</ce:para></ce:acknowledgment></body><tail><ce:bibliography id="bl0010"><ce:section-title id="st0070">References</ce:section-title><ce:bibliography-sec id="bs0010"><ce:bib-reference id="br0010"><ce:label>[1]</ce:label><sb:reference id="bib866BD36D5E8AFE73B567F49DDE14CD65s1"><sb:contribution><sb:authors><sb:collaboration>BRAHMS Collaboration</sb:collaboration><sb:author><ce:given-name>I.C.</ce:given-name><ce:surname>Arsene</ce:surname></sb:author><sb:et-al/></sb:authors><sb:title><sb:maintitle>Nuclear stopping and rapidity loss in Au+Au collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>62.4</mml:mn><mml:mspace width="0.2em"/><mml:mtext>GeV</mml:mtext></mml:math></sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Phys. 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C94 (2016) 024914, arXiv:1603.07375 [nucl-ex].</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> \ No newline at end of file +<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd" [<!ENTITY gr001 SYSTEM "gr001" NDATA IMAGE><!ENTITY gr002 SYSTEM "gr002" NDATA IMAGE><!ENTITY gr003 SYSTEM "gr003" NDATA IMAGE><!ENTITY gr004 SYSTEM "gr004" NDATA IMAGE><!ENTITY gr005 SYSTEM "gr005" NDATA IMAGE><!ENTITY gr006 SYSTEM "gr006" NDATA IMAGE>]><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="sco" xml:lang="en"><item-info><jid>PLB</jid><aid>137730</aid><ce:article-number>137730</ce:article-number><ce:pii>S0370-2693(23)00064-3</ce:pii><ce:doi>10.1016/j.physletb.2023.137730</ce:doi><ce:copyright year="2023" type="other">European Center of Nuclear Research, ALICE experiment</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>Experiments</ce:text></ce:doctopic></ce:doctopics><ce:preprint><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/preprint" xlink:href="arxiv:2204.10210" id="inf0010"/></ce:preprint></item-info><ce:floats><ce:figure id="fg0010"><ce:label>Fig. 1</ce:label><ce:caption id="cp0010"><ce:simple-para id="sp0010">Charged-particle pseudorapidity density in Pb<ce:glyph name="sbnd"/>Pb <ce:cross-ref refid="br0020" id="crf0010">[2]</ce:cross-ref> and p<ce:glyph name="sbnd"/>Pb for the 5% most central collisions, and for pp collisions with INEL>0 trigger class. For symmetric collision systems (Pb<ce:glyph name="sbnd"/>Pb and pp) the data has been symmetrised around <ce:italic>η</ce:italic> = 0 and points for <ce:italic>η</ce:italic> > 3.5 have been reflected around <ce:italic> η</ce:italic> = 0. The boxes around the points and to the right reflect the uncorrelated and correlated, with respect to pseudorapidity, systematic uncertainty, respectively. The relative correlated, normalisation, uncertainties are evaluated at d<ce:italic>N</ce:italic><ce:inf>ch</ce:inf>/d<ce:italic>η</ce:italic> |<ce:inf><ce:italic>η</ce:italic>=0</ce:inf>. The lines show fits of Eq. <ce:cross-ref refid="fm0030" id="crf0020">(1)</ce:cross-ref> (Pb<ce:glyph name="sbnd"/>Pb and pp) and Eq. <ce:cross-ref refid="fm0040" id="crf0030">(2)</ce:cross-ref> (p<ce:glyph name="sbnd"/>Pb) to the data (discussed in Section <ce:cross-ref refid="se0040" id="crf0040">4</ce:cross-ref>). Please note that the ordinate has been cut twice to accommodate for the very different ranges of the charged-particle pseudorapidity densities.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0010">Fig. 1</ce:alt-text><ce:link locator="gr001" xlink:type="simple" xlink:href="pii:S0370269323000643/gr001" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0010"/></ce:figure><ce:figure id="fg0020"><ce:label>Fig. 2</ce:label><ce:caption id="cp0020"><ce:simple-para id="sp0020">Charged-particle pseudorapidity density in p<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> in seven centrality classes based on the V0A and V0C estimators. The lines are obtained using a fit of a scaled, normal distribution in rapidity Eq. <ce:cross-ref refid="fm0040" id="crf0050">(2)</ce:cross-ref> to the data (discussed in Section <ce:cross-ref refid="se0040" id="crf0060">4</ce:cross-ref>).</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0020">Fig. 2</ce:alt-text><ce:link locator="gr002" xlink:type="simple" xlink:href="pii:S0370269323000643/gr002" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0020"/></ce:figure><ce:figure id="fg0030"><ce:label>Fig. 3</ce:label><ce:caption id="cp0030"><ce:simple-para id="sp0030">Ratio <ce:italic>r</ce:italic><ce:inf><ce:italic>X</ce:italic></ce:inf> of the charged-particle pseudorapidity density in Pb<ce:glyph name="sbnd"/>Pb (top) and p<ce:glyph name="sbnd"/>Pb (bottom) in different centrality classes to the charged-particle pseudorapidity density in pp in the INEL>0 event class. Note, for Pb<ce:glyph name="sbnd"/>Pb <ce:italic>η</ce:italic><ce:inf>lab</ce:inf> is the same as the centre-of-mass pseudorapidity.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0030">Fig. 3</ce:alt-text><ce:link locator="gr003" xlink:type="simple" xlink:href="pii:S0370269323000643/gr003" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0030"/></ce:figure><ce:figure id="fg0040"><ce:label>Fig. 4</ce:label><ce:caption id="cp0040"><ce:simple-para id="sp0040">The width (top) and effective <ce:italic>p</ce:italic><ce:inf> T</ce:inf>/<ce:italic>m</ce:italic> (bottom) fit parameters as a function of the mean number of participants in pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>. Vertical uncertainties are the standard error on the best-fit parameter values, while horizontal uncertainties reflect the uncertainty on 〈<ce:italic>N</ce:italic><ce:inf> part</ce:inf>〉 from the Glauber calculations. Also shown are similar fit parameters from the same parameterisation of EPOS-LHC calculations as well as direct calculations of the standard deviation of the d<ce:italic>N</ce:italic><ce:inf> ch</ce:inf>/d<ce:italic>y</ce:italic> distributions and the 〈<ce:italic>p</ce:italic><ce:inf> T</ce:inf>〉/〈<ce:italic>m</ce:italic>〉 ratio from the EPOS-LHC calculations.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0040">Fig. 4</ce:alt-text><ce:link locator="gr004" xlink:type="simple" xlink:href="pii:S0370269323000643/gr004" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0040"/></ce:figure><ce:figure id="fg0050"><ce:label>Fig. 5</ce:label><ce:caption id="cp0050"><ce:simple-para id="sp0050">The transverse area <ce:italic>S</ce:italic><ce:inf>T</ce:inf> as calculated in a numerical Glauber model for two extreme cases: a) only the exclusive overlap of nucleons is considered (∩, open markers) and b) the inclusive area of participating nucleons contribute (∪, closed markers) in both p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0050">Fig. 5</ce:alt-text><ce:link locator="gr005" xlink:type="simple" xlink:href="pii:S0370269323000643/gr005" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0050"/></ce:figure><ce:figure id="fg0060"><ce:label>Fig. 6</ce:label><ce:caption id="cp0060"><ce:simple-para id="sp0060">Estimate of the lower bound on the Bjorken transverse energy density in pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>, considering the exclusive (∩, open markers) and inclusive (∪, full markers) overlap area <ce:italic> S</ce:italic><ce:inf>T</ce:inf> of the nucleons. The expression <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si74.svg"><mml:mi>C</mml:mi><mml:mmultiscripts><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow><mml:none/><mml:none/><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:mmultiscripts></mml:math> is fitted to case ∪, and we find <ce:italic>C</ce:italic> = (0.8 ± 0.3) GeV/(fm<ce:sup> 2</ce:sup><ce:italic>c</ce:italic>) and <ce:italic>p</ce:italic> = 0.44 ± 0.08. Also shown is an estimate, via d<ce:italic>E</ce:italic><ce:inf>T</ce:inf>/d<ce:italic> y</ce:italic>, of <ce:italic>ε</ce:italic><ce:inf>Bj</ce:inf> from Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> (stars with uncertainty band) <ce:cross-ref refid="br0310" id="crf0070">[31]</ce:cross-ref>.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0060">Fig. 6</ce:alt-text><ce:link locator="gr006" xlink:type="simple" xlink:href="pii:S0370269323000643/gr006" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0060"/></ce:figure></ce:floats><head><ce:title id="ti0010">System-size dependence of the charged-particle pseudorapidity density at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> for pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions</ce:title><ce:author-group id="ag0010"><ce:collaboration id="co0010" collaboration-id="S0370269323000643-3bea72599603117cd9d18494a0279c47"><ce:text>ALICE Collaboration</ce:text><ce:cross-ref refid="fn0080" id="crf0080"><ce:sup>⋆</ce:sup></ce:cross-ref><ce:author-group id="ag0020"><ce:author id="au0010" author-id="S0370269323000643-e60c93a934b81cf9801254193264c6ee"><ce:given-name>S.</ce:given-name><ce:surname>Acharya</ce:surname><ce:cross-ref refid="aff1420" id="crf0090"><ce:sup>142</ce:sup></ce:cross-ref></ce:author><ce:author id="au0020" author-id="S0370269323000643-0eab85892b6d74b18661e74a7987c599"><ce:given-name>D.</ce:given-name><ce:surname>Adamová</ce:surname><ce:cross-ref refid="aff0960" id="crf0100"><ce:sup>96</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="aff1420" affiliation-id="S0370269323000643-f1ae52f852d4d7d99988b3e872f887e4"><ce:label>142</ce:label><ce:textfn>Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Variable Energy Cyclotron Centre</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0710">Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0960" affiliation-id="S0370269323000643-35f0d35e6b51b7c24d9b3bc939a45b30"><ce:label>96</ce:label><ce:textfn>Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Institute of the Czech Academy of Sciences</sa:organization><sa:city>Řež u Prahy</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0480">Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy, Czech Republic</ce:source-text></ce:affiliation></ce:author-group></ce:collaboration><ce:footnote id="fn0080"><ce:label>⋆</ce:label><ce:note-para id="np0080"><ce:italic>E-mail address:</ce:italic><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/text/html" xlink:href="mailto:alice-publications@cern.ch" id="inf0020"> alice-publications@cern.ch</ce:inter-ref>.</ce:note-para></ce:footnote></ce:author-group><ce:date-received day="20" month="5" year="2022"/><ce:date-revised day="17" month="1" year="2023"/><ce:date-accepted day="17" month="1" year="2023"/><ce:miscellaneous id="ms0010">Editor: M. Doser</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0070">We present the first systematic comparison of the charged-particle pseudorapidity densities for three widely different collision systems, pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb, at the top energy of the Large Hadron Collider (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>) measured over a wide pseudorapidity range (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.5</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5</mml:mn></mml:math>), the widest possible among the four experiments at that facility. The systematic uncertainties are minimised since the measurements are recorded by the same experimental apparatus (ALICE). The distributions for p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb collisions are determined as a function of the centrality of the collisions, while results from pp collisions are reported for inelastic events with at least one charged particle at midrapidity. The charged-particle pseudorapidity densities are, under simple and robust assumptions, transformed to charged-particle rapidity densities. This allows for the calculation and the presentation of the evolution of the width of the rapidity distributions and of a lower bound on the Bjorken energy density, as a function of the number of participants in all three collision systems. We find a decreasing width of the particle production, and roughly a smooth ten fold increase in the energy density, as the system size grows, which is consistent with a gradually higher dense phase of matter.</ce:simple-para></ce:abstract-sec></ce:abstract></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">The number of charged particles produced in energetic nuclear collisions is an important indicator for the strong interaction processes that determine the particle production at the sub-nucleonic level. In particular, the production of charged particles is expected to reflect the number of quark and gluon collisions occurring during the initial stages of the reaction. The total number of particles produced also provides information on the energy transfer available from the initial colliding beams to particle production, as a consequence of nuclear stopping <ce:cross-ref refid="br0010" id="crf10940">[1]</ce:cross-ref>. In order to help unravel this complex scenario it is important to compare the particle production amongst collision systems of different sizes over a wide kinematic range.</ce:para><ce:para id="pr0020">We present the measured charged-particle pseudorapidity density, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math>, for pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb (previously published <ce:cross-ref refid="br0020" id="crf10950">[2]</ce:cross-ref>) collisions at the same collision energy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> in the nucleon–nucleon centre-of-mass reference frame. This is, at present, the maximum available energy at CERN's Large Hadron Collider (LHC) for Pb<ce:glyph name="sbnd"/>Pb collisions. The measurements were carried out using ALICE at LHC (for earlier <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math> results see for example Refs. <ce:cross-refs refid="br0030 br0040 br0050" id="crs0010">[3–5]</ce:cross-refs>). The three studied reactions have different characteristics probing widely different particle production yields and mechanisms. In Pb<ce:glyph name="sbnd"/>Pb collisions, the total particle yield for central collisions is of the order 10<ce:sup> 4</ce:sup><ce:cross-ref refid="br0020" id="crf10960">[2]</ce:cross-ref>, and a strongly coupled plasma of quarks and gluons (sQGP) is formed <ce:cross-refs refid="br0060 br0070 br0080 br0090" id="crs0020">[6–9]</ce:cross-refs>, whose collective and transport properties are currently under intense study. On the other hand, pp collisions represent the simplest possible nuclear collision system, where the average total particle production is much smaller (≈80, by integrating the measured distributions), and is to first approximation much less subject to collective effects <ce:cross-ref refid="br0100" id="crf10970">[10]</ce:cross-ref>. The p<ce:glyph name="sbnd"/>Pb system is intermediate to the other reactions, corresponding to the situation where a single nucleon probes the nucleons in a narrow cylinder of the target nucleus. The extent to which p<ce:glyph name="sbnd"/>Pb is governed by the initial state cold nuclear matter of the lead ion or whether collective phenomena in the hot and dense medium play an important role is, at present, a matter under scrutiny by the community <ce:cross-refs refid="br0100 br0110" id="crs0030">[10,11]</ce:cross-refs>.</ce:para><ce:para id="pr0030">In this letter, we compare the three reactions and present the ratios of the charged-particle pseudorapidity density distributions (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math>) of the more complex reactions to the pp distribution. Owing to ALICE's unique large acceptance in pseudorapidity, and using simple and robust assumptions, we transform the measured charged-particle pseudorapidity density distributions into charged-particle rapidity density distributions (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math>). This allows us to calculate the width of the rapidity distributions as a function of the number of participating nucleons. The parameters of the transformation also allow us to estimate a lower bound on the energy density using the well-known formula from Bjorken <ce:cross-ref refid="br0120" id="crf10980">[12]</ce:cross-ref>. An energy density exceeding the critical energy density of roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math><ce:cross-ref refid="br0130" id="crf10990">[13]</ce:cross-ref> is a necessary condition for the formation of deconfined matter of quarks and gluons, and thus it is of the utmost interest to understand the development of these energy densities across different collision systems.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">Experimental set-up, data sample, analysis method, systematic uncertainties</ce:section-title><ce:para id="pr0040">A detailed description of the ALICE detector and its performance can be found elsewhere <ce:cross-refs refid="br0140 br0150" id="crs0040">[14,15]</ce:cross-refs>. The present analysis uses the Silicon Pixel Detector (SPD) to determine the pseudorapidity densities in the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math> and the Forward Multiplicity Detector (FMD) in the ranges <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.5</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1.8</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"><mml:mn>1.8</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5</mml:mn></mml:math>. The V0, comprised of two plastic scintillator discs covering <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.7</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1.7</mml:mn></mml:math> (V0C) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:mn>2.8</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5.1</mml:mn></mml:math> (V0A), and the ZDC, two zero-degree calorimeters located <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>112.5</mml:mn><mml:mspace width="0.2em"/><mml:mtext>m</mml:mtext></mml:math> from the interaction point, measurements determine the collision centrality and are used for offline event selection <ce:cross-ref refid="br0020" id="crf11000">[2]</ce:cross-ref>.</ce:para><ce:para id="pr0050">The results presented are based on data from collisions at a centre-of-mass energy per nucleon pair of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> as collected by ALICE during LHC Run 1 (2013) for p<ce:glyph name="sbnd"/>Pb, and during Run 2 (2015) for pp and Pb<ce:glyph name="sbnd"/>Pb. The FMD suffered high levels of background noise during the 2016 p<ce:glyph name="sbnd"/>Pb campaign, due to the high collision rate, and this data is therefore not used for the present analysis. About 10<ce:sup>5</ce:sup> events with a minimum bias trigger requirement <ce:cross-ref refid="br0020" id="crf11010">[2]</ce:cross-ref> were analysed in the centrality range from 0% to 90% and 0% to 100% of the visible cross section for Pb<ce:glyph name="sbnd"/>Pb and p<ce:glyph name="sbnd"/>Pb collisions, respectively. The minimum bias trigger for p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb collisions in ALICE was defined as a coincidence between the V0A and V0C sides of the V0 detector.</ce:para><ce:para id="pr0060">The data from the p<ce:glyph name="sbnd"/>Pb collisions were taken in two beam configurations: one where the lead ion travelled toward positive pseudorapidity and one where it travelled toward negative pseudorapidity. The results from the latter collisions are mirrored around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>. The centre-of-mass frame in p<ce:glyph name="sbnd"/>Pb collisions does not coincide with the laboratory frame, due to the single magnetic field in the LHC, and thus the rapidity of the centre-of-mass is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">CM</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo>±</mml:mo><mml:mn>0.465</mml:mn></mml:math> for the two directions, respectively, in the laboratory frame. For this reason, pseudorapidity, calculated with respect to the laboratory frame, is denoted <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub></mml:math> whenever p<ce:glyph name="sbnd"/>Pb results are presented.</ce:para><ce:para id="pr0070">Likewise, for the pp collisions, about 10<ce:sup>5</ce:sup> events with coincidence between V0A and V0C and at least one charged particle in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1</mml:mn></mml:math> were analysed. By requiring at least one charged particle at midrapidity, the so-called INEL>0 event class, the systematic uncertainty, related to the absolute normalisation to the full inelastic cross section, is reduced, while still sampling a large fraction (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>75</mml:mn><mml:mtext>%</mml:mtext></mml:math>) of the hadronic cross section <ce:cross-refs refid="br0160 br0170" id="crs0050">[16,17]</ce:cross-refs> .</ce:para><ce:para id="pr0080">The standard ALICE event selection <ce:cross-ref refid="br0180" id="crf11020">[18]</ce:cross-ref> and centrality estimator based on the V0 amplitude <ce:cross-refs refid="br0190 br0200" id="crs0060">[19,20]</ce:cross-refs> are used in this analysis. The event selection consists of: a) exclusion of background events using the timing information from the ZDC (for Pb<ce:glyph name="sbnd"/>Pb and p<ce:glyph name="sbnd"/>Pb, e.g., beam–gas interactions) and V0 detectors, b) verification of the trigger conditions, and c) a reconstructed position of the collision (primary vertex). In Pb<ce:glyph name="sbnd"/>Pb collisions, centrality is obtained from the sum amplitude in both V0 detector arrays (V0M). For p<ce:glyph name="sbnd"/>Pb only the amplitude in the array on the lead-going side (V0A or V0C) is used. In Pb<ce:glyph name="sbnd"/>Pb collisions, the 10% most peripheral collisions have substantial contributions from electromagnetic processes and are therefore not included in the results presented here <ce:cross-ref refid="br0190" id="crf11030">[19]</ce:cross-ref>.</ce:para><ce:para id="pr0090">A primary charged particle is defined as a charged particle with a mean proper lifetime <ce:italic>τ</ce:italic> larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mtext>cm</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mtext>c</mml:mtext></mml:math>, which is either a) produced directly in the interaction, or b) from decays of particles with <ce:italic> τ</ce:italic> smaller than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mtext>cm</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mtext>c</mml:mtext></mml:math><ce:cross-ref refid="br0210" id="crf11040">[21]</ce:cross-ref>. All quantities reported here are for primary, charged particles, though “primary” is omitted in the following for brevity.</ce:para><ce:para id="pr0100">The analysis method is identical to that of previous publications <ce:cross-ref refid="br0020" id="crf11050">[2]</ce:cross-ref>: the measurement of the charged-particle pseudorapidity density at midrapidity is obtained from counting particle trajectories determined using the two layers of the SPD. The SPD has a lower transverse momentum acceptance of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:mn>50</mml:mn><mml:mspace width="0.2em"/><mml:mtext>MeV</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="italic">c</mml:mi></mml:math>, and the yield is extrapolated down to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.2em"/><mml:mtext>MeV</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="italic">c</mml:mi></mml:math> via simulations. In the forward regions, the measurement is provided by the analysis of the deposited energy signal in the FMD and a statistical method is employed to calculate the inclusive number of charged particles. A data-driven correction <ce:cross-ref refid="br0220" id="crf11060">[22]</ce:cross-ref>, based on separate measurements exploiting displaced collision vertices, is applied to remove the background from secondary particles.</ce:para><ce:para id="pr0110">Systematic uncertainty estimations for the midrapidity measurements are detailed elsewhere <ce:cross-refs refid="br0020 br0160 br0200" id="crs0070">[2,16,20]</ce:cross-refs>, and are from background suppression, transverse momentum extrapolation, weak decays, and simulations. The estimates are obtained through variation of thresholds and simulation studies. For pp (p<ce:glyph name="sbnd"/>Pb), the total systematic uncertainty amounts to 1.5% (2.7%) over the whole pseudorapidity range; while for Pb<ce:glyph name="sbnd"/>Pb the total systematic uncertainty is 2.6% at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math> and 2.9% at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2</mml:mn></mml:math>. The systematic uncertainty is mostly correlated over pseudorapidity for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math>, and largely independent of centrality. The uncertainty in the forward region, estimated via variations of thresholds and simulation studies, is the same for all collision systems and is uncorrelated across <ce:italic>η</ce:italic>, amounting to 6.9% for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>3.5</mml:mn></mml:math> and 6.4% elsewhere within the forward regions <ce:cross-ref refid="br0220" id="crf11070">[22]</ce:cross-ref>. In the figures of this letter, uncorrelated, local in pseudorapidity, systematic uncertainties are indicated by open boxes on the data points, while correlated systematic uncertainties, those that affect the overall scale and typically from event classification and selection, are indicated by filled boxes to the right of the data. The systematic uncertainty on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math>, due to the centrality class definition in Pb<ce:glyph name="sbnd"/>Pb, is estimated to vary from 0.6% for the most central to 9.5% for the most peripheral class <ce:cross-ref refid="br0230" id="crf11080">[23]</ce:cross-ref>. The 80% to 90% centrality class has residual contamination from electromagnetic processes as detailed elsewhere <ce:cross-ref refid="br0190" id="crf11090">[19]</ce:cross-ref>, which gives rise to an additional 4% systematic uncertainty in the measurements. No overall systematic uncertainty has been estimated for p<ce:glyph name="sbnd"/>Pb collisions, as the centrality selection in that collision system is inherently difficult to map to the underlying dynamics of the collisions <ce:cross-ref refid="br0200" id="crf11100">[20]</ce:cross-ref>.</ce:para></ce:section><ce:section id="se0030" role="results"><ce:label>3</ce:label><ce:section-title id="st0040">Results</ce:section-title><ce:para id="pr0120"><ce:cross-ref refid="fg0010" id="crf11420">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/> shows the measured pseudorapidity densities in pp, and in central p<ce:glyph name="sbnd"/>Pb, and the previously published results for Pb<ce:glyph name="sbnd"/>Pb <ce:cross-ref refid="br0020" id="crf11120">[2]</ce:cross-ref> collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> for primary particles.</ce:para><ce:para id="pr0130">For the 5% most central Pb<ce:glyph name="sbnd"/>Pb collisions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:mo>≈</mml:mo><mml:mn>2000</mml:mn></mml:math> at midrapidity (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0</mml:mn></mml:math>) <ce:cross-ref refid="br0020" id="crf11130">[2]</ce:cross-ref>, while for p<ce:glyph name="sbnd"/>Pb collisions the distribution peaks at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mn>60</mml:mn></mml:math> around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>3</mml:mn></mml:math> in the lead-going direction (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0</mml:mn></mml:math>). For pp collisions with the INEL>0 trigger condition discussed above, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.7</mml:mn><mml:mo>±</mml:mo><mml:mn>0.2</mml:mn></mml:math> at midrapidity, consistent with previous results derived from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra <ce:cross-ref refid="br0240" id="crf11140">[24]</ce:cross-ref>.</ce:para><ce:para id="pr0140"><ce:cross-ref refid="fg0020" id="crf11430">Fig. 2</ce:cross-ref><ce:float-anchor refid="fg0020"/> shows, as a function of centrality, the measured charged-particle pseudorapidity densities for p<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>. The strategy of centrality selection for proton on nucleus reactions is explained elsewhere <ce:cross-ref refid="br0200" id="crf11160">[20]</ce:cross-ref>. The ALICE Collaboration has previously presented <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math> for Pb<ce:glyph name="sbnd"/>Pb collisions at this energy <ce:cross-ref refid="br0020" id="crf11170">[2]</ce:cross-ref>.</ce:para><ce:para id="pr0150">In <ce:cross-ref refid="fg0030" id="crf11180">Fig. 3</ce:cross-ref><ce:float-anchor refid="fg0030"/>, the charged-particle pseudorapidity densities in p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb reactions are divided by the pp distributions corresponding to the INEL>0 trigger class. The ratio is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pp</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>, where <ce:italic> X</ce:italic> labels p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb collisions, in centrality classes, as a function of pseudorapidity. In the ratios, systematic uncertainties, of common origin, are partially cancelled, and, as an estimate, the magnitude of the resulting systematic uncertainties are given only by the uncertainties in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:math> measurements, since the uncertainties are independent of the collision system. In p<ce:glyph name="sbnd"/>Pb collisions the rapidity of the centre-of-mass is non-zero, which is not taken into account in the ratios. Such a correction would require prior determination of the full Jacobian of the transformation from pseudorapidity to rapidity, which is not possible to perform reliably with the ALICE apparatus.</ce:para><ce:para id="pr0160">The ratio of the p<ce:glyph name="sbnd"/>Pb relative to the pp distributions increases with pseudorapidity from the p-going to the Pb-going direction for central collisions, which Brodsky et al. and Adil et al. <ce:cross-refs refid="br0250 br0260" id="crs0080">[25,26]</ce:cross-refs> suggest is a sign of scaling of the pp distribution with the increasing number of participants as the lead nucleus is probed by the incident proton, and thus independent proton–nucleon scatterings on the lead-ion side. A similar scaling, however, does not hold for the Pb<ce:glyph name="sbnd"/>Pb reaction. The ratios cannot be obtained by simple scaling of the elementary pp distributions. Instead, the ratio of the Pb<ce:glyph name="sbnd"/>Pb relative to the pp distributions exhibits an enhancement of particle production around midrapidity for the more central collisions which is indicative of the formation of the sQGP <ce:cross-ref refid="br0070" id="crf11190">[7]</ce:cross-ref>. Likewise, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="normal">pPb</mml:mtext></mml:mrow></mml:msub></mml:math> increases for all but the two most peripheral centrality classes as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:mn>3</mml:mn></mml:math>. In Pb<ce:glyph name="sbnd"/>Pb collisions it is seen that the various mechanisms behind the pseudorapidity distributions are more transversely directed than in pp collisions by the increase of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="normal">PbPb</mml:mtext></mml:mrow></mml:msub></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mn>0</mml:mn></mml:math></ce:para></ce:section><ce:section id="se0040"><ce:label>4</ce:label><ce:section-title id="st0050">Rapidity and energy-density dependence on system size and discussion</ce:section-title><ce:para id="pr0170">It has been shown that the charged-particle <ce:italic>rapidity</ce:italic> density (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math>) in Pb<ce:glyph name="sbnd"/>Pb collisions, to a good accuracy, follows a normal distribution over the considered rapidity interval (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo>≲</mml:mo><mml:mn>5</mml:mn></mml:math>) <ce:cross-refs refid="br0020 br0270" id="crs0090">[2,27]</ce:cross-refs>. Those results relied on calculating the average Jacobian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>J</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math> using the full <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra, at midrapidity, of charged pions and kaons as well as protons and antiprotons. Here, we use the approximation<ce:display><ce:formula id="fm0010"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mrow><mml:mi>y</mml:mi><mml:mo>≈</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">cos</mml:mi><mml:mo>⁡</mml:mo><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>ϑ</ce:italic> is the polar angle of emission, and identify <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mi>a</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>m</mml:mi></mml:math> with an effective ratio of transverse momentum over mass. With this, the effective Jacobian can be written as<ce:display><ce:formula id="fm0020"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">cosh</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>⁡</mml:mo><mml:mi>η</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0180">We further make the ansatz that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> is normal distributed for symmetric collision systems (pp and Pb<ce:glyph name="sbnd"/>Pb), so that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:math> can be parameterised as<ce:display><ce:formula id="fm0030"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mi>A</mml:mi><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:msqrt><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">{</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>A</ce:italic> and <ce:italic>σ</ce:italic> are the total integral and width of the distribution, respectively, and <ce:italic>y</ce:italic> the rapidity in the centre-of-mass frame. Motivated by the observed approximate linearity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pPb</mml:mi></mml:mrow></mml:msub></mml:math> (see lower panel of <ce:cross-ref refid="fg0030" id="crf11200">Fig. 3</ce:cross-ref>), we replace <ce:italic>A</ce:italic> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mi>y</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> for the asymmetric system (p<ce:glyph name="sbnd"/>Pb) and parameterise <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lab</mml:mi></mml:mrow></mml:msub></mml:math> as<ce:display><ce:formula id="fm0040"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo id="mmlbr0001" linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>α</mml:mi><mml:mi>y</mml:mi><mml:mo stretchy="false">{</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="true" linebreak="newline" indentalign="id" indenttarget="mmlbr0001" linebreakstyle="after">)</mml:mo></mml:mrow><mml:mspace width="1em"/><mml:mo>×</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:msqrt><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">exp</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">{</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">}</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">CM</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0190">The functions <ce:italic>f</ce:italic> and <ce:italic>g</ce:italic> defined in Eq. <ce:cross-ref refid="fm0030" id="crf11210">(1)</ce:cross-ref> and Eq. <ce:cross-ref refid="fm0040" id="crf11220">(2)</ce:cross-ref>, respectively, describe the measurements within the measured region with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> per degrees of freedom (<ce:italic>ν</ce:italic>) in the range of 0.1 to 0.5. The small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:msup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mi>ν</mml:mi></mml:math> values are a consequence of the relatively large uncorrelated systematic uncertainties on the measurements. That is, the charged-particle distributions for pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> follow a normal distribution in rapidity, with free parameters <ce:italic>A</ce:italic>, <ce:italic> a</ce:italic>, <ce:italic>σ</ce:italic>, and <ce:italic>α</ce:italic> in the asymmetric case.</ce:para><ce:para id="pr0200">The top panel of <ce:cross-ref refid="fg0040" id="crf11230">Fig. 4</ce:cross-ref><ce:float-anchor refid="fg0040"/> shows the best-fit parameter values of the normal width (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math>) for all three collision systems as a function of the average number of participating nucleons (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo></mml:math>) calculated using a Glauber model <ce:cross-ref refid="br0280" id="crf11240">[28]</ce:cross-ref>. The best-fit parameters are found taking statistical and uncorrelated systematic uncertainties into account. The result using the above procedure, for the most central Pb<ce:glyph name="sbnd"/>Pb collisions, is found to be compatible with previous results extracted by unfolding with the mean Jacobian estimated from transverse momentum spectra <ce:cross-ref refid="br0020" id="crf11250">[2]</ce:cross-ref>. The open points (crosses) and dashed lines on the figure are from evaluations of Eq. <ce:cross-ref refid="fm0030" id="crf11260">(1)</ce:cross-ref> and Eq. <ce:cross-ref refid="fm0040" id="crf11270">(2)</ce:cross-ref>, and direct calculations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math>, respectively, using model calculations with EPOS-LHC <ce:cross-ref refid="br0290" id="crf11280">[29]</ce:cross-ref>. EPOS-LHC was chosen as it provides predictions for all three collision systems. The parameterisation, in terms of the two functions, of this model calculation generally reproduces the widths of the charged-particle rapidity densities, except in the asymmetric case where a direct evaluation of the standard deviation is less motivated.</ce:para><ce:para id="pr0210">The general trend is that the widths decrease as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo></mml:math> increases, consistent with the behaviour of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="normal">PbPb</mml:mtext></mml:mrow></mml:msub></mml:math> ratios. Notably, the width of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> distributions in p<ce:glyph name="sbnd"/>Pb and Pb<ce:glyph name="sbnd"/>Pb, for low number of participant nucleons in the collisions, approaches the width of the pp distribution, which, presumably, is dominated by kinematic and phase space constraints.</ce:para><ce:para id="pr0220">The lower panel of <ce:cross-ref refid="fg0040" id="crf11290">Fig. 4</ce:cross-ref> shows the dependence of <ce:italic>a</ce:italic> on the average number of participants. The right-hand ordinate is the same, but multiplied by the average mass <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>0.215</mml:mn><mml:mo>±</mml:mo><mml:mn>0.001</mml:mn><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="0.2em"/><mml:mtext>GeV</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="italic">c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> estimated from measurements of identified particles in Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math><ce:cross-ref refid="br0300" id="crf11300">[30]</ce:cross-ref>. To better understand the parameter <ce:italic>a</ce:italic>, this parameter extracted from the EPOS-LHC calculations, using the above procedure, is also shown in the figure. The dotted lines show the average <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>m</mml:mi></mml:math> predicted by EPOS-LHC <ce:cross-ref refid="br0290" id="crf11310">[29]</ce:cross-ref>. The EPOS-LHC calculations indicate that the extracted effective transverse momentum to mass ratio <ce:italic> a</ce:italic> is consistently smaller than the ratio of the average transverse momentum to the average mass. Thus <ce:italic>a</ce:italic> gives a lower bound on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math>.</ce:para><ce:para id="pr0230">We can estimate the energy density that is reached in the collisions as a function of the number of participants for the three systems. A conventional approach is to use the model originally proposed by Bjorken <ce:cross-ref refid="br0120" id="crf11320">[12]</ce:cross-ref> in which the energy density (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub></mml:math>) depends on the rapidity density of particles and the volume of a longitudinal cylinder with cross sectional area determined by the overlap between the colliding partners and length determined by a characteristic particle formation time<ce:display><ce:formula id="fm0050"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>τ</mml:mi></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="true">〈</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">〉</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> Here, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mi>π</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>≈</mml:mo><mml:mi>π</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msubsup></mml:math> is the transverse area spanned by the participating nucleons, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> is the transverse-energy rapidity density, and <ce:italic>τ</ce:italic> is the formation time. While a formation time of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:mi>τ</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">c</mml:mi></mml:math> is often assumed, it is left as a free parameter here. With <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt></mml:math>, the transverse-energy rapidity density can be approximated by<ce:display><ce:formula id="fm0060"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg"><mml:mrow><mml:mrow><mml:mo stretchy="true">〈</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">〉</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:mrow></mml:mfrac><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0.55</mml:mn><mml:mo>±</mml:mo><mml:mn>0.01</mml:mn></mml:math>, the ratio of charged particles to all particles <ce:cross-ref refid="br0310" id="crf11330">[31]</ce:cross-ref>, accounts for neutral particles not measured in the experiment, and is assumed the same for all collision systems. Substituting the derived <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>y</mml:mi></mml:math> and the effective <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si69.svg"><mml:mi>a</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">/</mml:mo><mml:mi>m</mml:mi><mml:mo>≲</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math> results in a lower bound estimate for the Bjorken energy density (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">LB</mml:mi></mml:mrow></mml:msub></mml:math> )<ce:display><ce:formula id="fm0070"><ce:label>(3)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si71.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi><mml:mo>≥</mml:mo><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">LB</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">cosh</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>⁡</mml:mo><mml:mi>η</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>η</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <ce:italic>a</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mo stretchy="false">〈</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math> are as in the top panel of <ce:cross-ref refid="fg0040" id="crf11340">Fig. 4</ce:cross-ref> .</ce:para><ce:para id="pr0240">The transverse area <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> is estimated in a numerical Glauber model <ce:cross-refs refid="br0320 br0330" id="crs0100">[32,33]</ce:cross-refs> as shown in <ce:cross-ref refid="fg0050" id="crf11350">Fig. 5</ce:cross-ref><ce:float-anchor refid="fg0050"/>. We consider two extremes for the transverse area spanned by the participating nucleons: a) the <ce:italic>exclusive</ce:italic> (or direct) overlap between participating nucleons, ∩ and open markers in <ce:cross-ref refid="fg0050" id="crf11360">Fig. 5</ce:cross-ref>, and b) the <ce:italic> inclusive</ce:italic> (or full) area of all participating nucleons, ∪ and full markers in <ce:cross-ref refid="fg0050" id="crf11370">Fig. 5</ce:cross-ref>.</ce:para><ce:para id="pr0250"><ce:cross-ref refid="fg0060" id="crf11440">Fig. 6</ce:cross-ref><ce:float-anchor refid="fg0060"/> shows the lower-bound energy density estimate, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si76.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">LB</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi></mml:math>, as a function of the number of participants, which reaches values between 10 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si77.svg"><mml:mn>20</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> in the most central Pb<ce:glyph name="sbnd"/>Pb collisions. The uncertainties are from standard error propagation of Eq. <ce:cross-ref refid="fm0070" id="crf11390">(3)</ce:cross-ref> of uncertainties on the best-fit parameter values, the number of participants, mean mass, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si78.svg"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">total</mml:mi></mml:mrow></mml:msub></mml:math>. A rise from roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si79.svg"><mml:mn>1</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> to over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si80.svg"><mml:mn>10</mml:mn><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is observed if the transverse area is assumed to be the inclusive area of participating nucleons. This trend is illustrated by a power-law (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si81.svg"><mml:mi>C</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">part</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msubsup></mml:math>) fit to the data in the figure, with the parameter values <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si82.svg"><mml:mi>C</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>0.8</mml:mn><mml:mo>±</mml:mo><mml:mn>0.3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.2em"/><mml:mrow><mml:mi mathvariant="normal">GeV</mml:mi></mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">fm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">c</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si83.svg"><mml:mi>p</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>0.44</mml:mn><mml:mo>±</mml:mo><mml:mn>0.08</mml:mn></mml:math>. On the other hand, if the transverse area is assumed to be the smaller exclusive overlap area, we observe a substantially larger lower bound on the energy density, but a less dramatic increase with increasing number of participating nucleons. Also shown in the figure are estimates of the Bjorken energy density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg"><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Bj</mml:mi></mml:mrow></mml:msub><mml:mi>τ</mml:mi></mml:math> for Pb<ce:glyph name="sbnd"/>Pb reactions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math><ce:cross-ref refid="br0310" id="crf11400">[31]</ce:cross-ref>. These results where obtained from measurements of the transverse energy in the collisions and using the inclusive estimate of the transverse area <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>. The trend of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> results is similar to these earlier results. Bearing in mind that for the largest LHC collision energy we show a lower bound estimate of the energy density in <ce:cross-ref refid="fg0060" id="crf11410">Fig. 6</ce:cross-ref>, we find a likely overall increase in the energy density from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math>.</ce:para></ce:section><ce:section id="se0050"><ce:label>5</ce:label><ce:section-title id="st0060">Summary and conclusions</ce:section-title><ce:para id="pr0260">We have measured the charged particle pseudorapidity density in pp, p<ce:glyph name="sbnd"/>Pb, and Pb<ce:glyph name="sbnd"/>Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn><mml:mspace width="0.2em"/><mml:mtext>TeV</mml:mtext></mml:math> over the widest possible pseudorapidity range available at the LHC. The distributions where determined using the same experimental apparatus and methods, and systematic uncertainties have been minimised to within the capabilities of the set-up. While the particle production in central Pb<ce:glyph name="sbnd"/>Pb collisions clearly exhibits an enhancement as compared to pp collisions, particle production in p<ce:glyph name="sbnd"/>Pb collisions is consistent with dominantly incoherent nucleon–nucleon collisions. By transforming the measured pseudorapidity distributions to rapidity distributions we have obtained systematic trends for the width of the rapidity distributions and a lower bound on the energy density, which shows a clear scaling behaviour as a function of the average number of participant nucleons. The decreasing width of the deduced rapidity distributions with increasing participant number suggests that the kinematic spread of particles, including longitudinal degrees of freedom, is reduced due to interactions in the early stages of the collisions. This is also reflected in the accompanying growth of the energy density. Both observations are consistent with the gradual establishment of a high-density phase of matter with increasing size of the collision domain.</ce:para></ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0080">Declaration of Competing Interest</ce:section-title><ce:para id="pr0270">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0090">Acknowledgements</ce:section-title><ce:para id="pr0280">The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: <ce:grant-sponsor id="gsp0010">A. I. 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<ce:grant-sponsor id="gsp0690" sponsor-id="https://doi.org/10.13039/100020381">Turkish Energy, Nuclear and Mineral Research Agency</ce:grant-sponsor> (TENMAK), Turkey; <ce:grant-sponsor id="gsp0700" sponsor-id="https://doi.org/10.13039/501100004742">National Academy of Sciences of Ukraine</ce:grant-sponsor>, Ukraine; <ce:grant-sponsor id="gsp0710" sponsor-id="https://doi.org/10.13039/501100000271">Science and Technology Facilities Council</ce:grant-sponsor> (STFC), United Kingdom; National Science Foundation of the United States of America (<ce:grant-sponsor id="gsp0720" sponsor-id="https://doi.org/10.13039/100000001">NSF</ce:grant-sponsor>) and United States Department of Energy, Office of Nuclear Physics (<ce:grant-sponsor id="gsp0730" sponsor-id="https://doi.org/10.13039/100006209">DOE NP</ce:grant-sponsor>), United States of America.</ce:para></ce:acknowledgment></body><tail><ce:bibliography id="bl0010"><ce:section-title id="st0070">References</ce:section-title><ce:bibliography-sec id="bs0010"><ce:bib-reference id="br0010"><ce:label>[1]</ce:label><sb:reference id="bib866BD36D5E8AFE73B567F49DDE14CD65s1"><sb:contribution><sb:authors><sb:collaboration>BRAHMS Collaboration</sb:collaboration><sb:author><ce:given-name>I.C.</ce:given-name><ce:surname>Arsene</ce:surname></sb:author><sb:et-al/></sb:authors><sb:title><sb:maintitle>Nuclear stopping and rapidity loss in Au+Au collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si88.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>62.4</mml:mn><mml:mspace width="0.2em"/><mml:mtext>GeV</mml:mtext></mml:math></sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Phys. 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C94 (2016) 024914, arXiv:1603.07375 [nucl-ex].</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> diff --git a/tests/units/elsevier/data/main_rjjlr.xml b/tests/units/elsevier/data/main_rjjlr.xml index 928b68c4..4034b408 100644 --- a/tests/units/elsevier/data/main_rjjlr.xml +++ b/tests/units/elsevier/data/main_rjjlr.xml @@ -1 +1 @@ -<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd" [<!ENTITY gr001 SYSTEM "gr001" NDATA IMAGE><!ENTITY gr002 SYSTEM "gr002" NDATA IMAGE><!ENTITY gr003 SYSTEM "gr003" NDATA IMAGE><!ENTITY gr004 SYSTEM "gr004" NDATA IMAGE><!ENTITY gr005 SYSTEM "gr005" NDATA IMAGE>]><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="sco" xml:lang="en"><item-info><jid>PLB</jid><aid>137649</aid><ce:article-number>137649</ce:article-number><ce:pii>S0370-2693(22)00783-3</ce:pii><ce:doi>10.1016/j.physletb.2022.137649</ce:doi><ce:copyright year="2023" type="other">The Author(s)</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>Experiments</ce:text></ce:doctopic></ce:doctopics><ce:preprint><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/preprint" xlink:href="arxiv:2204.10157" id="inf0010"/></ce:preprint><ce:associated-resource><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/research-data" xlink:href="https://www.hepdata.net/" id="inf0540">https://www.hepdata.net/</ce:inter-ref></ce:associated-resource></item-info><ce:floats><ce:figure id="fg0010"><ce:label>Fig. 1</ce:label><ce:caption id="cp0010"><ce:simple-para id="sp0010">Illustration of toward, away and transverse regions with respect to the leading particle in a collision.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0010">Fig. 1</ce:alt-text><ce:link locator="gr001" xlink:type="simple" xlink:href="pii:S0370269322007833/gr001" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0010"/></ce:figure><ce:figure id="fg0020"><ce:label>Fig. 2</ce:label><ce:caption id="cp0020"><ce:simple-para id="sp0020">Top panels: transverse momentum spectra of charged particles in the transverse region for different multiplicity classes in pp (left), p–Pb (middle) and Pb–Pb (right) collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra are measured at mid pseudorapidity (|<ce:italic>η</ce:italic>| < 0.8). Lower panels: Ratio of <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra in different multiplicity classes to the <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectrum in the 0−100% multiplicity class for the corresponding collision systems. The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0020">Fig. 2</ce:alt-text><ce:link locator="gr002" xlink:type="simple" xlink:href="pii:S0370269322007833/gr002" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0020"/></ce:figure><ce:figure id="fg0030"><ce:label>Fig. 3</ce:label><ce:caption id="cp0030"><ce:simple-para id="sp0030">Transverse momentum spectra of charged particles in Toward-Transverse, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> (top plot) and Away-Transverse, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> (bottom plot) regions for different multiplicity classes in pp (left), p–Pb (middle) and Pb–Pb (right) collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra are measured at mid pseudorapidity (|<ce:italic>η</ce:italic>| < 0.8). The lower panels of both plots show the ratio to minimum bias pp collisions. The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0030">Fig. 3</ce:alt-text><ce:link locator="gr003" xlink:type="simple" xlink:href="pii:S0370269322007833/gr003" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0030"/></ce:figure><ce:figure id="fg0040"><ce:label>Fig. 4</ce:label><ce:caption id="cp0040"><ce:simple-para id="sp0040">The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> (left) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> (right) as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> in 4 <<ce:italic>p</ce:italic><ce:inf>T</ce:inf>< 6 GeV/<ce:italic>c</ce:italic> for different multiplicity classes in pp, p–Pb and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. Pb–Pb results are shown assuming a flat background (filled markers), and assuming a <ce:italic>v</ce:italic><ce:inf>2</ce:inf>-modulated background (empty markers). The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0040">Fig. 4</ce:alt-text><ce:link locator="gr004" xlink:type="simple" xlink:href="pii:S0370269322007833/gr004" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0040"/></ce:figure><ce:figure id="fg0050"><ce:label>Fig. 5</ce:label><ce:caption id="cp0050"><ce:simple-para id="sp0050">Comparison of the measured the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> (left) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> (right) in 4 <<ce:italic>p</ce:italic><ce:inf>T</ce:inf>< 6 GeV/<ce:italic>c</ce:italic> with model predictions. The results are shown as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> for different multiplicity classes in pp (top panel), p–Pb (middle panel) and Pb–Pb (bottom panel) collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The red and magenta lines show the <ce:small-caps>PYTHIA</ce:small-caps> 8 (Monash) <ce:cross-ref refid="br0280" id="crf0010">[28]</ce:cross-ref> and <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr <ce:cross-ref refid="br0280" id="crf0020">[28]</ce:cross-ref> predictions, respectively. The blue lines show the EPOS-LHC <ce:cross-ref refid="br0210" id="crf0030">[21]</ce:cross-ref> results. The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0050">Fig. 5</ce:alt-text><ce:link locator="gr005" xlink:type="simple" xlink:href="pii:S0370269322007833/gr005" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0050"/></ce:figure><ce:table xmlns="http://www.elsevier.com/xml/common/cals/dtd" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" id="tbl0010" frame="topbot" rowsep="0" colsep="0"><ce:label>Table 1</ce:label><ce:caption id="cp0060"><ce:simple-para id="sp0060">Contributions to the relative (%) systematic uncertainty on the <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra of primary charged particles in pp, p–Pb, and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. Just for illustration, the range in the table corresponds to the lowest and highest relative systematic uncertainty in the considered <ce:italic>p</ce:italic><ce:inf>T</ce:inf> range. The individual contributions are summed in quadrature to obtain the total uncertainty.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0060">Table 1</ce:alt-text><tgroup cols="4"><colspec colnum="1" colname="col1" align="left"/><colspec colnum="2" colname="col2" align="left"/><colspec colnum="3" colname="col3" align="left"/><colspec colnum="4" colname="col4" align="left"/><thead valign="top"><row rowsep="1"><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Source of uncertainty</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">pp</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">p–Pb</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">Pb–Pb</entry></row></thead><tbody valign="top"><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Track selection</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.1–8.2</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.4–5.8</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.0–9.9</entry></row><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Particle composition</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.3–1.8</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.5–1.9</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.3–2.4</entry></row><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Secondary particles</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.0–0.4</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.0–2.4</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.0–1.9</entry></row><row rowsep="1"><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Matching efficiency</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.0–4.2</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.7–3.7</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.6–3.7</entry></row><row rowsep="1"><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Total</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.2–8.8</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.6–6.3</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.5–10.0</entry></row><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Total (<ce:italic>N</ce:italic><ce:inf>ch</ce:inf>-dependent)</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.0–4.5</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">1.7–4.0</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">1.1–3.7</entry></row></tbody></tgroup></ce:table></ce:floats><head><ce:title id="ti0010">Study of charged particle production at high <ce:italic>p</ce:italic><ce:inf>T</ce:inf> using event topology in pp, p–Pb and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV</ce:title><ce:author-group id="ag0010"><ce:collaboration id="co0010" collaboration-id="S0370269322007833-3bea72599603117cd9d18494a0279c47"><ce:text>ALICE Collaboration</ce:text><ce:cross-ref refid="fn0080" id="crf0040"><ce:sup>⋆</ce:sup></ce:cross-ref><ce:author-group id="ag0020"><ce:author orcid="0000-0002-9213-5329" id="au0010" author-id="S0370269322007833-e60c93a934b81cf9801254193264c6ee"><ce:given-name>S.</ce:given-name><ce:surname>Acharya</ce:surname><ce:cross-ref refid="aff1240" id="crf0050"><ce:sup>124</ce:sup></ce:cross-ref><ce:cross-ref refid="aff1310" id="crf0060"><ce:sup>131</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-0504-7428" id="au0020" author-id="S0370269322007833-0eab85892b6d74b18661e74a7987c599"><ce:given-name>D.</ce:given-name><ce:surname>Adamová</ce:surname><ce:cross-ref refid="aff0860" id="crf0070"><ce:sup>86</ce:sup></ce:cross-ref></ce:author><ce:author id="au0030" author-id="S0370269322007833-e83a30ae1d5f89088c60ca2ee154d714"><ce:given-name>A.</ce:given-name><ce:surname>Adler</ce:surname><ce:cross-ref refid="aff0690" id="crf0080"><ce:sup>69</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-9611-3696" id="au0040" author-id="S0370269322007833-ed2d58d89990c41bb43c091d01e5029a"><ce:given-name>G.</ce:given-name><ce:surname>Aglieri Rinella</ce:surname><ce:cross-ref refid="aff0320" id="crf0090"><ce:sup>32</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-0760-5075" id="au0050" author-id="S0370269322007833-0c7a7863b7384aa5fdf06a0f187949c8"><ce:given-name>M.</ce:given-name><ce:surname>Agnello</ce:surname><ce:cross-ref refid="aff0290" id="crf0100"><ce:sup>29</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0003-0348-9836" id="au0060" author-id="S0370269322007833-bbdbb014653d7bdacb111c613a0fcbe0"><ce:given-name>N.</ce:given-name><ce:surname>Agrawal</ce:surname><ce:cross-ref refid="aff0500" id="crf0110"><ce:sup>50</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0001-5241-7412" id="au0070" author-id="S0370269322007833-2059a9508121069139ea49dbf566539a"><ce:given-name>Z.</ce:given-name><ce:surname>Ahammed</ce:surname><ce:cross-ref refid="aff1310" id="crf0120"><ce:sup>131</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0003-0497-5705" id="au0080" author-id="S0370269322007833-397c2b4743f367cb4aceb69d448bb6c6"><ce:given-name>S.</ce:given-name><ce:surname>Ahmad</ce:surname><ce:cross-ref refid="aff0150" id="crf0130"><ce:sup>15</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0001-8847-489X" id="au0090" author-id="S0370269322007833-f7c3cbd2e9a0f545640d2202ad1ddbf3"><ce:given-name>S.U.</ce:given-name><ce:surname>Ahn</ce:surname><ce:cross-ref refid="aff0700" id="crf0140"><ce:sup>70</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-4417-1392" id="au0100" author-id="S0370269322007833-07322fd13596772e8f99876f081be003"><ce:given-name>I.</ce:given-name><ce:surname>Ahuja</ce:surname><ce:cross-ref refid="aff0370" id="crf0150"><ce:sup>37</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-7388-3022" id="au0110" author-id="S0370269322007833-e1ca35714b53c8e677e37d627a5bbf38"><ce:given-name>A.</ce:given-name><ce:surname>Akindinov</ce:surname><ce:cross-ref refid="aff1390" id="crf0160"><ce:sup>139</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-8071-4497" id="au0120" author-id="S0370269322007833-92548b9c2ec6b0a94be4b4532227aed9"><ce:given-name>M.</ce:given-name><ce:surname>Al-Turany</ce:surname><ce:cross-ref refid="aff0980" 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author-id="S0370269322007833-617c42c28cb1b1d3dab680558c293767"><ce:given-name>Y.</ce:given-name><ce:surname>Zhi</ce:surname><ce:cross-ref refid="aff0100" id="crf10800"><ce:sup>10</ce:sup></ce:cross-ref></ce:author><ce:author id="au10260" author-id="S0370269322007833-3038f7cd79250ffa05fa0aff7644d2b3"><ce:given-name>N.</ce:given-name><ce:surname>Zhigareva</ce:surname><ce:cross-ref refid="aff1390" id="crf10810"><ce:sup>139</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0009-0009-2528-906X" id="au10270" author-id="S0370269322007833-e9bc0465b352cd706eac4fc8b795a4c2"><ce:given-name>D.</ce:given-name><ce:surname>Zhou</ce:surname><ce:cross-ref refid="aff0060" id="crf10820"><ce:sup>6</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-7868-6706" id="au10280" author-id="S0370269322007833-4a0cdba9c352af9310759dd83e906db9"><ce:given-name>Y.</ce:given-name><ce:surname>Zhou</ce:surname><ce:cross-ref refid="aff0830" id="crf10830"><ce:sup>83</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0001-9358-5762" id="au10290" author-id="S0370269322007833-b564bd6149c66f06bbea2780670b8f50"><ce:given-name>J.</ce:given-name><ce:surname>Zhu</ce:surname><ce:cross-ref refid="aff0980" id="crf10840"><ce:sup>98</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0060" id="crf10850"><ce:sup>6</ce:sup></ce:cross-ref></ce:author><ce:author id="au10300" author-id="S0370269322007833-1fb0ab925cfcdaacb160193d82c1a978"><ce:given-name>Y.</ce:given-name><ce:surname>Zhu</ce:surname><ce:cross-ref refid="aff0060" id="crf10860"><ce:sup>6</ce:sup></ce:cross-ref></ce:author><ce:author id="au10310" author-id="S0370269322007833-2ad055472e2fac995111c51c08396d0c"><ce:given-name>G.</ce:given-name><ce:surname>Zinovjev</ce:surname><ce:cross-ref refid="aff0030" id="crf10870"><ce:sup>3</ce:sup></ce:cross-ref><ce:cross-ref refid="fn0010" id="crf10880"><ce:sup>I</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-7478-2493" id="au10320" author-id="S0370269322007833-92b5ecf5612e2e849fc1bba72c6cff99"><ce:given-name>N.</ce:given-name><ce:surname>Zurlo</ce:surname><ce:cross-ref refid="aff1300" id="crf10890"><ce:sup>130</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0540" id="crf10900"><ce:sup>54</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="aff0010" affiliation-id="S0370269322007833-79d30baa35325e84d46378ba6ce12c18"><ce:label>1</ce:label><ce:textfn>A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia</ce:textfn><sa:affiliation><sa:organization>A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation</sa:organization><sa:city>Yerevan</sa:city><sa:country>Armenia</sa:country></sa:affiliation><ce:source-text id="srct0005">A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia</ce:source-text></ce:affiliation><ce:affiliation id="aff0020" affiliation-id="S0370269322007833-65754d218cf7f84bf1e02306b80caca0"><ce:label>2</ce:label><ce:textfn>AGH University of Science and Technology, Cracow, Poland</ce:textfn><sa:affiliation><sa:organization>AGH University of Science and Technology</sa:organization><sa:city>Cracow</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0010">AGH University of Science and Technology, Cracow, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff0030" affiliation-id="S0370269322007833-e916a2f48a17bc32220b61ae0e9b8e05"><ce:label>3</ce:label><ce:textfn>Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine</ce:textfn><sa:affiliation><sa:organization>Bogolyubov Institute for Theoretical Physics</sa:organization><sa:organization>National Academy of Sciences of Ukraine</sa:organization><sa:city>Kiev</sa:city><sa:country>Ukraine</sa:country></sa:affiliation><ce:source-text id="srct0015">Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine</ce:source-text></ce:affiliation><ce:affiliation id="aff0040" affiliation-id="S0370269322007833-9869d95133b2275e34836b8ad2f235c3"><ce:label>4</ce:label><ce:textfn>Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Bose Institute</sa:organization><sa:organization>Department of Physics</sa:organization><sa:organization>Centre for Astroparticle Physics and Space Science (CAPSS)</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0020">Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0050" affiliation-id="S0370269322007833-b7f796ef6c934d79a497288cdec192f6"><ce:label>5</ce:label><ce:textfn>California Polytechnic State University, San Luis Obispo, CA, United States</ce:textfn><sa:affiliation><sa:organization>California Polytechnic State University</sa:organization><sa:city>San Luis Obispo</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0025">California Polytechnic State University, San Luis Obispo, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0060" affiliation-id="S0370269322007833-3dcd6ffc2e8f27d6ed2f6237a209a384"><ce:label>6</ce:label><ce:textfn>Central China Normal University, Wuhan, China</ce:textfn><sa:affiliation><sa:organization>Central China Normal University</sa:organization><sa:city>Wuhan</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0030">Central China Normal University, Wuhan, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0070" affiliation-id="S0370269322007833-110460c7f2fbb319ec6d52e7cc3fc1d5"><ce:label>7</ce:label><ce:textfn>Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba</ce:textfn><sa:affiliation><sa:organization>Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN)</sa:organization><sa:city>Havana</sa:city><sa:country>Cuba</sa:country></sa:affiliation><ce:source-text id="srct0035">Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba</ce:source-text></ce:affiliation><ce:affiliation id="aff0080" affiliation-id="S0370269322007833-641aa526558990d110c090840e6f3d0c"><ce:label>8</ce:label><ce:textfn>Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico</ce:textfn><sa:affiliation><sa:organization>Centro de Investigación y de Estudios Avanzados (CINVESTAV)</sa:organization><sa:city>Mexico City and Mérida</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0040">Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0090" affiliation-id="S0370269322007833-9c73ece39ab638447cd10f268e329b2e"><ce:label>9</ce:label><ce:textfn>Chicago State University, Chicago, IL, United States</ce:textfn><sa:affiliation><sa:organization>Chicago State University</sa:organization><sa:city>Chicago</sa:city><sa:state>IL</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0045">Chicago State University, Chicago, Illinois, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0100" affiliation-id="S0370269322007833-d7da93bf3f02be0a46bee74f439b5930"><ce:label>10</ce:label><ce:textfn>China Institute of Atomic Energy, Beijing, China</ce:textfn><sa:affiliation><sa:organization>China Institute of Atomic Energy</sa:organization><sa:city>Beijing</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0050">China Institute of Atomic Energy, Beijing, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0110" affiliation-id="S0370269322007833-d477851cd020cd39da135784676a96ff"><ce:label>11</ce:label><ce:textfn>Chungbuk National University, Cheongju, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Chungbuk National University</sa:organization><sa:city>Cheongju</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0055">Chungbuk National University, Cheongju, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0120" affiliation-id="S0370269322007833-b03a6e652a70daee257206602eb6f83f"><ce:label>12</ce:label><ce:textfn>Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Comenius University Bratislava</sa:organization><sa:organization>Faculty of Mathematics, Physics and Informatics</sa:organization><sa:city>Bratislava</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0060">Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0130" affiliation-id="S0370269322007833-78cc8333b14d598fc5e3c0230d1de22e"><ce:label>13</ce:label><ce:textfn>COMSATS University Islamabad, Islamabad, Pakistan</ce:textfn><sa:affiliation><sa:organization>COMSATS University Islamabad</sa:organization><sa:city>Islamabad</sa:city><sa:country>Pakistan</sa:country></sa:affiliation><ce:source-text id="srct0065">COMSATS University Islamabad, Islamabad, Pakistan</ce:source-text></ce:affiliation><ce:affiliation id="aff0140" affiliation-id="S0370269322007833-42161ea7466da89325c5b37601107c55"><ce:label>14</ce:label><ce:textfn>Creighton University, Omaha, NE, United States</ce:textfn><sa:affiliation><sa:organization>Creighton University</sa:organization><sa:city>Omaha</sa:city><sa:state>NE</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0070">Creighton University, Omaha, Nebraska, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0150" affiliation-id="S0370269322007833-bd5e6b818668501c1deea76e96cac833"><ce:label>15</ce:label><ce:textfn>Department of Physics, Aligarh Muslim University, Aligarh, India</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Aligarh Muslim University</sa:organization><sa:city>Aligarh</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0075">Department of Physics, Aligarh Muslim University, Aligarh, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0160" affiliation-id="S0370269322007833-ebe7875fb705d3c179953d196aa8b94b"><ce:label>16</ce:label><ce:textfn>Department of Physics, Pusan National University, Pusan, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Pusan National University</sa:organization><sa:city>Pusan</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0080">Department of Physics, Pusan National University, Pusan, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0170" affiliation-id="S0370269322007833-24ddc81b37e640977b88412ba28336a4"><ce:label>17</ce:label><ce:textfn>Department of Physics, Sejong University, Seoul, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Sejong University</sa:organization><sa:city>Seoul</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0085">Department of Physics, Sejong University, Seoul, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0180" affiliation-id="S0370269322007833-66245837acb0d7fb7ada51de0fd95043"><ce:label>18</ce:label><ce:textfn>Department of Physics, University of California, Berkeley, CA, United States</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>University of California</sa:organization><sa:city>Berkeley</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0090">Department of Physics, University of California, Berkeley, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0190" affiliation-id="S0370269322007833-63776eabafa9e32856b35562692a4488"><ce:label>19</ce:label><ce:textfn>Department of Physics, University of Oslo, Oslo, Norway</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>University of Oslo</sa:organization><sa:city>Oslo</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0095">Department of Physics, University of Oslo, Oslo, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0200" affiliation-id="S0370269322007833-020451a59d5e257505166bdb7847cdd7"><ce:label>20</ce:label><ce:textfn>Department of Physics and Technology, University of Bergen, Bergen, Norway</ce:textfn><sa:affiliation><sa:organization>Department of Physics and Technology</sa:organization><sa:organization>University of Bergen</sa:organization><sa:city>Bergen</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0100">Department of Physics and Technology, University of Bergen, Bergen, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0210" affiliation-id="S0370269322007833-f2de76852013198101844806b89d3836"><ce:label>21</ce:label><ce:textfn>Dipartimento di Fisica, Università di Pavia, Pavia, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica</sa:organization><sa:organization>Università di Pavia</sa:organization><sa:city>Pavia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0105">Dipartimento di Fisica, Università di Pavia, Pavia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0220" affiliation-id="S0370269322007833-81d87d2cd8c9f6aa2a1c7f4b13115ef3"><ce:label>22</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Cagliari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0110">Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0230" affiliation-id="S0370269322007833-849d0ba7888fcb2a09aad7ac1095ec15"><ce:label>23</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Trieste</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0115">Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0240" affiliation-id="S0370269322007833-68c63fd1cb9e8623af62118bbef39c9e"><ce:label>24</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0120">Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0250" affiliation-id="S0370269322007833-7a40dab487121429c63e59c3de15ccfa"><ce:label>25</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bologna</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0125">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0260" affiliation-id="S0370269322007833-72e3fe624de7d3d3223574366a22ef30"><ce:label>26</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Catania</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0130">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0270" affiliation-id="S0370269322007833-6145c67e3009fb117e82f5429b2af282"><ce:label>27</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Padova</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0135">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0280" affiliation-id="S0370269322007833-a001bc692b1f1f84377401c9e2632e51"><ce:label>28</ce:label><ce:textfn>Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università and Gruppo Collegato INFN, Salerno, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università</sa:organization><sa:organization>Gruppo Collegato INFN</sa:organization><sa:city>Salerno</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0140">Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università and Gruppo Collegato INFN, Salerno, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0290" affiliation-id="S0370269322007833-362983c57dbe79a62add2550bfe565dc"><ce:label>29</ce:label><ce:textfn>Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento DISAT del Politecnico</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0145">Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0300" affiliation-id="S0370269322007833-7a82e32411929fc5768d11b72609a4d8"><ce:label>30</ce:label><ce:textfn>Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Scienze MIFT</sa:organization><sa:organization>Università di Messina</sa:organization><sa:city>Messina</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0150">Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0310" affiliation-id="S0370269322007833-0bfba0b176e2b6e1b67e6564c87308b5"><ce:label>31</ce:label><ce:textfn>Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento Interateneo di Fisica ‘M. Merlin’</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0155">Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0320" affiliation-id="S0370269322007833-44f92095d23d6e3d2fb8e8fc998c51f2"><ce:label>32</ce:label><ce:textfn>European Organization for Nuclear Research (CERN), Geneva, Switzerland</ce:textfn><sa:affiliation><sa:organization>European Organization for Nuclear Research (CERN)</sa:organization><sa:city>Geneva</sa:city><sa:country>Switzerland</sa:country></sa:affiliation><ce:source-text id="srct0160">European Organization for Nuclear Research (CERN), Geneva, Switzerland</ce:source-text></ce:affiliation><ce:affiliation id="aff0330" affiliation-id="S0370269322007833-f220870dd6ed2747e8ec11b2ba624bf5"><ce:label>33</ce:label><ce:textfn>Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia</ce:textfn><sa:affiliation><sa:organization>Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture</sa:organization><sa:organization>University of Split</sa:organization><sa:city>Split</sa:city><sa:country>Croatia</sa:country></sa:affiliation><ce:source-text id="srct0165">Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia</ce:source-text></ce:affiliation><ce:affiliation id="aff0340" affiliation-id="S0370269322007833-5bf4b9d297f0544e6037addfb7689840"><ce:label>34</ce:label><ce:textfn>Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway</ce:textfn><sa:affiliation><sa:organization>Faculty of Engineering and Science</sa:organization><sa:organization>Western Norway University of Applied Sciences</sa:organization><sa:city>Bergen</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0170">Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0350" affiliation-id="S0370269322007833-56d412e1c3fe114d9a18225fa76274c9"><ce:label>35</ce:label><ce:textfn>Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Faculty of Nuclear Sciences and Physical Engineering</sa:organization><sa:organization>Czech Technical University in Prague</sa:organization><sa:city>Prague</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0175">Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0360" affiliation-id="S0370269322007833-de312990ec5b7745ed2a9609074bb450"><ce:label>36</ce:label><ce:textfn>Faculty of Physics, Sofia University, Sofia, Bulgaria</ce:textfn><sa:affiliation><sa:organization>Faculty of Physics</sa:organization><sa:organization>Sofia University</sa:organization><sa:city>Sofia</sa:city><sa:country>Bulgaria</sa:country></sa:affiliation><ce:source-text id="srct0180">Faculty of Physics, Sofia University, Sofia, Bulgaria</ce:source-text></ce:affiliation><ce:affiliation id="aff0370" affiliation-id="S0370269322007833-0879c60c6550c16bd60686725f5c6938"><ce:label>37</ce:label><ce:textfn>Faculty of Science, P.J. Šafárik University, Košice, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Faculty of Science</sa:organization><sa:organization>P.J. Šafárik University</sa:organization><sa:city>Košice</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0185">Faculty of Science, P.J. Šafárik University, Košice, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0380" affiliation-id="S0370269322007833-a18716885f5bc83f5d1aee8d0d80c7af"><ce:label>38</ce:label><ce:textfn>Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Frankfurt Institute for Advanced Studies</sa:organization><sa:organization>Johann Wolfgang Goethe-Universität Frankfurt</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0190">Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0390" affiliation-id="S0370269322007833-f0b0a2b18fef5547dcd39253b5714404"><ce:label>39</ce:label><ce:textfn>Fudan University, Shanghai, China</ce:textfn><sa:affiliation><sa:organization>Fudan University</sa:organization><sa:city>Shanghai</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0195">Fudan University, Shanghai, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0400" affiliation-id="S0370269322007833-a2398937a38c48ebf6ac3ff5e5d60b95"><ce:label>40</ce:label><ce:textfn>Gangneung-Wonju National University, Gangneung, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Gangneung-Wonju National University</sa:organization><sa:city>Gangneung</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0200">Gangneung-Wonju National University, Gangneung, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0410" affiliation-id="S0370269322007833-73bc80872cd116960a09bc477e035838"><ce:label>41</ce:label><ce:textfn>Gauhati University, Department of Physics, Guwahati, India</ce:textfn><sa:affiliation><sa:organization>Gauhati University</sa:organization><sa:organization>Department of Physics</sa:organization><sa:city>Guwahati</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0205">Gauhati University, Department of Physics, Guwahati, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0420" affiliation-id="S0370269322007833-9fd27eebf76465366fe59cb8d9620ae8"><ce:label>42</ce:label><ce:textfn>Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany</ce:textfn><sa:affiliation><sa:organization>Helmholtz-Institut für Strahlen- und Kernphysik</sa:organization><sa:organization>Rheinische Friedrich-Wilhelms-Universität Bonn</sa:organization><sa:city>Bonn</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0210">Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0430" affiliation-id="S0370269322007833-66b9f8889421b091fa908925e638d91e"><ce:label>43</ce:label><ce:textfn>Helsinki Institute of Physics (HIP), Helsinki, Finland</ce:textfn><sa:affiliation><sa:organization>Helsinki Institute of Physics (HIP)</sa:organization><sa:city>Helsinki</sa:city><sa:country>Finland</sa:country></sa:affiliation><ce:source-text id="srct0215">Helsinki Institute of Physics (HIP), Helsinki, Finland</ce:source-text></ce:affiliation><ce:affiliation id="aff0440" affiliation-id="S0370269322007833-6968018d6fe1bd12a4413918be70ab85"><ce:label>44</ce:label><ce:textfn>High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico</ce:textfn><sa:affiliation><sa:organization>High Energy Physics Group</sa:organization><sa:organization>Universidad Autónoma de Puebla</sa:organization><sa:city>Puebla</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0220">High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0450" affiliation-id="S0370269322007833-16bdf6e3577a480da99159f5d54452b1"><ce:label>45</ce:label><ce:textfn>Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>Horia Hulubei National Institute of Physics and Nuclear Engineering</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0225">Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff0460" affiliation-id="S0370269322007833-78df0ad3e050545612fd8d71a4776a19"><ce:label>46</ce:label><ce:textfn>Indian Institute of Technology Bombay (IIT), Mumbai, India</ce:textfn><sa:affiliation><sa:organization>Indian Institute of Technology Bombay (IIT)</sa:organization><sa:city>Mumbai</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0230">Indian Institute of Technology Bombay (IIT), Mumbai, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0470" affiliation-id="S0370269322007833-b8ad6c9375b7a89b768adb13f27427b4"><ce:label>47</ce:label><ce:textfn>Indian Institute of Technology Indore, Indore, India</ce:textfn><sa:affiliation><sa:organization>Indian Institute of Technology Indore</sa:organization><sa:city>Indore</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0235">Indian Institute of Technology Indore, Indore, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0480" affiliation-id="S0370269322007833-25658fba725d22058ae2a8649ceeb084"><ce:label>48</ce:label><ce:textfn>INFN, Laboratori Nazionali di Frascati, Frascati, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Laboratori Nazionali di Frascati</sa:organization><sa:city>Frascati</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0240">INFN, Laboratori Nazionali di Frascati, Frascati, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0490" affiliation-id="S0370269322007833-5d9dee68bdf16e34f2f3f01d930f367d"><ce:label>49</ce:label><ce:textfn>INFN, Sezione di Bari, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Bari</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0245">INFN, Sezione di Bari, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0500" affiliation-id="S0370269322007833-340d750afed4ff705cf1dd72bf688ca4"><ce:label>50</ce:label><ce:textfn>INFN, Sezione di Bologna, Bologna, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Bologna</sa:organization><sa:city>Bologna</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0250">INFN, Sezione di Bologna, Bologna, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0510" affiliation-id="S0370269322007833-436e8c989d6c10cae74944a81714a4e2"><ce:label>51</ce:label><ce:textfn>INFN, Sezione di Cagliari, Cagliari, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Cagliari</sa:organization><sa:city>Cagliari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0255">INFN, Sezione di Cagliari, Cagliari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0520" affiliation-id="S0370269322007833-f0607c03a8b381da2bb83a11f8d899c8"><ce:label>52</ce:label><ce:textfn>INFN, Sezione di Catania, Catania, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Catania</sa:organization><sa:city>Catania</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0260">INFN, Sezione di Catania, Catania, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0530" affiliation-id="S0370269322007833-cb33711bf32ecc9cc4697cbbea10fa88"><ce:label>53</ce:label><ce:textfn>INFN, Sezione di Padova, Padova, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Padova</sa:organization><sa:city>Padova</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0265">INFN, Sezione di Padova, Padova, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0540" affiliation-id="S0370269322007833-5c70f16aa0d389aa33e2589db6fcd5d6"><ce:label>54</ce:label><ce:textfn>INFN, Sezione di Pavia, Pavia, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Pavia</sa:organization><sa:city>Pavia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0270">INFN, Sezione di Pavia, Pavia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0550" affiliation-id="S0370269322007833-f1faa0de67f6d8d1ece0914257c7635c"><ce:label>55</ce:label><ce:textfn>INFN, Sezione di Torino, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Torino</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0275">INFN, Sezione di Torino, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0560" affiliation-id="S0370269322007833-0035d662cbe7eef98bed21545fd04324"><ce:label>56</ce:label><ce:textfn>INFN, Sezione di Trieste, Trieste, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Trieste</sa:organization><sa:city>Trieste</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0280">INFN, Sezione di Trieste, Trieste, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0570" affiliation-id="S0370269322007833-22e3a6d8462b39bb6c155bce0c9b21cd"><ce:label>57</ce:label><ce:textfn>Inha University, Incheon, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Inha University</sa:organization><sa:city>Incheon</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0285">Inha University, Incheon, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0580" affiliation-id="S0370269322007833-d4941ebd67ab9bbf96b1e6a906f4397c"><ce:label>58</ce:label><ce:textfn>Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands</ce:textfn><sa:affiliation><sa:organization>Institute for Gravitational and Subatomic Physics (GRASP)</sa:organization><sa:organization>Utrecht University/Nikhef</sa:organization><sa:city>Utrecht</sa:city><sa:country>Netherlands</sa:country></sa:affiliation><ce:source-text id="srct0290">Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands</ce:source-text></ce:affiliation><ce:affiliation id="aff0590" affiliation-id="S0370269322007833-f6d8becf2ef35885577b8164313401cb"><ce:label>59</ce:label><ce:textfn>Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Institute of Experimental Physics</sa:organization><sa:organization>Slovak Academy of Sciences</sa:organization><sa:city>Košice</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0295">Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0600" affiliation-id="S0370269322007833-6fa96ff1fcc4aaff09633c1217f06ff0"><ce:label>60</ce:label><ce:textfn>Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India</ce:textfn><sa:affiliation><sa:organization>Institute of Physics</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Bhubaneswar</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0300">Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0610" affiliation-id="S0370269322007833-9f77e70dcbde10c6ed3d37b3b0071107"><ce:label>61</ce:label><ce:textfn>Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Institute of Physics of the Czech Academy of Sciences</sa:organization><sa:city>Prague</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0305">Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0620" affiliation-id="S0370269322007833-3309ea65bfe3ae2f96b232545cb67043"><ce:label>62</ce:label><ce:textfn>Institute of Space Science (ISS), Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>Institute of Space Science (ISS)</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0310">Institute of Space Science (ISS), Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff0630" affiliation-id="S0370269322007833-2759820a41f8b0ece8c42aa319465ed5"><ce:label>63</ce:label><ce:textfn>Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Institut für Kernphysik</sa:organization><sa:organization>Johann Wolfgang Goethe-Universität Frankfurt</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0315">Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0640" affiliation-id="S0370269322007833-b18018b82a469dc4e4eb94f595cf4812"><ce:label>64</ce:label><ce:textfn>Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:textfn><sa:affiliation><sa:organization>Instituto de Ciencias Nucleares</sa:organization><sa:organization>Universidad Nacional Autónoma de México</sa:organization><sa:city>Mexico City</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0320">Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0650" affiliation-id="S0370269322007833-2aa194eda46d62198ea9b929240200b8"><ce:label>65</ce:label><ce:textfn>Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil</ce:textfn><sa:affiliation><sa:organization>Instituto de Física</sa:organization><sa:organization>Universidade Federal do Rio Grande do Sul (UFRGS)</sa:organization><sa:city>Porto Alegre</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0325">Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff0660" affiliation-id="S0370269322007833-9e9b95fa2c082308cb0efc3488541c67"><ce:label>66</ce:label><ce:textfn>Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:textfn><sa:affiliation><sa:organization>Instituto de Física</sa:organization><sa:organization>Universidad Nacional Autónoma de México</sa:organization><sa:city>Mexico City</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0330">Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0670" affiliation-id="S0370269322007833-4ef9aeae6b2f74e366edd12aafbea4cf"><ce:label>67</ce:label><ce:textfn>iThemba LABS, National Research Foundation, Somerset West, South Africa</ce:textfn><sa:affiliation><sa:organization>iThemba LABS</sa:organization><sa:organization>National Research Foundation</sa:organization><sa:city>Somerset West</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0335">iThemba LABS, National Research Foundation, Somerset West, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff0680" affiliation-id="S0370269322007833-2f4c4db0447d27fd2e43117b90bb74d4"><ce:label>68</ce:label><ce:textfn>Jeonbuk National University, Jeonju, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Jeonbuk National University</sa:organization><sa:city>Jeonju</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0340">Jeonbuk National University, Jeonju, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0690" affiliation-id="S0370269322007833-81ec7d4c8c69486a57ef1ca6a567076c"><ce:label>69</ce:label><ce:textfn>Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik</sa:organization><sa:organization>Fachbereich Informatik und Mathematik</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0345">Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0700" affiliation-id="S0370269322007833-c1d849875c0de0db6476f9cc96b07dd2"><ce:label>70</ce:label><ce:textfn>Korea Institute of Science and Technology Information, Daejeon, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Korea Institute of Science and Technology Information</sa:organization><sa:city>Daejeon</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0350">Korea Institute of Science and Technology Information, Daejeon, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0710" affiliation-id="S0370269322007833-c5cd420fa3d6b7c79dc3ab9e7c2dfb06"><ce:label>71</ce:label><ce:textfn>KTO Karatay University, Konya, Turkey</ce:textfn><sa:affiliation><sa:organization>KTO Karatay University</sa:organization><sa:city>Konya</sa:city><sa:country>Turkey</sa:country></sa:affiliation><ce:source-text id="srct0355">KTO Karatay University, Konya, Turkey</ce:source-text></ce:affiliation><ce:affiliation id="aff0720" affiliation-id="S0370269322007833-0bdd954451acb47f491c8c4996fa87da"><ce:label>72</ce:label><ce:textfn>Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France</ce:textfn><sa:affiliation><sa:organization>Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie</sa:organization><sa:city>Orsay</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0360">Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France</ce:source-text></ce:affiliation><ce:affiliation id="aff0730" affiliation-id="S0370269322007833-1487532e5bfe30325bc10c0583f7c38e"><ce:label>73</ce:label><ce:textfn>Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France</ce:textfn><sa:affiliation><sa:organization>Laboratoire de Physique Subatomique et de Cosmologie</sa:organization><sa:organization>Université Grenoble-Alpes</sa:organization><sa:organization>CNRS-IN2P3</sa:organization><sa:city>Grenoble</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0365">Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France</ce:source-text></ce:affiliation><ce:affiliation id="aff0740" affiliation-id="S0370269322007833-dd5512fbf2faf90b56635e0b411d44a2"><ce:label>74</ce:label><ce:textfn>Lawrence Berkeley National Laboratory, Berkeley, CA, United States</ce:textfn><sa:affiliation><sa:organization>Lawrence Berkeley National Laboratory</sa:organization><sa:city>Berkeley</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0370">Lawrence Berkeley National Laboratory, Berkeley, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0750" affiliation-id="S0370269322007833-ed03eefd58007822249c697d882deffc"><ce:label>75</ce:label><ce:textfn>Lund University Department of Physics, Division of Particle Physics, Lund, Sweden</ce:textfn><sa:affiliation><sa:organization>Lund University Department of Physics</sa:organization><sa:organization>Division of Particle Physics</sa:organization><sa:city>Lund</sa:city><sa:country>Sweden</sa:country></sa:affiliation><ce:source-text id="srct0375">Lund University Department of Physics, Division of Particle Physics, Lund, Sweden</ce:source-text></ce:affiliation><ce:affiliation id="aff0760" affiliation-id="S0370269322007833-5b47ca06644b7f88e2267e9accbe867b"><ce:label>76</ce:label><ce:textfn>Nagasaki Institute of Applied Science, Nagasaki, Japan</ce:textfn><sa:affiliation><sa:organization>Nagasaki Institute of Applied Science</sa:organization><sa:city>Nagasaki</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0380">Nagasaki Institute of Applied Science, Nagasaki, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0770" affiliation-id="S0370269322007833-61cd55f0b96e9831d6063318d9967d43"><ce:label>77</ce:label><ce:textfn>Nara Women's University (NWU), Nara, Japan</ce:textfn><sa:affiliation><sa:organization>Nara Women's University (NWU)</sa:organization><sa:city>Nara</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0385">Nara Women's University (NWU), Nara, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0780" affiliation-id="S0370269322007833-7f50d99eb340e408fbc323cbce0c8905"><ce:label>78</ce:label><ce:textfn>National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece</ce:textfn><sa:affiliation><sa:organization>National and Kapodistrian University of Athens</sa:organization><sa:organization>School of Science</sa:organization><sa:organization>Department of Physics</sa:organization><sa:city>Athens</sa:city><sa:country>Greece</sa:country></sa:affiliation><ce:source-text id="srct0390">National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece</ce:source-text></ce:affiliation><ce:affiliation id="aff0790" affiliation-id="S0370269322007833-58192de9f95a93cb8bf485b9a5647bf5"><ce:label>79</ce:label><ce:textfn>National Centre for Nuclear Research, Warsaw, Poland</ce:textfn><sa:affiliation><sa:organization>National Centre for Nuclear Research</sa:organization><sa:city>Warsaw</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0395">National Centre for Nuclear Research, Warsaw, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff0800" affiliation-id="S0370269322007833-5c73c93f2c9aba1b7fcfa30741ce17ca"><ce:label>80</ce:label><ce:textfn>National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India</ce:textfn><sa:affiliation><sa:organization>National Institute of Science Education and Research</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Jatni</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0400">National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0810" affiliation-id="S0370269322007833-1e229a9758219f877a7f903c2dde9c2b"><ce:label>81</ce:label><ce:textfn>National Nuclear Research Center, Baku, Azerbaijan</ce:textfn><sa:affiliation><sa:organization>National Nuclear Research Center</sa:organization><sa:city>Baku</sa:city><sa:country>Azerbaijan</sa:country></sa:affiliation><ce:source-text id="srct0405">National Nuclear Research Center, Baku, Azerbaijan</ce:source-text></ce:affiliation><ce:affiliation id="aff0820" affiliation-id="S0370269322007833-ab3fe404fa49046cc1d6ed593a4d5eb8"><ce:label>82</ce:label><ce:textfn>National Research and Innovation Agency - BRIN, Jakarta, Indonesia</ce:textfn><sa:affiliation><sa:organization>National Research and Innovation Agency - BRIN</sa:organization><sa:city>Jakarta</sa:city><sa:country>Indonesia</sa:country></sa:affiliation><ce:source-text id="srct0410">National Research and Innovation Agency - BRIN, Jakarta, Indonesia</ce:source-text></ce:affiliation><ce:affiliation id="aff0830" affiliation-id="S0370269322007833-3e7cb9222cae4e75df0d93d2ea96329f"><ce:label>83</ce:label><ce:textfn>Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark</ce:textfn><sa:affiliation><sa:organization>Niels Bohr Institute</sa:organization><sa:organization>University of Copenhagen</sa:organization><sa:city>Copenhagen</sa:city><sa:country>Denmark</sa:country></sa:affiliation><ce:source-text id="srct0415">Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark</ce:source-text></ce:affiliation><ce:affiliation id="aff0840" affiliation-id="S0370269322007833-11b144edfb6d992e108d17991cc7681b"><ce:label>84</ce:label><ce:textfn>Nikhef, National institute for subatomic physics, Amsterdam, Netherlands</ce:textfn><sa:affiliation><sa:organization>Nikhef, National institute for subatomic physics</sa:organization><sa:city>Amsterdam</sa:city><sa:country>Netherlands</sa:country></sa:affiliation><ce:source-text id="srct0420">Nikhef, National institute for subatomic physics, Amsterdam, Netherlands</ce:source-text></ce:affiliation><ce:affiliation id="aff0850" affiliation-id="S0370269322007833-7ffc71d9a245ae3927bb4d373ff478ed"><ce:label>85</ce:label><ce:textfn>Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Group</sa:organization><sa:organization>STFC Daresbury Laboratory</sa:organization><sa:city>Daresbury</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0425">Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff0860" affiliation-id="S0370269322007833-4ea73349ab4e79a7b6570dd0fd42cd13"><ce:label>86</ce:label><ce:textfn>Nuclear Physics Institute of the Czech Academy of Sciences, Husinec-Řež, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Institute of the Czech Academy of Sciences</sa:organization><sa:city>Husinec-Řež</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0430">Nuclear Physics Institute of the Czech Academy of Sciences, Husinec-Řež, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0870" affiliation-id="S0370269322007833-43c0b745ad096562d858417552718575"><ce:label>87</ce:label><ce:textfn>Oak Ridge National Laboratory, Oak Ridge, TN, United States</ce:textfn><sa:affiliation><sa:organization>Oak Ridge National Laboratory</sa:organization><sa:city>Oak Ridge</sa:city><sa:state>TN</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0435">Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0880" affiliation-id="S0370269322007833-e4e32a64c7e436b0f2a688687dedfe5e"><ce:label>88</ce:label><ce:textfn>Ohio State University, Columbus, OH, United States</ce:textfn><sa:affiliation><sa:organization>Ohio State University</sa:organization><sa:city>Columbus</sa:city><sa:state>OH</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0440">Ohio State University, Columbus, Ohio, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0890" affiliation-id="S0370269322007833-3c4eb171ede7112c64f1bc2c164f4732"><ce:label>89</ce:label><ce:textfn>Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia</ce:textfn><sa:affiliation><sa:organization>Physics department</sa:organization><sa:organization>Faculty of science, University of Zagreb</sa:organization><sa:city>Zagreb</sa:city><sa:country>Croatia</sa:country></sa:affiliation><ce:source-text id="srct0445">Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia</ce:source-text></ce:affiliation><ce:affiliation id="aff0900" affiliation-id="S0370269322007833-30d6dbb96747c914840798f88bf250cb"><ce:label>90</ce:label><ce:textfn>Physics Department, Panjab University, Chandigarh, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>Panjab University</sa:organization><sa:city>Chandigarh</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0450">Physics Department, Panjab University, Chandigarh, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0910" affiliation-id="S0370269322007833-712bc80495e97050498604da603cb5cc"><ce:label>91</ce:label><ce:textfn>Physics Department, University of Jammu, Jammu, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>University of Jammu</sa:organization><sa:city>Jammu</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0455">Physics Department, University of Jammu, Jammu, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0920" affiliation-id="S0370269322007833-500771749243cadcbaeeac3f726e6a62"><ce:label>92</ce:label><ce:textfn>Physics Department, University of Rajasthan, Jaipur, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>University of Rajasthan</sa:organization><sa:city>Jaipur</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0460">Physics Department, University of Rajasthan, Jaipur, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0930" affiliation-id="S0370269322007833-87f0c1fe6b138b8300b7695dc3391666"><ce:label>93</ce:label><ce:textfn>Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan</ce:textfn><sa:affiliation><sa:organization>Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2)</sa:organization><sa:organization>Hiroshima University</sa:organization><sa:city>Hiroshima</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0465">Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0940" affiliation-id="S0370269322007833-0886f70da565231fd0e43398021fd604"><ce:label>94</ce:label><ce:textfn>Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany</ce:textfn><sa:affiliation><sa:organization>Physikalisches Institut</sa:organization><sa:organization>Eberhard-Karls-Universität Tübingen</sa:organization><sa:city>Tübingen</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0470">Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0950" affiliation-id="S0370269322007833-1c67678e124982de924355cdd6d6b91b"><ce:label>95</ce:label><ce:textfn>Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany</ce:textfn><sa:affiliation><sa:organization>Physikalisches Institut</sa:organization><sa:organization>Ruprecht-Karls-Universität Heidelberg</sa:organization><sa:city>Heidelberg</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0475">Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0960" affiliation-id="S0370269322007833-7a5765cc3928149c8b77128cd52462ae"><ce:label>96</ce:label><ce:textfn>Physik Department, Technische Universität München, Munich, Germany</ce:textfn><sa:affiliation><sa:organization>Physik Department</sa:organization><sa:organization>Technische Universität München</sa:organization><sa:city>Munich</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0480">Physik Department, Technische Universität München, Munich, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0970" affiliation-id="S0370269322007833-0a06f4f5fb60a25c8f1936ed71c96cfe"><ce:label>97</ce:label><ce:textfn>Politecnico di Bari and Sezione INFN, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>Politecnico di Bari</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0485">Politecnico di Bari and Sezione INFN, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0980" affiliation-id="S0370269322007833-3451d358e90b759225813c9b9a6f098c"><ce:label>98</ce:label><ce:textfn>Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany</ce:textfn><sa:affiliation><sa:organization>Research Division</sa:organization><sa:organization>ExtreMe Matter Institute EMMI</sa:organization><sa:organization>GSI Helmholtzzentrum für Schwerionenforschung GmbH</sa:organization><sa:city>Darmstadt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0490">Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0990" affiliation-id="S0370269322007833-2771a2ae295804be11a4a937b894a546"><ce:label>99</ce:label><ce:textfn>Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Saha Institute of Nuclear Physics</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0495">Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1000" affiliation-id="S0370269322007833-b602cde70213041ca40ec5720e4e3e75"><ce:label>100</ce:label><ce:textfn>School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom</ce:textfn><sa:affiliation><sa:organization>School of Physics and Astronomy</sa:organization><sa:organization>University of Birmingham</sa:organization><sa:city>Birmingham</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0500">School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff1010" affiliation-id="S0370269322007833-1a1f0f3ae33ba0323ad4da08a216f4c3"><ce:label>101</ce:label><ce:textfn>Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru</ce:textfn><sa:affiliation><sa:organization>Sección Física</sa:organization><sa:organization>Departamento de Ciencias</sa:organization><sa:organization>Pontificia Universidad Católica del Perú</sa:organization><sa:city>Lima</sa:city><sa:country>Peru</sa:country></sa:affiliation><ce:source-text id="srct0505">Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru</ce:source-text></ce:affiliation><ce:affiliation id="aff1020" affiliation-id="S0370269322007833-24347400090c9a1efb87e7537af7461b"><ce:label>102</ce:label><ce:textfn>Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria</ce:textfn><sa:affiliation><sa:organization>Stefan Meyer Institut für Subatomare Physik (SMI)</sa:organization><sa:city>Vienna</sa:city><sa:country>Austria</sa:country></sa:affiliation><ce:source-text id="srct0510">Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria</ce:source-text></ce:affiliation><ce:affiliation id="aff1030" affiliation-id="S0370269322007833-317714469d8c208c77efd6f026942633"><ce:label>103</ce:label><ce:textfn>SUBATECH, IMT Atlantique, Nantes Université, CNRS-IN2P3, Nantes, France</ce:textfn><sa:affiliation><sa:organization>SUBATECH</sa:organization><sa:organization>IMT Atlantique</sa:organization><sa:organization>Nantes Université</sa:organization><sa:organization>CNRS-IN2P3</sa:organization><sa:city>Nantes</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0515">SUBATECH, IMT Atlantique, Nantes Université, CNRS-IN2P3, Nantes, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1040" affiliation-id="S0370269322007833-c3972f6af24eef12cc2ae3a53d6a7623"><ce:label>104</ce:label><ce:textfn>Suranaree University of Technology, Nakhon Ratchasima, Thailand</ce:textfn><sa:affiliation><sa:organization>Suranaree University of Technology</sa:organization><sa:city>Nakhon Ratchasima</sa:city><sa:country>Thailand</sa:country></sa:affiliation><ce:source-text id="srct0520">Suranaree University of Technology, Nakhon Ratchasima, Thailand</ce:source-text></ce:affiliation><ce:affiliation id="aff1050" affiliation-id="S0370269322007833-9186de958ef0b51277f8f34872e5c71b"><ce:label>105</ce:label><ce:textfn>Technical University of Košice, Košice, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Technical University of Košice</sa:organization><sa:city>Košice</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0525">Technical University of Košice, Košice, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff1060" affiliation-id="S0370269322007833-425836ada61fe0d3ee4ebf5b87598919"><ce:label>106</ce:label><ce:textfn>The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland</ce:textfn><sa:affiliation><sa:organization>The Henryk Niewodniczanski Institute of Nuclear Physics</sa:organization><sa:organization>Polish Academy of Sciences</sa:organization><sa:city>Cracow</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0530">The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff1070" affiliation-id="S0370269322007833-302a66584ead3636c3e74b86b51cf474"><ce:label>107</ce:label><ce:textfn>The University of Texas at Austin, Austin, TX, United States</ce:textfn><sa:affiliation><sa:organization>The University of Texas at Austin</sa:organization><sa:city>Austin</sa:city><sa:state>TX</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0535">The University of Texas at Austin, Austin, Texas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1080" affiliation-id="S0370269322007833-48ea3700cc067946e6c3032832f0ce0a"><ce:label>108</ce:label><ce:textfn>Universidad Autónoma de Sinaloa, Culiacán, Mexico</ce:textfn><sa:affiliation><sa:organization>Universidad Autónoma de Sinaloa</sa:organization><sa:city>Culiacán</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0540">Universidad Autónoma de Sinaloa, Culiacán, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff1090" affiliation-id="S0370269322007833-310945a13cafb9845fe651fc7aa661b6"><ce:label>109</ce:label><ce:textfn>Universidade de São Paulo (USP), São Paulo, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade de São Paulo (USP)</sa:organization><sa:city>São Paulo</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0545">Universidade de São Paulo (USP), São Paulo, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1100" affiliation-id="S0370269322007833-226f6a475b7a1634e1530e2077d7590e"><ce:label>110</ce:label><ce:textfn>Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade Estadual de Campinas (UNICAMP)</sa:organization><sa:city>Campinas</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0550">Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1110" affiliation-id="S0370269322007833-7582533209ee7a368c4d249782e93c03"><ce:label>111</ce:label><ce:textfn>Universidade Federal do ABC, Santo Andre, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade Federal do ABC</sa:organization><sa:city>Santo Andre</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0555">Universidade Federal do ABC, Santo Andre, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1120" affiliation-id="S0370269322007833-f37f0b36132124cce5539b1f9c518079"><ce:label>112</ce:label><ce:textfn>University of Cape Town, Cape Town, South Africa</ce:textfn><sa:affiliation><sa:organization>University of Cape Town</sa:organization><sa:city>Cape Town</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0560">University of Cape Town, Cape Town, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff1130" affiliation-id="S0370269322007833-58fe35a5cc9d91eea34fd6e19293599b"><ce:label>113</ce:label><ce:textfn>University of Houston, Houston, TX, United States</ce:textfn><sa:affiliation><sa:organization>University of Houston</sa:organization><sa:city>Houston</sa:city><sa:state>TX</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0565">University of Houston, Houston, Texas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1140" affiliation-id="S0370269322007833-e9be2b1885c8285b7cb7b6534ca93d98"><ce:label>114</ce:label><ce:textfn>University of Jyväskylä, Jyväskylä, Finland</ce:textfn><sa:affiliation><sa:organization>University of Jyväskylä</sa:organization><sa:city>Jyväskylä</sa:city><sa:country>Finland</sa:country></sa:affiliation><ce:source-text id="srct0570">University of Jyväskylä, Jyväskylä, Finland</ce:source-text></ce:affiliation><ce:affiliation id="aff1150" affiliation-id="S0370269322007833-6e2074f6fd0b88987501b22e5d74f9c9"><ce:label>115</ce:label><ce:textfn>University of Kansas, Lawrence, KS, United States</ce:textfn><sa:affiliation><sa:organization>University of Kansas</sa:organization><sa:city>Lawrence</sa:city><sa:state>KS</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0575">University of Kansas, Lawrence, Kansas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1160" affiliation-id="S0370269322007833-f538fc5a59ec1d1c8f86d2a83cb4ced6"><ce:label>116</ce:label><ce:textfn>University of Liverpool, Liverpool, United Kingdom</ce:textfn><sa:affiliation><sa:organization>University of Liverpool</sa:organization><sa:city>Liverpool</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0580">University of Liverpool, Liverpool, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff1170" affiliation-id="S0370269322007833-5507e3344bf7b46ba698c2c045a8aba6"><ce:label>117</ce:label><ce:textfn>University of Science and Technology of China, Hefei, China</ce:textfn><sa:affiliation><sa:organization>University of Science and Technology of China</sa:organization><sa:city>Hefei</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0585">University of Science and Technology of China, Hefei, China</ce:source-text></ce:affiliation><ce:affiliation id="aff1180" affiliation-id="S0370269322007833-5825f1740b6b974441dd90d9c8ad0a53"><ce:label>118</ce:label><ce:textfn>University of South-Eastern Norway, Kongsberg, Norway</ce:textfn><sa:affiliation><sa:organization>University of South-Eastern Norway</sa:organization><sa:city>Kongsberg</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0590">University of South-Eastern Norway, Kongsberg, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff1190" affiliation-id="S0370269322007833-d6620449e5365c80224ff22fe1ca4e4a"><ce:label>119</ce:label><ce:textfn>University of Tennessee, Knoxville, TN, United States</ce:textfn><sa:affiliation><sa:organization>University of Tennessee</sa:organization><sa:city>Knoxville</sa:city><sa:state>TN</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0595">University of Tennessee, Knoxville, Tennessee, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1200" affiliation-id="S0370269322007833-6d43022152ccaa62119e4287bf3b5270"><ce:label>120</ce:label><ce:textfn>University of the Witwatersrand, Johannesburg, South Africa</ce:textfn><sa:affiliation><sa:organization>University of the Witwatersrand</sa:organization><sa:city>Johannesburg</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0600">University of the Witwatersrand, Johannesburg, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff1210" affiliation-id="S0370269322007833-9b9071fc23073da0e4317b3863c91b9e"><ce:label>121</ce:label><ce:textfn>University of Tokyo, Tokyo, Japan</ce:textfn><sa:affiliation><sa:organization>University of Tokyo</sa:organization><sa:city>Tokyo</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0605">University of Tokyo, Tokyo, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff1220" affiliation-id="S0370269322007833-da3fb88e8779358429638ee8f8dc4cb5"><ce:label>122</ce:label><ce:textfn>University of Tsukuba, Tsukuba, Japan</ce:textfn><sa:affiliation><sa:organization>University of Tsukuba</sa:organization><sa:city>Tsukuba</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0610">University of Tsukuba, Tsukuba, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff1230" affiliation-id="S0370269322007833-9fc1715eadb7b2ba79b1a87cef8015cd"><ce:label>123</ce:label><ce:textfn>University Politehnica of Bucharest, Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>University Politehnica of Bucharest</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0615">University Politehnica of Bucharest, Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff1240" affiliation-id="S0370269322007833-43e2ac82e11c4ca2fabbc7e457c74b37"><ce:label>124</ce:label><ce:textfn>Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France</ce:textfn><sa:affiliation><sa:organization>Université Clermont Auvergne</sa:organization><sa:organization>CNRS/IN2P3</sa:organization><sa:organization>LPC</sa:organization><sa:city>Clermont-Ferrand</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0620">Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1250" affiliation-id="S0370269322007833-08f13150d05b32d440951efec6f80d29"><ce:label>125</ce:label><ce:textfn>Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France</ce:textfn><sa:affiliation><sa:organization>Université de Lyon</sa:organization><sa:organization>CNRS/IN2P3</sa:organization><sa:organization>Institut de Physique des 2 Infinis de Lyon</sa:organization><sa:city>Lyon</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0625">Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1260" affiliation-id="S0370269322007833-766d0075165899c16752f5f7ff1b8771"><ce:label>126</ce:label><ce:textfn>Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France</ce:textfn><sa:affiliation><sa:organization>Université de Strasbourg</sa:organization><sa:organization>CNRS</sa:organization><sa:organization>IPHC UMR 7178</sa:organization><sa:city>Strasbourg</sa:city><sa:postal-code>F-67000</sa:postal-code><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0630">Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1270" affiliation-id="S0370269322007833-36117066550bb6faaf9708b1b0ea670e"><ce:label>127</ce:label><ce:textfn>Université Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France</ce:textfn><sa:affiliation><sa:organization>Université Paris-Saclay Centre d'Etudes de Saclay (CEA)</sa:organization><sa:organization>IRFU</sa:organization><sa:organization>Départment de Physique Nucléaire (DPhN)</sa:organization><sa:city>Saclay</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0635">Université Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1280" affiliation-id="S0370269322007833-47ef552cdaf90d05d791574b14a8dcf9"><ce:label>128</ce:label><ce:textfn>Università degli Studi di Foggia, Foggia, Italy</ce:textfn><sa:affiliation><sa:organization>Università degli Studi di Foggia</sa:organization><sa:city>Foggia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0640">Università degli Studi di Foggia, Foggia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1290" affiliation-id="S0370269322007833-79fbcbe2a42e9099094360b5bf7bd602"><ce:label>129</ce:label><ce:textfn>Università del Piemonte Orientale, Vercelli, Italy</ce:textfn><sa:affiliation><sa:organization>Università del Piemonte Orientale</sa:organization><sa:city>Vercelli</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0645">Università del Piemonte Orientale, Vercelli, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1300" affiliation-id="S0370269322007833-83cd0c0929f483f88e3d2cb59f064287"><ce:label>130</ce:label><ce:textfn>Università di Brescia, Brescia, Italy</ce:textfn><sa:affiliation><sa:organization>Università di Brescia</sa:organization><sa:city>Brescia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0650">Università di Brescia, Brescia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1310" affiliation-id="S0370269322007833-f1ae52f852d4d7d99988b3e872f887e4"><ce:label>131</ce:label><ce:textfn>Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Variable Energy Cyclotron Centre</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0655">Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1320" affiliation-id="S0370269322007833-bfd4fe0d3b8675ea55d96ce64033d817"><ce:label>132</ce:label><ce:textfn>Warsaw University of Technology, Warsaw, Poland</ce:textfn><sa:affiliation><sa:organization>Warsaw University of Technology</sa:organization><sa:city>Warsaw</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0660">Warsaw University of Technology, Warsaw, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff1330" affiliation-id="S0370269322007833-4e11d38f810a3540206ffb5151a35d3b"><ce:label>133</ce:label><ce:textfn>Wayne State University, Detroit, MI, United States</ce:textfn><sa:affiliation><sa:organization>Wayne State University</sa:organization><sa:city>Detroit</sa:city><sa:state>MI</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0665">Wayne State University, Detroit, Michigan, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1340" affiliation-id="S0370269322007833-0b76517a6de7ff0579ec14780f79d81b"><ce:label>134</ce:label><ce:textfn>Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany</ce:textfn><sa:affiliation><sa:organization>Westfälische Wilhelms-Universität Münster</sa:organization><sa:organization>Institut für Kernphysik</sa:organization><sa:city>Münster</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0670">Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1350" affiliation-id="S0370269322007833-6b1ae45228f1f040c55767dc107b589d"><ce:label>135</ce:label><ce:textfn>Wigner Research Centre for Physics, Budapest, Hungary</ce:textfn><sa:affiliation><sa:organization>Wigner Research Centre for Physics</sa:organization><sa:city>Budapest</sa:city><sa:country>Hungary</sa:country></sa:affiliation><ce:source-text id="srct0675">Wigner Research Centre for Physics, Budapest, Hungary</ce:source-text></ce:affiliation><ce:affiliation id="aff1360" affiliation-id="S0370269322007833-96e79e7381eab77664732c3553e247e8"><ce:label>136</ce:label><ce:textfn>Yale University, New Haven, CT, United States</ce:textfn><sa:affiliation><sa:organization>Yale University</sa:organization><sa:city>New Haven</sa:city><sa:state>CT</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0680">Yale University, New Haven, Connecticut, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1370" affiliation-id="S0370269322007833-f3db17ceaf6ea4190e2176a6a129c96b"><ce:label>137</ce:label><ce:textfn>Yonsei University, Seoul, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Yonsei University</sa:organization><sa:city>Seoul</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0685">Yonsei University, Seoul, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff1380" affiliation-id="S0370269322007833-2e935afd39a812eb118b95cc53ef2ac3"><ce:label>138</ce:label><ce:textfn>Zentrum für Technologie und Transfer (ZTT), Worms, Germany</ce:textfn><sa:affiliation><sa:organization>Zentrum für Technologie und Transfer (ZTT)</sa:organization><sa:city>Worms</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0690">Zentrum für Technologie und Transfer (ZTT), Worms, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1390" affiliation-id="S0370269322007833-7972cea8d4a6e5b7150baf4a74e03970"><ce:label>139</ce:label><ce:textfn>Affiliated with an institute covered by a cooperation agreement with CERN</ce:textfn><sa:affiliation><sa:address-line>Affiliated with an institute covered by a cooperation agreement with CERN</sa:address-line></sa:affiliation><ce:source-text id="srct0695">Affiliated with an institute covered by a cooperation agreement with CERN</ce:source-text></ce:affiliation><ce:affiliation id="aff1400" affiliation-id="S0370269322007833-3c2eaa2a494293a99583fccbf5d95e9f"><ce:label>140</ce:label><ce:textfn>Affiliated with an international laboratory covered by a cooperation agreement with CERN</ce:textfn><sa:affiliation><sa:address-line>Affiliated with an international laboratory covered by a cooperation agreement with CERN</sa:address-line></sa:affiliation><ce:source-text id="srct0700">Affiliated with an international laboratory covered by a cooperation agreement with CERN</ce:source-text></ce:affiliation><ce:footnote id="fn0010"><ce:label>I</ce:label><ce:note-para id="np0010">Deceased.</ce:note-para></ce:footnote><ce:footnote id="fn0020"><ce:label>II</ce:label><ce:note-para id="np0020">Also at: Max-Planck-Institut für Physik, Munich, Germany.</ce:note-para></ce:footnote><ce:footnote id="fn0030"><ce:label>III</ce:label><ce:note-para id="np0030">Also at: Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA), Bologna, Italy.</ce:note-para></ce:footnote><ce:footnote id="fn0040"><ce:label>IV</ce:label><ce:note-para id="np0040">Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy.</ce:note-para></ce:footnote><ce:footnote id="fn0050"><ce:label>V</ce:label><ce:note-para id="np0050">Also at: Department of Applied Physics, Aligarh Muslim University, Aligarh, India.</ce:note-para></ce:footnote><ce:footnote id="fn0060"><ce:label>VI</ce:label><ce:note-para id="np0060">Also at: Institute of Theoretical Physics, University of Wroclaw, Poland.</ce:note-para></ce:footnote><ce:footnote id="fn0070"><ce:label>VII</ce:label><ce:note-para id="np0070">Also at: An institution covered by a cooperation agreement with CERN.</ce:note-para></ce:footnote></ce:author-group></ce:collaboration><ce:footnote id="fn0080"><ce:label>⋆</ce:label><ce:note-para id="np0080"><ce:italic>E-mail address:</ce:italic> <ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/text/html" xlink:href="mailto:alice-publications@cern.ch" id="inf0020">alice-publications@cern.ch</ce:inter-ref>.</ce:note-para></ce:footnote></ce:author-group><ce:date-received day="5" month="5" year="2022"/><ce:date-revised day="2" month="12" year="2022"/><ce:date-accepted day="23" month="12" year="2022"/><ce:miscellaneous id="ms0010">Editor: M. Pierini</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0070">This letter reports measurements which characterize the underlying event associated with hard scatterings at mid-pseudorapidity (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math>) in pp, p–Pb and Pb–Pb collisions at centre-of-mass energy per nucleon pair, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The measurements are performed with ALICE at the LHC. Different multiplicity classes are defined based on the event activity measured at forward rapidities. The hard scatterings are identified by the leading particle defined as the charged particle with the largest transverse momentum (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) in the collision and having 8 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mo linebreak="badbreak" linebreakstyle="after"><</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra of associated particles (0.5 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) are measured in different azimuthal regions defined with respect to the leading particle direction: toward, transverse, and away. The associated charged particle yields in the transverse region are subtracted from those of the away and toward regions. The remaining jet-like yields are reported as a function of the multiplicity measured in the transverse region. The measurements show a suppression of the jet-like yield in the away region and an enhancement of high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> associated particles in the toward region in central Pb–Pb collisions, as compared to minimum-bias pp collisions. These observations are consistent with previous measurements that used two-particle correlations, and with an interpretation in terms of parton energy loss in a high-density quark gluon plasma. These yield modifications vanish in peripheral Pb–Pb collisions and are not observed in either high-multiplicity pp or p–Pb collisions.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0080">Data availability</ce:section-title><ce:para id="pr0200">This manuscript has associated data in a HEPData repository at <ce:inter-ref xlink:href="https://www.hepdata.net/" xlink:role="http://www.elsevier.com/xml/linking-roles/research-data" id="inf0550">https://www.hepdata.net/</ce:inter-ref>.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">In proton-proton (pp) collisions, jets, originating from partonic scatterings with large momentum transfer, are accompanied by particles produced by initial- and final-state radiation (ISR and FSR, respectively), as well as, by a plethora of other mechanisms. These include proton break-up, and, in a scenario incorporating multi-parton interactions (MPI) <ce:cross-refs refid="br0010 br0020" id="crs0010">[1,2]</ce:cross-refs>, several semi-hard parton-parton scatterings in a single pp collision. These jet-accompanying particles experimentally make up the underlying event (UE) and are commonly studied via azimuthal separations from the jets to minimise the influence of hard scatterings. The present study follows the strategy originally introduced by the CDF collaboration <ce:cross-ref refid="br0030" id="crf10910">[3]</ce:cross-ref>. First, the leading charged particle in the event is found, i.e., the charged particle with the highest transverse momentum in the collision (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math>). Secondly, the associated particles (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math>) are measured in three topological regions depending on their azimuthal angle relative to the leading particle, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>φ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msup><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">assoc</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">|</mml:mo></mml:math>, see <ce:cross-ref refid="fg0010" id="crf10920">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/>.</ce:para><ce:para id="pr0020">The toward region contains the primary jet within the acceptance of the detector, while the away region contains the back-scattered particles of the recoil jet <ce:cross-ref refid="br0040" id="crf10930">[4]</ce:cross-ref>. In contrast, the transverse region is dominated by the underlying-event dynamics, but it also includes contributions from ISR and FSR <ce:cross-ref refid="br0050" id="crf10940">[5]</ce:cross-ref>.</ce:para><ce:para id="pr0030">The measurements performed at RHIC and LHC in small systems (pp, p–A, and d–A collisions) have shown for high particle multiplicities similar phenomena as were originally observed only in A–A collisions and have been attributed there to the formation of the strongly interacting quark gluon plasma <ce:cross-refs refid="br0060 br0070" id="crs0020">[6,7]</ce:cross-refs>, namely, long range angular correlations and collectivity <ce:cross-ref refid="br0080" id="crf10950">[8]</ce:cross-ref>. The origin of these effects in small systems is still an open question; on one hand, hydrodynamical calculations describe some aspects of the data <ce:cross-ref refid="br0090" id="crf10960">[9]</ce:cross-ref>; on the other hand, mechanisms like colour reconnection <ce:cross-ref refid="br0100" id="crf10970">[10]</ce:cross-ref>, rope hadronisation <ce:cross-ref refid="br0110" id="crf10980">[11]</ce:cross-ref>, and string shoving <ce:cross-ref refid="br0120" id="crf10990">[12]</ce:cross-ref> can produce collective-like effects in Monte Carlo event generators such as <ce:small-caps>PYTHIA</ce:small-caps> 8 <ce:cross-ref refid="br0130" id="crf11000">[13]</ce:cross-ref>. Thus, investigating pp collisions as a function of the charged particle multiplicity has become ever more pertinent <ce:cross-refs refid="br0090 br0140 br0150 br0160 br0170 br0180" id="crs0030">[9,14–18]</ce:cross-refs>. The interpretation of the results from the analysis of high-multiplicity pp collisions is challenging due to the selection biases of the sample towards events in which partonic scatterings with large momentum transfer (hard scatterings) occurred. To mitigate this inherent bias, Martin et al. <ce:cross-ref refid="br0190" id="crf11010">[19]</ce:cross-ref> suggested to use the charged-particle multiplicity in the transverse region (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>) as a classifier of the activity in the collisions, since the correlation between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math> and the hardest scattering in the collision is small. The ALICE collaboration has reported the first <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math> spectra measured in pp collisions at centre-of-mass energy, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:msqrt><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>13</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV <ce:cross-ref refid="br0200" id="crf11020">[20]</ce:cross-ref>. Event generators, such as <ce:small-caps>PYTHIA</ce:small-caps> 8 <ce:cross-ref refid="br0130" id="crf11030">[13]</ce:cross-ref> and EPOS-LHC <ce:cross-ref refid="br0210" id="crf11040">[21]</ce:cross-ref>, do not provide a good description of the measured distribution of the ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>/<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> is the event-averaged charged-particle multiplicity in the transverse region, underestimating in particular the number of collisions with large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:math>. In the framework of MPI-based models, like those implemented in <ce:small-caps>PYTHIA</ce:small-caps> 8 and <ce:small-caps>HERWIG</ce:small-caps> 7 <ce:cross-ref refid="br0220" id="crf11050">[22]</ce:cross-ref>, the probability for a hard scattering in the collision increases with decreasing impact parameter<ce:cross-ref refid="fn0090" id="crf11060"><ce:sup>VIII</ce:sup></ce:cross-ref><ce:footnote id="fn0090"><ce:label>VIII</ce:label><ce:note-para id="np0090">In event generators like <ce:small-caps>PYTHIA</ce:small-caps> 8 the impact parameter profile is described by an overlap matter distribution of the two incoming hadrons.</ce:note-para></ce:footnote> between the colliding protons. Thus, requiring a high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> particle (e.g., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>8</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) in a given pp collision biases the selection of collisions towards those with a smaller impact parameter <ce:cross-ref refid="br0230" id="crf11070">[23]</ce:cross-ref>, which in turn biases the selection towards pp collisions with more MPI <ce:cross-ref refid="br0200" id="crf11080">[20]</ce:cross-ref>. This feature of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>-based analysis is important for the isolation of potential MPI and colour reconnection effects, which according to <ce:small-caps>PYTHIA</ce:small-caps> 8, produce effects resembling collective behaviour <ce:cross-ref refid="br0100" id="crf11090">[10]</ce:cross-ref>. By construction, MPI and colour reconnection effects are expected to be more relevant in the transverse region than in the away and toward regions <ce:cross-ref refid="br0240" id="crf11100">[24]</ce:cross-ref>. It is worth mentioning that the MPI picture has been used to explain the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in p–Pb collisions and peripheral Pb–Pb collisions <ce:cross-refs refid="br0250 br0260 br0270" id="crs0040">[25–27]</ce:cross-refs>. Studies, as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>, are therefore important to the understanding of the effects observed in high-multiplicity pp collisions. Last but not least, measurements of UE observables are also important to tune event generators <ce:cross-ref refid="br0280" id="crf11110">[28]</ce:cross-ref> that include hard partonic scatterings and MPI.</ce:para><ce:para id="pr0040">This letter reports the inclusive charged-particle transverse momentum spectra in pp, p–Pb and Pb–Pb collisions at centre-of-mass energy per nucleon pair <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV containing a high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> leading particle within the kinematic intervals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>8</mml:mn><mml:mo>≤</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math>. This guarantees the selection of collisions in which the average activity in the transverse region is roughly flat as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math> <ce:cross-ref refid="br0200" id="crf11120">[20]</ce:cross-ref>, and therefore, any additional selection on the charged particle multiplicity will only modulate the UE activity. The measurements are performed considering different event classes defined in terms of the multiplicity registered in the forward detectors. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra of associated charged particles (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mn>0.5</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math>) are measured in the toward, away, and transverse regions as a function of the average charged particle multiplicity in the transverse region. To further investigate the possible modification of the particles produced in the hard scattering in pp, p–Pb, and Pb–Pb collisions, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> distributions in the toward (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) and away (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) regions obtained after the subtraction of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in the transverse region (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) are also reported. The subtracted yields (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) are further normalised to those measured in minimum-bias (MB) pp collisions,<ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pp</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">MB</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pp</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">MB</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>X</ce:italic> indicates the collision system and the event multiplicity class. In this way, the hard process <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in the toward and away regions are isolated, and thus allowing us to study possible modifications to the produced particles due to medium effects in high-multiplicity pp, p–Pb, and Pb–Pb collisions. In heavy-ion collisions, this ratio is sensitive to the same effects which were studied using the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math> quantity <ce:cross-refs refid="br0290 br0300 br0310" id="crs0050">[29–31]</ce:cross-refs>, where jets produced in the early stage of the collision propagate through the hot and dense quark–gluon plasma. Their interaction with the coloured medium lead to parton-energy loss (jet quenching) <ce:cross-ref refid="br0320" id="crf11130">[32]</ce:cross-ref> which, for example, results in the suppression of the charged-particle yield at high <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> <ce:cross-ref refid="br0330" id="crf11140">[33]</ce:cross-ref>, and the suppression of the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> yield in the away region <ce:cross-refs refid="br0290 br0300" id="crs0060">[29,30]</ce:cross-refs>. It is worth mentioning that jet quenching effects have not been observed so far in small systems <ce:cross-refs refid="br0330 br0340" id="crs0070">[33,34]</ce:cross-refs>.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">Experiment and data analysis</ce:section-title><ce:para id="pr0050">This analysis is based on the data recorded by the ALICE apparatus during the pp and Pb–Pb runs at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV in 2015, and the p–Pb run at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV in 2016. The present study uses the V0 detector, and the Silicon Pixel Detector (SPD) for triggering and background rejection. The V0 consists of two arrays of scintillating tiles placed on each side of the interaction point covering the full azimuthal acceptance and the pseudorapidity intervals of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:mn>2.8</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5.1</mml:mn></mml:math> (V0A) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.7</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1.7</mml:mn></mml:math> (V0C). The SPD is the innermost part of the Inner Tracking System (ITS) and it is the closest detector to the interaction point. It consists of two cylindrical silicon pixel layers at radial distances of 3.9 and 7.6 cm from the beam line and the pseudorapidity coverages of the two layers are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1.4</mml:mn></mml:math>, respectively. The data were collected using a minimum-bias trigger, which required a signal in both V0A and V0C detectors. The offline event selection was optimised to reject beam-induced background in all collision systems by utilising the timing signals in the two V0 detectors. In Pb–Pb collisions, the beam-induced background is further suppressed by correlating the timing signals of the neutron zero degree calorimeters, which are positioned on both sides of the interaction point at 112.5<ce:hsp sp="0.20"/>m distance along the beam axis <ce:cross-ref refid="br0350" id="crf11150">[35]</ce:cross-ref>. The signals from the zero degree calorimeters are also used to suppress the contamination from electromagnetic interactions. This is performed by requesting the coincidence of the signals coming from both side zero degree calorimeters by which the background due to single nucleus electromagnetic dissociation processes is excluded. A criterion based on the offline reconstruction of multiple primary vertices in the SPD is applied to reduce the pileup caused by multiple interactions in the same bunch crossing <ce:cross-ref refid="br0360" id="crf11160">[36]</ce:cross-ref>. The results presented in this letter are for minimum-bias triggered pp collisions having at least one charged particle in the pseudorapidity interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1</mml:mn></mml:math> (INEL>0). The INEL>0 event class corresponds to about 75% of the total inelastic cross section <ce:cross-ref refid="br0370" id="crf11170">[37]</ce:cross-ref>. For pp and Pb–Pb collisions, the sample is subdivided into different multiplicity classes based on the total charge deposited in both V0 sub-detectors, which is termed as V0M amplitude <ce:cross-ref refid="br0380" id="crf11180">[38]</ce:cross-ref>. For p–Pb collisions, the sample is subdivided based on the total charge deposited in V0A sub-detector (V0A amplitude) <ce:cross-ref refid="br0390" id="crf11190">[39]</ce:cross-ref>, which is located in the Pb-going direction. The V0A estimator has been implemented in previous measurements that used p–Pb data (see e.g. <ce:cross-ref refid="br0400" id="crf11200">[40]</ce:cross-ref>). This allows for comparisons with other observables for similar V0A multiplicity classes. To ensure that a hard scattering took place in the collision, events are required to have a trigger particle within <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>8</mml:mn><mml:mo>≤</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math> GeV/<ce:italic>c</ce:italic>. In this <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math> interval, the momentum resolution effects are negligible on the extracted yields, and therefore, no <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math> resolution correction is applied. The total number of analysed collisions before the trigger particle selection are about 10<ce:sup>8</ce:sup>, 10<ce:sup>8</ce:sup>, and 10<ce:sup>7</ce:sup> for pp, p–Pb, and Pb–Pb collisions, respectively.</ce:para><ce:para id="pr0060">The transverse momentum of particles is determined from measurements in the central barrel with the ITS and the Time Projection Chamber (TPC). The ITS is a tracking detector which consists of six cylindrical layers of silicon detectors. The TPC is a cylindrical drift detector which covers a radial distance of 85-247<ce:hsp sp="0.20"/>cm from the beam axis and it has longitudinal dimension extending from about -250<ce:hsp sp="0.20"/>cm to +250<ce:hsp sp="0.20"/>cm around the nominal interaction point. Primary charged particles are measured in the pseudorapidity range of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math> and with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0.5</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, where <ce:italic>η</ce:italic> is measured in the laboratory frame for the three collision systems. The configuration for p–Pb collisions with protons at 4<ce:hsp sp="0.20"/>TeV energy colliding with Pb ions that have per-nucleon energies of (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mi>Z</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>A</mml:mi></mml:math>) × 4<ce:hsp sp="0.20"/>TeV ∼ 1.58<ce:hsp sp="0.20"/>TeV results in a shift in the rapidity of the nucleon–nucleon centre-of-mass system by 0.465 in the direction of the proton beam (negative z-direction). Here <ce:italic>Z</ce:italic> and <ce:italic>A</ce:italic> are the atomic and mass numbers of the Pb ion, respectively. Therefore, the detector coverage <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math> corresponds to roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>0.3</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cms</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1.3</mml:mn></mml:math> for p–Pb collisions. The particles with mean proper lifetime larger than 1<ce:hsp sp="0.20"/>cm/<ce:italic>c</ce:italic>, which are either produced directly in the interaction or from decays of particles with mean proper lifetime smaller than 1<ce:hsp sp="0.20"/>cm/<ce:italic>c</ce:italic> are termed as primary particles <ce:cross-ref refid="br0410" id="crf11210">[41]</ce:cross-ref>. The track selection follows a procedure similar to the one described in Ref. <ce:cross-ref refid="br0420" id="crf11220">[42]</ce:cross-ref> and only few specific details are reported here. Tracks (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">tracks</mml:mi></mml:mrow></mml:msub></mml:math>) are required to have two hits in the ITS, out of which at least one should be in either of the two innermost layers. The geometrical track length <ce:italic>L</ce:italic> is calculated in the TPC readout plane, excluding the information from the pads at the sector boundaries (≈3<ce:hsp sp="0.20"/>cm from the sector edges). The trajectory lengths built from radial segments, i.e. the crossed TPC pad rows, traversed in the TPC by a particle are required to be larger than 85% of the geometrical track length. The pad rows are made of at least 3 neighbouring individual observations (clusters), and their height varies from 7.5<ce:hsp sp="0.20"/>mm to 15<ce:hsp sp="0.20"/>mm <ce:cross-ref refid="br0430" id="crf11230">[43]</ce:cross-ref>. The trajectory lengths built from clusters (one cluster per pad row) is required to be larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mn>0.7</mml:mn><mml:mo>×</mml:mo><mml:mi>L</mml:mi></mml:math>. The fraction of TPC clusters shared with another track is required to be lower than 0.4. The fit quality for the ITS and TPC track points must satisfy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">hits</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>36</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">clusters</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>4</mml:mn></mml:math>, respectively, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">hits</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">clusters</mml:mi></mml:mrow></mml:msub></mml:math> are the numbers of hits in the ITS and the number of clusters in the TPC, respectively. Only tracks with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>36</mml:mn></mml:math> are included in the analysis, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> is calculated comparing the track parameters from the combined ITS and TPC track reconstruction to that derived only from the TPC and constrained to the interaction point. The definition of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> can be found in Ref. <ce:cross-ref refid="br0440" id="crf11240">[44]</ce:cross-ref>. To reduce the contamination from secondary particles, tracks are accepted if their distance-of-closest-approach (DCA) to the reconstructed primary interaction vertex satisfies in the longitudinal (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow></mml:msub></mml:math>) and transverse (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">xy</mml:mi></mml:mrow></mml:msub></mml:math>) directions the conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math><ce:hsp sp="0.20"/>cm and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">xy</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.018</mml:mn></mml:math><ce:hsp sp="0.20"/>cm + 0.035<ce:hsp sp="0.20"/>(cm×GeV/<ce:italic>c</ce:italic>)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>.</ce:para><ce:para id="pr0070">The measurement of the transverse momentum spectra of charged particles follows the standard procedure of the ALICE collaboration <ce:cross-refs refid="br0420 br0450" id="crs0080">[42,45]</ce:cross-refs>. The raw yields are corrected for efficiency and contamination from secondary particles. The efficiency correction is calculated from Monte Carlo simulations with GEANT3 <ce:cross-ref refid="br0460" id="crf11250">[46]</ce:cross-ref> transport code, which made use of PYTHIA 8 (Monash) <ce:cross-ref refid="br0280" id="crf11260">[28]</ce:cross-ref>, EPOS-LHC <ce:cross-ref refid="br0210" id="crf11270">[21]</ce:cross-ref> and HIJING <ce:cross-ref refid="br0470" id="crf11280">[47]</ce:cross-ref> event generators for pp, p–Pb and Pb–Pb collisions, respectively and incorporated a detailed description of the detector material, geometry and response. Since the event generators do not reproduce the relative abundances of different particle species in the real data, the efficiency obtained from the simulations is re-weighted considering the particle composition from data as outlined in <ce:cross-ref refid="br0420" id="crf11290">[42]</ce:cross-ref>. A multi-component template fit based on the DCA distributions from the simulation is used for the estimation of secondary contamination <ce:cross-ref refid="br0420" id="crf11300">[42]</ce:cross-ref>.</ce:para><ce:para id="pr0080">The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra for the toward and away regions include contributions from the jet fragmentation, ISR, and FSR, as well as, the contribution from the underlying event. In order to increase the sensitivity to the hardest process of the event, the particle yields measured in the transverse region are subtracted from the corresponding yields in both the toward and away regions: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>. This approach assumes that the background (UE, ISR, and FSR) in the toward and away regions is similar to the activity in the transverse region. However, one has to keep in mind that in Pb–Pb collisions two-particle correlations are affected by anisotropic transverse flow. In particular, the main contribution is due to the elliptic flow, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>, which is the second order coefficient in the Fourier expansion of the azimuthal distribution of the particle momenta <ce:cross-ref refid="br0480" id="crf11310">[48]</ce:cross-ref>. This elliptic azimuthal anisotropy modulates the background according to:<ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi>B</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>φ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant="normal">cos</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>φ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> is approximately given by the product of anisotropic flow coefficients for trigger and associated particles at their respective momenta i.e. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant="normal">assoc</mml:mi></mml:mrow></mml:msubsup></mml:math>. The existing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> measurements over a broad transverse momentum range <ce:cross-ref refid="br0490" id="crf11320">[49]</ce:cross-ref> suggest that the effect of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulation of background should be more relevant in semi-central Pb–Pb collisions. The effect is expected to be important at low and intermediate transverse momenta and decreases for high transverse momentum particles <ce:cross-ref refid="br0300" id="crf11330">[30]</ce:cross-ref>. In the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> region of interest for the jet quenching studies, namely <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>4</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, the effect of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulation is estimated to be small (about 5%) for Pb–Pb collisions. Given that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> effect is larger in Pb–Pb collisions than in pp and p–Pb collisions, no correction for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulation is applied for pp and p–Pb collisions since its effect is smaller than the other sources of systematic uncertainty.</ce:para><ce:para id="pr0090">The results are shown as a function of the average number of charged particles in the transverse region <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math>. The values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> are extracted in each multiplicity class from the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">tracks</mml:mi></mml:mrow></mml:msub></mml:math> distributions in the transverse region that are corrected for detector effects using a Bayesian unfolding <ce:cross-ref refid="br0500" id="crf11340">[50]</ce:cross-ref>. The Bayesian unfolding requires the multiplicity response matrix, which is built from the correlation between the measured multiplicity and the multiplicity at generator level (without detector effects) in the transverse region. This has been obtained from MC simulations which include the propagation of particles through the detector using GEANT 3. As a crosscheck, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> values are also calculated by integrating the transverse momentum distributions in the interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mn>0.5</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>8</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>. The difference between the results from the two strategies is assigned as the systematic uncertainty on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math>, where the effects related to the discrepancy between data and MC in the particle composition and secondary contamination are considered. This uncertainty amounts up to 3.5%, 4% and 6.5% for pp, p–Pb and Pb–Pb collisions, respectively.</ce:para><ce:para id="pr0100">The systematic uncertainties related to the track selection criteria were studied by repeating the analysis varying one-by-one the track selection criteria <ce:cross-refs refid="br0420 br0450" id="crs0090">[42,45]</ce:cross-refs>. In particular, the upper limits of the track fit quality parameters in the ITS (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">hits</mml:mi></mml:mrow></mml:msub></mml:math>) and in the TPC (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">clusters</mml:mi></mml:mrow></mml:msub></mml:math>) were varied in the ranges of 25–49 and 3–5, respectively. The maximum fraction of shared TPC clusters was varied between 0.2 to 1 and the maximum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow></mml:msub></mml:math> was varied between 1 and 5<ce:hsp sp="0.20"/>cm <ce:cross-ref refid="br0420" id="crf11350">[42]</ce:cross-ref>. We have also quantified the impact of not including the ITS hit requirement in the track selection. The systematic uncertainty on the primary particle composition was estimated using a procedure similar to the one described in <ce:cross-ref refid="br0420" id="crf11360">[42]</ce:cross-ref>. To quantify the uncertainty due to the imperfect simulation of the detector response, the track matching between the TPC and the ITS information in the data and in the simulation were compared. To achieve this, the fraction of secondary particles was rescaled according to fits to the measured DCA distributions. After this rescaling, the agreement between data and model was found to be within 3% for all collision systems. This value was assigned as an additional systematic uncertainty <ce:cross-ref refid="br0420" id="crf11370">[42]</ce:cross-ref>. The systematic uncertainty on the secondary particle contamination considers the imperfection of the method (multi-component template fit) used to extract the correction. The fit ranges were varied and the fit was repeated using templates with two (primaries, secondaries) or three (primaries, secondaries from material, secondaries from weak decays) components. The maximum spread among these variations was assigned as the systematic uncertainty on the secondary contamination. This contribution dominates at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>. The density of materials used in simulations of the experimental setup was varied by ± 4.5% <ce:cross-ref refid="br0350" id="crf11380">[35]</ce:cross-ref>, resulting in a negligible systematic uncertainty in the considered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> range of 0.5 to 6.0 GeV/<ce:italic>c</ce:italic>. For the estimation of total systematic uncertainty, all the above listed contributions were summed in quadrature. The systematic uncertainties are independent of the difference between the azimuthal angle of the associated particle and that of the trigger particle. The estimated systematic uncertainties on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra significantly depend on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>, while the dependence on the multiplicity classes is mild. The ranges of systematic uncertainties in the three considered collision systems are reported in <ce:cross-ref refid="tbl0010" id="crf11390">Table 1</ce:cross-ref><ce:float-anchor refid="tbl0010"/> for the various sources described above.</ce:para></ce:section><ce:section id="se0030" role="results"><ce:label>3</ce:label><ce:section-title id="st0040">Results and discussion</ce:section-title><ce:para id="pr0110">The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra measured in the transverse region for pp, p–Pb, and Pb–Pb collisions are shown in <ce:cross-ref refid="fg0020" id="crf11400">Fig. 2</ce:cross-ref><ce:float-anchor refid="fg0020"/> (top panel). Results are presented for different multiplicity classes. The ratios between the spectra in the individual multiplicity classes and the MB (0−100%) one are shown in the bottom panel. In the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:mn>0.5</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, the ratios for the highest multiplicity class (0−5%) are larger than unity and show an increasing trend with increasing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:mo linebreak="badbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) followed at higher <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> by a slow decrease. Instead, for the lowest multiplicity classes (40−60% and 60−90%) the ratios are lower than unity and follow an opposite trend with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>, decreasing at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> and increasing for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>3</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>. The behaviour of the ratios as a function of the event activity is reminiscent of analogous ratios as a function of the number of MPI in pp collisions simulated with <ce:small-caps>PYTHIA</ce:small-caps> 8, including colour reconnection <ce:cross-ref refid="br0510" id="crf11410">[51]</ce:cross-ref>. In particular, at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:math> GeV/<ce:italic>c</ce:italic> the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectrum of pp collisions with large MPI activity exhibits an enhancement with respect to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectrum of MB pp collisions. The effect was not observed before in data because, in contrast to the present analysis, the jet contribution was included in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra <ce:cross-ref refid="br0450" id="crf11420">[45]</ce:cross-ref>.</ce:para><ce:para id="pr0120">The top (bottom) panel of <ce:cross-ref refid="fg0030" id="crf11430">Fig. 3</ce:cross-ref><ce:float-anchor refid="fg0030"/> shows the charged particle yields for the toward (away) region after the subtraction of the yields measured in the transverse region in pp, p–Pb and Pb–Pb collisions. Results are compared with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra measured for MB pp collisions (0−100% V0M pp event class) quantified with the ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math>, as defined in Eq. <ce:cross-ref refid="fm0010" id="crf11440">(1)</ce:cross-ref>. At low transverse momenta, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>4</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is close to unity in pp and p–Pb collisions. In contrast, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> in Pb–Pb collisions exhibits a strong multiplicity dependence over the whole measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> interval. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> magnitude is larger for semi-peripheral Pb–Pb collisions, the maximum is observed for 20−40% Pb–Pb collisions, and is smaller for the most central and most peripheral classes. Given that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> contribution is not subtracted from the jet-like yields reported in <ce:cross-ref refid="fg0030" id="crf11450">Fig. 3</ce:cross-ref>, the centrality dependence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> follows the behaviour of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> as a function of collision centrality and particle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> in Pb–Pb collisions at LHC energies <ce:cross-ref refid="br0520" id="crf11460">[52]</ce:cross-ref>.</ce:para><ce:para id="pr0130"><ce:cross-ref refid="fg0040" id="crf11660">Fig. 4</ce:cross-ref><ce:float-anchor refid="fg0040"/> shows the measured values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> in the transverse momentum interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mn>4</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> as a function of the average multiplicity in the transverse region for all the multiplicity classes considered in pp, p–Pb and Pb–Pb collisions. The figure shows that, within uncertainties, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are close to unity for all the multiplicity classes measured in pp and p–Pb collisions. This indicates that effects induced by possible energy loss in these systems are not observed within uncertainties. This result is consistent with previous studies of nuclear modification factor <ce:cross-ref refid="br0330" id="crf11480">[33]</ce:cross-ref> and hadron-jet recoil measurements <ce:cross-ref refid="br0340" id="crf11490">[34]</ce:cross-ref>. By contrast, for Pb–Pb collisions the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are compatible to unity for peripheral collisions, and show a gradual enhancement (reduction) with the increase in multiplicity for the toward (away) region. The behaviour is the same for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values measured either assuming a flat background or a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>-modulated background. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>-modulated background was estimated following the approach depicted in Eq. <ce:cross-ref refid="fm0020" id="crf11500">(2)</ce:cross-ref> and using the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> data reported in <ce:cross-ref refid="br0490" id="crf11510">[49]</ce:cross-ref>. This behaviour is qualitatively similar to the di-hadron correlation results reported by the STAR and ALICE collaborations <ce:cross-refs refid="br0290 br0300" id="crs0100">[29,30]</ce:cross-refs>. In Pb–Pb collisions, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> provides information about the fragmenting jet leaving the medium, while on the away side, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> reflects the survival probability of the recoiling parton during passage through the medium. Thus a suppression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> would indicate that fewer partons survive the passage through the medium and is expected from the strong in-medium energy loss. On the other hand, the enhancement observed in the toward region is also subject to medium effects. The ratio is sensitive to a) a possible change of the fragmentation functions, b) a possible modification of the quark to gluon jet ratio in the final state due to different coupling with medium, and c) a possible bias on the parton spectrum due to trigger particle selection. Moreover, given that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is sensitive to the same effects as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math>, the interpretation of the results is similar to that reported in <ce:cross-ref refid="br0300" id="crf11520">[30]</ce:cross-ref>. It is likely that all three effects play a role <ce:cross-ref refid="br0300" id="crf11530">[30]</ce:cross-ref>. A detailed quantification of the contribution of each effect is beyond the scope of the present paper.</ce:para><ce:para id="pr0140">In order to get further insight into the effect, the measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are compared in <ce:cross-ref refid="fg0050" id="crf11540">Fig. 5</ce:cross-ref><ce:float-anchor refid="fg0050"/> with model predictions. Following the similar treatment of the experimental data, for the models, the total sample is subdivided into different V0M classes and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> is calculated for each class. For high-multiplicity pp collisions, although <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is close to unity, a small trend with multiplicity is visible, which is not seen at similar multiplicities (20−90% V0A) in p–Pb data. To understand the source of these slight deviations from unity, the data are compared with the predictions from the <ce:small-caps>PYTHIA</ce:small-caps> 8 (Monash tune <ce:cross-ref refid="br0280" id="crf11550">[28]</ce:cross-ref>) and EPOS-LHC <ce:cross-ref refid="br0210" id="crf11560">[21]</ce:cross-ref> event generators. In PYTHIA, the hadronization of quarks is simulated using the Lund string fragmentation model <ce:cross-ref refid="br0530" id="crf11570">[53]</ce:cross-ref>. Various PYTHIA tunes have been developed through extensive comparison of Monte Carlo distributions with the minimum-bias data from different experiments. The Monash tune of <ce:small-caps>PYTHIA</ce:small-caps> 8 is tuned to LHC data and uses an updated set of hadronization parameters compared to the previous tunes <ce:cross-ref refid="br0280" id="crf11580">[28]</ce:cross-ref>. EPOS-LHC is built on the Parton-Based Gribov Regge Theory. Utilising the colour exchange mechanism of string excitation, the model is tuned to LHC data <ce:cross-ref refid="br0210" id="crf11590">[21]</ce:cross-ref>. In this model, a part of the collision system which has high parton densities becomes a “core” region that evolves hydrodynamically as a quark–gluon plasma and it is surrounded by a more dilute “corona” for which fragmentation occurs in the vacuum. The upper panel of <ce:cross-ref refid="fg0050" id="crf11600">Fig. 5</ce:cross-ref> shows <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> for different multiplicity classes. The observed deviations from unity are reproduced by <ce:small-caps>PYTHIA</ce:small-caps> 8 for both the toward and away regions. Given that <ce:small-caps>PYTHIA</ce:small-caps> 8 does not incorporate any jet quenching mechanism, the origin of the effect in high <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> collisions is related to a remaining bias towards harder fragmentation and more activity from initial and final state radiation <ce:cross-ref refid="br0540" id="crf11610">[54]</ce:cross-ref>. These effects enhance the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> yield in the toward region, and produce a broadening in the away region <ce:cross-ref refid="br0550" id="crf11620">[55]</ce:cross-ref>. The EPOS-LHC results in the away region are similar to both data and <ce:small-caps>PYTHIA</ce:small-caps> 8. However, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> EPOS-LHC exhibits a trend with a maximum at intermediate multiplicity and a reduction toward low and high multiplicities, which is not consistent with the measurements.</ce:para><ce:para id="pr0150">The middle and bottom panels of <ce:cross-ref refid="fg0050" id="crf11630">Fig. 5</ce:cross-ref> show <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> measured for p–Pb and Pb–Pb collisions, respectively. The data are compared to <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr <ce:cross-ref refid="br0560" id="crf11640">[56]</ce:cross-ref> and EPOS-LHC predictions. The Angantyr model in <ce:small-caps>PYTHIA</ce:small-caps> 8 extrapolates the dynamics from pp collisions to p–Pb and Pb–Pb collisions, generalising the formalism adopted for pp collisions by including a description of the nucleon positions within the colliding nuclei and utilising the Glauber model to calculate the number of interacting nucleons and binary nucleon–nucleon collisions. <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr, which does not include jet quenching effects, predicts <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values consistent with unity for all the multiplicity classes in Pb–Pb collisions. Whereas for p–Pb collisions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is consistent with unity, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> is slightly below unity. In EPOS-LHC, a certain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> cutoff is defined in such a way that, above this cutoff, a particle loses part of its momentum in the core but survives as an independent particle produced by a flux tube. Soft particles, which are below the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> cutoff, get completely absorbed and form the core. This sort of energy loss mechanism implemented in EPOS-LHC depends on the system size <ce:cross-refs refid="br0210 br0570 br0580" id="crs0110">[21,57,58]</ce:cross-refs>. <ce:cross-ref refid="fg0050" id="crf11650">Fig. 5</ce:cross-ref> (middle) shows that for p–Pb collisions, EPOS-LHC does not describe either the magnitude or the trend of the multiplicity dependence of the measured ratio in the toward region, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math>. However, the model is in reasonable agreement with data in the away region. For Pb–Pb collisions, EPOS-LHC predicts a significant enhancement of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> for low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> ranges and deviates significantly from the experimental results.</ce:para><ce:para id="pr0160">In summary, while the data from Pb–Pb collisions are in qualitative agreement with expectations from parton energy loss due to the presence of a hot and dense medium, pp and p–Pb data do not show any hint of medium effects in the multiplicity range which is reported.</ce:para></ce:section><ce:section id="se0040"><ce:label>4</ce:label><ce:section-title id="st0050">Summary</ce:section-title><ce:para id="pr0170">The transverse momentum spectra (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mn>0.5</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) of primary charged particles in three azimuthal regions (toward, away and transverse) defined with respect to the direction of the particle with the highest transverse momentum in the event (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>8</mml:mn><mml:mo>≤</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) are reported. The spectra are studied in intervals of the multiplicity measured at forward pseudorapidities for pp, p–Pb, and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in the transverse region are subtracted from those of the away and toward regions. This is based on the assumption that the transverse side provides a good estimation of the underlying event contribution in both the toward and away regions. However, for the interpretation of the results one has to keep in mind that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulates the background and this effect is important for semi-central Pb–Pb collisions and for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>4</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> the effect is less than 5% in central and peripheral Pb–Pb collisions. Ratios to MB pp (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math>), i.e., the multiplicity dependent yields normalised to the yield measured in MB pp collisions, are reported. At low transverse momentum (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>), within 20%, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are multiplicity independent for both the toward and away regions in pp and p–Pb collisions. In contrast, in Pb–Pb collisions for both toward and away regions the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values exhibit a centrality dependence which is expected given the residual presence of elliptic flow. In the highest transverse momentum interval (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mn>4</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>), the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values in pp collisions are closer to unity but they exhibit a small reduction (increase) towards high V0 activity in pp collisions. This trend is well reproduced by <ce:small-caps>PYTHIA</ce:small-caps> 8. In the model, it is due to a selection bias towards pp collisions with harder fragmentation and larger activity from initial and final state radiation. For p–Pb collisions, within uncertainties, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are consistent with unity and do not show a multiplicity dependence. <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr fairly describes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math>, but it underestimates by about 10% the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> values in the low multiplicity classes (40−90% V0A event class). For Pb–Pb collisions, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are close to unity for peripheral collisions, and show a gradual increase (reduction) in the toward (away) region with increasing multiplicity. A similar observable, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math>, based on the per-trigger yield of associated particles in di-hadron correlation has been studied for central and peripheral Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The behaviour of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> exhibits the same features as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math>: in central collisions, on the away-side, a suppression is observed as expected from strong in-medium energy loss. In the toward region, an enhancement is observed. <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr predicts <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:math> for all multiplicity intervals, and it does not reproduce the observed away-side suppression or toward-side enhancement. Generally, EPOS-LHC does not describe the measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> ratios.</ce:para><ce:para id="pr0180">In summary, within the multiplicity reach reported in this paper, no jet quenching effects are observed in pp and p–Pb collisions within uncertainties. Further studies are required to extend the present analysis to higher multiplicities, which are currently limited by the event selection based on the forward V0 detector. The analysis of future pp and p–Pb collisions with much larger integrated luminosity may remove this limitation.</ce:para> </ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0090">Declaration of Competing Interest</ce:section-title><ce:para id="pr0210">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0060">Acknowledgements</ce:section-title><ce:para id="pr0190">The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: <ce:grant-sponsor id="gsp0010">A.I. 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National Research and Innovation Agency - <ce:grant-sponsor id="gsp0380" sponsor-id="https://doi.org/10.13039/100020473">BRIN</ce:grant-sponsor>, Indonesia; Istituto Nazionale di Fisica Nucleare (<ce:grant-sponsor id="gsp0390" sponsor-id="https://doi.org/10.13039/501100004007">INFN</ce:grant-sponsor>), Italy; Japanese <ce:grant-sponsor id="gsp0400" sponsor-id="https://doi.org/10.13039/501100001700">Ministry of Education, Culture, Sports, Science and Technology</ce:grant-sponsor> (MEXT) and <ce:grant-sponsor id="gsp0410" sponsor-id="https://doi.org/10.13039/501100001691">Japan Society for the Promotion of Science</ce:grant-sponsor> (JSPS) KAKENHI, Japan; Consejo Nacional de Ciencia (<ce:grant-sponsor id="gsp0420" sponsor-id="https://doi.org/10.13039/501100003141">CONACYT</ce:grant-sponsor>) y Tecnología, through <ce:grant-sponsor id="gsp0430" sponsor-id="https://doi.org/10.13039/501100007709">Fondo de Cooperación Internacional en Ciencia y Tecnología</ce:grant-sponsor> (FONCICYT) and <ce:grant-sponsor id="gsp0440" sponsor-id="https://doi.org/10.13039/501100006087">Dirección General de Asuntos del Personal Académico</ce:grant-sponsor> (DGAPA), Mexico; <ce:grant-sponsor id="gsp0450" sponsor-id="https://doi.org/10.13039/501100003246">Nederlandse Organisatie voor Wetenschappelijk Onderzoek</ce:grant-sponsor> (NWO), Netherlands; The <ce:grant-sponsor id="gsp0460" sponsor-id="https://doi.org/10.13039/501100005416">Research Council of Norway</ce:grant-sponsor>, Norway; <ce:grant-sponsor id="gsp0470">Commission on Science and Technology for Sustainable Development in the South</ce:grant-sponsor> (COMSATS), Pakistan; <ce:grant-sponsor id="gsp0480" sponsor-id="https://doi.org/10.13039/501100011871">Pontificia Universidad Católica del Perú</ce:grant-sponsor>, Peru; <ce:grant-sponsor id="gsp0490">Ministry of Education and Science</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0500" sponsor-id="https://doi.org/10.13039/501100004281">National Science Centre</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0510">WUT ID-UB</ce:grant-sponsor>, Poland; <ce:grant-sponsor id="gsp0520" sponsor-id="https://doi.org/10.13039/501100003708">Korea Institute of Science and Technology Information</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0530" sponsor-id="https://doi.org/10.13039/501100003725">National Research Foundation of Korea</ce:grant-sponsor> (NRF), Republic of Korea; <ce:grant-sponsor id="gsp0540">Ministry of Education and Scientific Research</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0550" sponsor-id="https://doi.org/10.13039/501100019278">Institute of Atomic Physics</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0560" sponsor-id="https://doi.org/10.13039/501100015622">Ministry of Research and Innovation</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0570" sponsor-id="https://doi.org/10.13039/501100019278">Institute of Atomic Physics</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0580">University Politehnica of Bucharest</ce:grant-sponsor>, Romania; <ce:grant-sponsor id="gsp0590" sponsor-id="https://doi.org/10.13039/501100003193">Ministry of Education, Science, Research and Sport of the Slovak Republic</ce:grant-sponsor>, Slovakia; <ce:grant-sponsor id="gsp0600">National Research Foundation of South Africa</ce:grant-sponsor>, South Africa; <ce:grant-sponsor id="gsp0610" sponsor-id="https://doi.org/10.13039/501100004359">Swedish Research Council</ce:grant-sponsor> (VR) and <ce:grant-sponsor id="gsp0620">Knut & Alice Wallenberg Foundation</ce:grant-sponsor> (KAW), Sweden; <ce:grant-sponsor id="gsp0630" sponsor-id="https://doi.org/10.13039/100012470">European Organization for Nuclear Research</ce:grant-sponsor>, Switzerland; <ce:grant-sponsor id="gsp0640" sponsor-id="https://doi.org/10.13039/501100004352">Suranaree University of Technology</ce:grant-sponsor> (SUT), <ce:grant-sponsor id="gsp0650" sponsor-id="https://doi.org/10.13039/501100004192">National Science and Technology Development Agency</ce:grant-sponsor> (NSTDA), <ce:grant-sponsor id="gsp0660" sponsor-id="https://doi.org/10.13039/501100017170">Thailand Science Research and Innovation</ce:grant-sponsor> (TSRI) and <ce:grant-sponsor id="gsp0670">National Science, Research and Innovation Fund</ce:grant-sponsor> (NSRF), Thailand; <ce:grant-sponsor id="gsp0680" sponsor-id="https://doi.org/10.13039/100020381">Turkish Energy, Nuclear and Mineral Research Agency</ce:grant-sponsor> (TENMAK), Turkey; <ce:grant-sponsor id="gsp0690" sponsor-id="https://doi.org/10.13039/501100004742">National Academy of Sciences of Ukraine</ce:grant-sponsor>, Ukraine; <ce:grant-sponsor id="gsp0700" sponsor-id="https://doi.org/10.13039/501100000271">Science and Technology Facilities Council</ce:grant-sponsor> (STFC), United Kingdom; National Science Foundation of the United States of America (<ce:grant-sponsor id="gsp0710" sponsor-id="https://doi.org/10.13039/100000001">NSF</ce:grant-sponsor>) and <ce:grant-sponsor id="gsp0720" sponsor-id="https://doi.org/10.13039/100000015">United States Department of Energy</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0730" sponsor-id="https://doi.org/10.13039/100006209">Office of Nuclear Physics</ce:grant-sponsor> (DOE NP), United States of America. In addition, individual groups or members have received support from: Marie Skłodowska Curie, Strong 2020 - <ce:grant-sponsor id="gsp0740" sponsor-id="https://doi.org/10.13039/100010661">Horizon 2020</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0750" sponsor-id="https://doi.org/10.13039/501100000781">European Research Council</ce:grant-sponsor> (grant nos. <ce:grant-number refid="gsp0750">824093</ce:grant-number>, <ce:grant-number refid="gsp0750">896850</ce:grant-number>, <ce:grant-number refid="gsp0750">950692</ce:grant-number>), <ce:grant-sponsor id="gsp0760" sponsor-id="https://doi.org/10.13039/501100000780">European Union</ce:grant-sponsor>; <ce:grant-sponsor id="gsp0770" sponsor-id="https://doi.org/10.13039/501100002341">Academy of Finland</ce:grant-sponsor> (Center of Excellence in Quark Matter) (grant nos. <ce:grant-number refid="gsp0770">346327</ce:grant-number>, <ce:grant-number refid="gsp0770">346328</ce:grant-number>), Finland; <ce:grant-sponsor id="gsp0780">Programa de Apoyos para la Superación del Personal Académico</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0790" sponsor-id="https://doi.org/10.13039/501100005739">UNAM</ce:grant-sponsor>, Mexico.</ce:para></ce:acknowledgment></body><tail><ce:bibliography id="bl0010"><ce:section-title id="st0070">References</ce:section-title><ce:bibliography-sec id="bs0010"><ce:bib-reference id="br0010"><ce:label>[1]</ce:label><sb:reference id="bibD17CEA27EFFA6FC144ED8C3DC8A6A4C2s1"><sb:contribution><sb:authors><sb:author><ce:given-name>T.</ce:given-name><ce:surname>Sjöstrand</ce:surname></sb:author><sb:author><ce:given-name>M.</ce:given-name><ce:surname>van Zijl</ce:surname></sb:author></sb:authors><sb:title><sb:maintitle>A multiple interaction model for the event structure in hadron collisions</sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Phys. 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Peigne, “Collisional Energy Loss of a Fast Parton in a QGP”, AIP Conf. Proc. 1038 no. 1, (2008) 139–148, arXiv:0806.0242 [hep-ph].</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> \ No newline at end of file +<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article PUBLIC "-//ES//DTD journal article DTD version 5.6.0//EN//XML" "art560.dtd" [<!ENTITY gr001 SYSTEM "gr001" NDATA IMAGE><!ENTITY gr002 SYSTEM "gr002" NDATA IMAGE><!ENTITY gr003 SYSTEM "gr003" NDATA IMAGE><!ENTITY gr004 SYSTEM "gr004" NDATA IMAGE><!ENTITY gr005 SYSTEM "gr005" NDATA IMAGE>]><article xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" docsubtype="sco" xml:lang="en"><item-info><jid>PLB</jid><aid>137649</aid><ce:article-number>137649</ce:article-number><ce:pii>S0370-2693(22)00783-3</ce:pii><ce:doi>10.1016/j.physletb.2022.137649</ce:doi><ce:copyright year="2023" type="other">The Author(s)</ce:copyright><ce:doctopics><ce:doctopic id="doc0010"><ce:text>Experiments</ce:text></ce:doctopic></ce:doctopics><ce:preprint><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/preprint" xlink:href="arxiv:2204.10157" id="inf0010"/></ce:preprint><ce:associated-resource><ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/research-data" xlink:href="https://www.hepdata.net/" id="inf0540">https://www.hepdata.net/</ce:inter-ref></ce:associated-resource></item-info><ce:floats><ce:figure id="fg0010"><ce:label>Fig. 1</ce:label><ce:caption id="cp0010"><ce:simple-para id="sp0010">Illustration of toward, away and transverse regions with respect to the leading particle in a collision.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0010">Fig. 1</ce:alt-text><ce:link locator="gr001" xlink:type="simple" xlink:href="pii:S0370269322007833/gr001" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0010"/></ce:figure><ce:figure id="fg0020"><ce:label>Fig. 2</ce:label><ce:caption id="cp0020"><ce:simple-para id="sp0020">Top panels: transverse momentum spectra of charged particles in the transverse region for different multiplicity classes in pp (left), p–Pb (middle) and Pb–Pb (right) collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra are measured at mid pseudorapidity (|<ce:italic>η</ce:italic>| < 0.8). Lower panels: Ratio of <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra in different multiplicity classes to the <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectrum in the 0−100% multiplicity class for the corresponding collision systems. The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0020">Fig. 2</ce:alt-text><ce:link locator="gr002" xlink:type="simple" xlink:href="pii:S0370269322007833/gr002" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0020"/></ce:figure><ce:figure id="fg0030"><ce:label>Fig. 3</ce:label><ce:caption id="cp0030"><ce:simple-para id="sp0030">Transverse momentum spectra of charged particles in Toward-Transverse, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si57.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> (top plot) and Away-Transverse, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si58.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> (bottom plot) regions for different multiplicity classes in pp (left), p–Pb (middle) and Pb–Pb (right) collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra are measured at mid pseudorapidity (|<ce:italic>η</ce:italic>| < 0.8). The lower panels of both plots show the ratio to minimum bias pp collisions. The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0030">Fig. 3</ce:alt-text><ce:link locator="gr003" xlink:type="simple" xlink:href="pii:S0370269322007833/gr003" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0030"/></ce:figure><ce:figure id="fg0040"><ce:label>Fig. 4</ce:label><ce:caption id="cp0040"><ce:simple-para id="sp0040">The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> (left) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> (right) as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> in 4 <<ce:italic>p</ce:italic><ce:inf>T</ce:inf>< 6 GeV/<ce:italic>c</ce:italic> for different multiplicity classes in pp, p–Pb and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. Pb–Pb results are shown assuming a flat background (filled markers), and assuming a <ce:italic>v</ce:italic><ce:inf>2</ce:inf>-modulated background (empty markers). The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0040">Fig. 4</ce:alt-text><ce:link locator="gr004" xlink:type="simple" xlink:href="pii:S0370269322007833/gr004" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0040"/></ce:figure><ce:figure id="fg0050"><ce:label>Fig. 5</ce:label><ce:caption id="cp0050"><ce:simple-para id="sp0050">Comparison of the measured the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> (left) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> (right) in 4 <<ce:italic>p</ce:italic><ce:inf>T</ce:inf>< 6 GeV/<ce:italic>c</ce:italic> with model predictions. The results are shown as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> for different multiplicity classes in pp (top panel), p–Pb (middle panel) and Pb–Pb (bottom panel) collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The red and magenta lines show the <ce:small-caps>PYTHIA</ce:small-caps> 8 (Monash) <ce:cross-ref refid="br0280" id="crf0010">[28]</ce:cross-ref> and <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr <ce:cross-ref refid="br0280" id="crf0020">[28]</ce:cross-ref> predictions, respectively. The blue lines show the EPOS-LHC <ce:cross-ref refid="br0210" id="crf0030">[21]</ce:cross-ref> results. The statistical and systematic uncertainties are shown by bars and boxes, respectively.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0050">Fig. 5</ce:alt-text><ce:link locator="gr005" xlink:type="simple" xlink:href="pii:S0370269322007833/gr005" xlink:role="http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4" id="ln0050"/></ce:figure><ce:table xmlns="http://www.elsevier.com/xml/common/cals/dtd" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" id="tbl0010" frame="topbot" rowsep="0" colsep="0"><ce:label>Table 1</ce:label><ce:caption id="cp0060"><ce:simple-para id="sp0060">Contributions to the relative (%) systematic uncertainty on the <ce:italic>p</ce:italic><ce:inf>T</ce:inf> spectra of primary charged particles in pp, p–Pb, and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. Just for illustration, the range in the table corresponds to the lowest and highest relative systematic uncertainty in the considered <ce:italic>p</ce:italic><ce:inf>T</ce:inf> range. The individual contributions are summed in quadrature to obtain the total uncertainty.</ce:simple-para></ce:caption><ce:alt-text role="short" id="at0060">Table 1</ce:alt-text><tgroup cols="4"><colspec colnum="1" colname="col1" align="left"/><colspec colnum="2" colname="col2" align="left"/><colspec colnum="3" colname="col3" align="left"/><colspec colnum="4" colname="col4" align="left"/><thead valign="top"><row rowsep="1"><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Source of uncertainty</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">pp</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">p–Pb</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">Pb–Pb</entry></row></thead><tbody valign="top"><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Track selection</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.1–8.2</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.4–5.8</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.0–9.9</entry></row><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Particle composition</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.3–1.8</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.5–1.9</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.3–2.4</entry></row><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Secondary particles</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.0–0.4</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.0–2.4</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.0–1.9</entry></row><row rowsep="1"><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Matching efficiency</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.0–4.2</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.7–3.7</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">0.6–3.7</entry></row><row rowsep="1"><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Total</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.2–8.8</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.6–6.3</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">3.5–10.0</entry></row><row><entry xmlns="http://www.elsevier.com/xml/common/dtd" role="rowhead">Total (<ce:italic>N</ce:italic><ce:inf>ch</ce:inf>-dependent)</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">2.0–4.5</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">1.7–4.0</entry><entry xmlns="http://www.elsevier.com/xml/common/dtd">1.1–3.7</entry></row></tbody></tgroup></ce:table></ce:floats><head><ce:title id="ti0010">Study of charged particle production at high <ce:italic>p</ce:italic><ce:inf>T</ce:inf> using event topology in pp, p–Pb and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV</ce:title><ce:author-group id="ag0010"><ce:collaboration id="co0010" collaboration-id="S0370269322007833-3bea72599603117cd9d18494a0279c47"><ce:text>ALICE Collaboration</ce:text><ce:cross-ref refid="fn0080" id="crf0040"><ce:sup>⋆</ce:sup></ce:cross-ref><ce:author-group id="ag0020"><ce:author orcid="0000-0002-9213-5329" id="au0010" author-id="S0370269322007833-e60c93a934b81cf9801254193264c6ee"><ce:given-name>S.</ce:given-name><ce:surname>Acharya</ce:surname><ce:cross-ref refid="aff1240" id="crf0050"><ce:sup>124</ce:sup></ce:cross-ref><ce:cross-ref refid="aff1310" id="crf0060"><ce:sup>131</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-0504-7428" id="au0020" author-id="S0370269322007833-0eab85892b6d74b18661e74a7987c599"><ce:given-name>D.</ce:given-name><ce:surname>Adamová</ce:surname><ce:cross-ref refid="aff0860" id="crf0070"><ce:sup>86</ce:sup></ce:cross-ref></ce:author><ce:author id="au0030" author-id="S0370269322007833-e83a30ae1d5f89088c60ca2ee154d714"><ce:given-name>A.</ce:given-name><ce:surname>Adler</ce:surname><ce:cross-ref refid="aff0690" id="crf0080"><ce:sup>69</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-9611-3696" id="au0040" author-id="S0370269322007833-ed2d58d89990c41bb43c091d01e5029a"><ce:given-name>G.</ce:given-name><ce:surname>Aglieri Rinella</ce:surname><ce:cross-ref refid="aff0320" id="crf0090"><ce:sup>32</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-0760-5075" id="au0050" author-id="S0370269322007833-0c7a7863b7384aa5fdf06a0f187949c8"><ce:given-name>M.</ce:given-name><ce:surname>Agnello</ce:surname><ce:cross-ref refid="aff0290" 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id="crf10830"><ce:sup>83</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0001-9358-5762" id="au10290" author-id="S0370269322007833-b564bd6149c66f06bbea2780670b8f50"><ce:given-name>J.</ce:given-name><ce:surname>Zhu</ce:surname><ce:cross-ref refid="aff0980" id="crf10840"><ce:sup>98</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0060" id="crf10850"><ce:sup>6</ce:sup></ce:cross-ref></ce:author><ce:author id="au10300" author-id="S0370269322007833-1fb0ab925cfcdaacb160193d82c1a978"><ce:given-name>Y.</ce:given-name><ce:surname>Zhu</ce:surname><ce:cross-ref refid="aff0060" id="crf10860"><ce:sup>6</ce:sup></ce:cross-ref></ce:author><ce:author id="au10310" author-id="S0370269322007833-2ad055472e2fac995111c51c08396d0c"><ce:given-name>G.</ce:given-name><ce:surname>Zinovjev</ce:surname><ce:cross-ref refid="aff0030" id="crf10870"><ce:sup>3</ce:sup></ce:cross-ref><ce:cross-ref refid="fn0010" id="crf10880"><ce:sup>I</ce:sup></ce:cross-ref></ce:author><ce:author orcid="0000-0002-7478-2493" id="au10320" author-id="S0370269322007833-92b5ecf5612e2e849fc1bba72c6cff99"><ce:given-name>N.</ce:given-name><ce:surname>Zurlo</ce:surname><ce:cross-ref refid="aff1300" id="crf10890"><ce:sup>130</ce:sup></ce:cross-ref><ce:cross-ref refid="aff0540" id="crf10900"><ce:sup>54</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="aff0010" affiliation-id="S0370269322007833-79d30baa35325e84d46378ba6ce12c18"><ce:label>1</ce:label><ce:textfn>A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia</ce:textfn><sa:affiliation><sa:organization>A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation</sa:organization><sa:city>Yerevan</sa:city><sa:country>Armenia</sa:country></sa:affiliation><ce:source-text id="srct0005">A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia</ce:source-text></ce:affiliation><ce:affiliation id="aff0020" affiliation-id="S0370269322007833-65754d218cf7f84bf1e02306b80caca0"><ce:label>2</ce:label><ce:textfn>AGH University of Science and Technology, Cracow, Poland</ce:textfn><sa:affiliation><sa:organization>AGH University of Science and Technology</sa:organization><sa:city>Cracow</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0010">AGH University of Science and Technology, Cracow, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff0030" affiliation-id="S0370269322007833-e916a2f48a17bc32220b61ae0e9b8e05"><ce:label>3</ce:label><ce:textfn>Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine</ce:textfn><sa:affiliation><sa:organization>Bogolyubov Institute for Theoretical Physics</sa:organization><sa:organization>National Academy of Sciences of Ukraine</sa:organization><sa:city>Kiev</sa:city><sa:country>Ukraine</sa:country></sa:affiliation><ce:source-text id="srct0015">Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine</ce:source-text></ce:affiliation><ce:affiliation id="aff0040" affiliation-id="S0370269322007833-9869d95133b2275e34836b8ad2f235c3"><ce:label>4</ce:label><ce:textfn>Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Bose Institute</sa:organization><sa:organization>Department of Physics</sa:organization><sa:organization>Centre for Astroparticle Physics and Space Science (CAPSS)</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0020">Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0050" affiliation-id="S0370269322007833-b7f796ef6c934d79a497288cdec192f6"><ce:label>5</ce:label><ce:textfn>California Polytechnic State University, San Luis Obispo, CA, United States</ce:textfn><sa:affiliation><sa:organization>California Polytechnic State University</sa:organization><sa:city>San Luis Obispo</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0025">California Polytechnic State University, San Luis Obispo, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0060" affiliation-id="S0370269322007833-3dcd6ffc2e8f27d6ed2f6237a209a384"><ce:label>6</ce:label><ce:textfn>Central China Normal University, Wuhan, China</ce:textfn><sa:affiliation><sa:organization>Central China Normal University</sa:organization><sa:city>Wuhan</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0030">Central China Normal University, Wuhan, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0070" affiliation-id="S0370269322007833-110460c7f2fbb319ec6d52e7cc3fc1d5"><ce:label>7</ce:label><ce:textfn>Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba</ce:textfn><sa:affiliation><sa:organization>Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN)</sa:organization><sa:city>Havana</sa:city><sa:country>Cuba</sa:country></sa:affiliation><ce:source-text id="srct0035">Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba</ce:source-text></ce:affiliation><ce:affiliation id="aff0080" affiliation-id="S0370269322007833-641aa526558990d110c090840e6f3d0c"><ce:label>8</ce:label><ce:textfn>Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico</ce:textfn><sa:affiliation><sa:organization>Centro de Investigación y de Estudios Avanzados (CINVESTAV)</sa:organization><sa:city>Mexico City and Mérida</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0040">Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0090" affiliation-id="S0370269322007833-9c73ece39ab638447cd10f268e329b2e"><ce:label>9</ce:label><ce:textfn>Chicago State University, Chicago, IL, United States</ce:textfn><sa:affiliation><sa:organization>Chicago State University</sa:organization><sa:city>Chicago</sa:city><sa:state>IL</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0045">Chicago State University, Chicago, Illinois, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0100" affiliation-id="S0370269322007833-d7da93bf3f02be0a46bee74f439b5930"><ce:label>10</ce:label><ce:textfn>China Institute of Atomic Energy, Beijing, China</ce:textfn><sa:affiliation><sa:organization>China Institute of Atomic Energy</sa:organization><sa:city>Beijing</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0050">China Institute of Atomic Energy, Beijing, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0110" affiliation-id="S0370269322007833-d477851cd020cd39da135784676a96ff"><ce:label>11</ce:label><ce:textfn>Chungbuk National University, Cheongju, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Chungbuk National University</sa:organization><sa:city>Cheongju</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0055">Chungbuk National University, Cheongju, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0120" affiliation-id="S0370269322007833-b03a6e652a70daee257206602eb6f83f"><ce:label>12</ce:label><ce:textfn>Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Comenius University Bratislava</sa:organization><sa:organization>Faculty of Mathematics, Physics and Informatics</sa:organization><sa:city>Bratislava</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0060">Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0130" affiliation-id="S0370269322007833-78cc8333b14d598fc5e3c0230d1de22e"><ce:label>13</ce:label><ce:textfn>COMSATS University Islamabad, Islamabad, Pakistan</ce:textfn><sa:affiliation><sa:organization>COMSATS University Islamabad</sa:organization><sa:city>Islamabad</sa:city><sa:country>Pakistan</sa:country></sa:affiliation><ce:source-text id="srct0065">COMSATS University Islamabad, Islamabad, Pakistan</ce:source-text></ce:affiliation><ce:affiliation id="aff0140" affiliation-id="S0370269322007833-42161ea7466da89325c5b37601107c55"><ce:label>14</ce:label><ce:textfn>Creighton University, Omaha, NE, United States</ce:textfn><sa:affiliation><sa:organization>Creighton University</sa:organization><sa:city>Omaha</sa:city><sa:state>NE</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0070">Creighton University, Omaha, Nebraska, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0150" affiliation-id="S0370269322007833-bd5e6b818668501c1deea76e96cac833"><ce:label>15</ce:label><ce:textfn>Department of Physics, Aligarh Muslim University, Aligarh, India</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Aligarh Muslim University</sa:organization><sa:city>Aligarh</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0075">Department of Physics, Aligarh Muslim University, Aligarh, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0160" affiliation-id="S0370269322007833-ebe7875fb705d3c179953d196aa8b94b"><ce:label>16</ce:label><ce:textfn>Department of Physics, Pusan National University, Pusan, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Pusan National University</sa:organization><sa:city>Pusan</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0080">Department of Physics, Pusan National University, Pusan, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0170" affiliation-id="S0370269322007833-24ddc81b37e640977b88412ba28336a4"><ce:label>17</ce:label><ce:textfn>Department of Physics, Sejong University, Seoul, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>Sejong University</sa:organization><sa:city>Seoul</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0085">Department of Physics, Sejong University, Seoul, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0180" affiliation-id="S0370269322007833-66245837acb0d7fb7ada51de0fd95043"><ce:label>18</ce:label><ce:textfn>Department of Physics, University of California, Berkeley, CA, United States</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>University of California</sa:organization><sa:city>Berkeley</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0090">Department of Physics, University of California, Berkeley, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0190" affiliation-id="S0370269322007833-63776eabafa9e32856b35562692a4488"><ce:label>19</ce:label><ce:textfn>Department of Physics, University of Oslo, Oslo, Norway</ce:textfn><sa:affiliation><sa:organization>Department of Physics</sa:organization><sa:organization>University of Oslo</sa:organization><sa:city>Oslo</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0095">Department of Physics, University of Oslo, Oslo, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0200" affiliation-id="S0370269322007833-020451a59d5e257505166bdb7847cdd7"><ce:label>20</ce:label><ce:textfn>Department of Physics and Technology, University of Bergen, Bergen, Norway</ce:textfn><sa:affiliation><sa:organization>Department of Physics and Technology</sa:organization><sa:organization>University of Bergen</sa:organization><sa:city>Bergen</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0100">Department of Physics and Technology, University of Bergen, Bergen, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0210" affiliation-id="S0370269322007833-f2de76852013198101844806b89d3836"><ce:label>21</ce:label><ce:textfn>Dipartimento di Fisica, Università di Pavia, Pavia, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica</sa:organization><sa:organization>Università di Pavia</sa:organization><sa:city>Pavia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0105">Dipartimento di Fisica, Università di Pavia, Pavia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0220" affiliation-id="S0370269322007833-81d87d2cd8c9f6aa2a1c7f4b13115ef3"><ce:label>22</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Cagliari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0110">Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0230" affiliation-id="S0370269322007833-849d0ba7888fcb2a09aad7ac1095ec15"><ce:label>23</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Trieste</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0115">Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0240" affiliation-id="S0370269322007833-68c63fd1cb9e8623af62118bbef39c9e"><ce:label>24</ce:label><ce:textfn>Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0120">Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0250" affiliation-id="S0370269322007833-7a40dab487121429c63e59c3de15ccfa"><ce:label>25</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bologna</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0125">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0260" affiliation-id="S0370269322007833-72e3fe624de7d3d3223574366a22ef30"><ce:label>26</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Catania</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0130">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0270" affiliation-id="S0370269322007833-6145c67e3009fb117e82f5429b2af282"><ce:label>27</ce:label><ce:textfn>Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica e Astronomia dell'Università</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Padova</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0135">Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0280" affiliation-id="S0370269322007833-a001bc692b1f1f84377401c9e2632e51"><ce:label>28</ce:label><ce:textfn>Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università and Gruppo Collegato INFN, Salerno, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università</sa:organization><sa:organization>Gruppo Collegato INFN</sa:organization><sa:city>Salerno</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0140">Dipartimento di Fisica ‘E.R. Caianiello’ dell'Università and Gruppo Collegato INFN, Salerno, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0290" affiliation-id="S0370269322007833-362983c57dbe79a62add2550bfe565dc"><ce:label>29</ce:label><ce:textfn>Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento DISAT del Politecnico</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0145">Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0300" affiliation-id="S0370269322007833-7a82e32411929fc5768d11b72609a4d8"><ce:label>30</ce:label><ce:textfn>Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento di Scienze MIFT</sa:organization><sa:organization>Università di Messina</sa:organization><sa:city>Messina</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0150">Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0310" affiliation-id="S0370269322007833-0bfba0b176e2b6e1b67e6564c87308b5"><ce:label>31</ce:label><ce:textfn>Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>Dipartimento Interateneo di Fisica ‘M. Merlin’</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0155">Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0320" affiliation-id="S0370269322007833-44f92095d23d6e3d2fb8e8fc998c51f2"><ce:label>32</ce:label><ce:textfn>European Organization for Nuclear Research (CERN), Geneva, Switzerland</ce:textfn><sa:affiliation><sa:organization>European Organization for Nuclear Research (CERN)</sa:organization><sa:city>Geneva</sa:city><sa:country>Switzerland</sa:country></sa:affiliation><ce:source-text id="srct0160">European Organization for Nuclear Research (CERN), Geneva, Switzerland</ce:source-text></ce:affiliation><ce:affiliation id="aff0330" affiliation-id="S0370269322007833-f220870dd6ed2747e8ec11b2ba624bf5"><ce:label>33</ce:label><ce:textfn>Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia</ce:textfn><sa:affiliation><sa:organization>Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture</sa:organization><sa:organization>University of Split</sa:organization><sa:city>Split</sa:city><sa:country>Croatia</sa:country></sa:affiliation><ce:source-text id="srct0165">Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia</ce:source-text></ce:affiliation><ce:affiliation id="aff0340" affiliation-id="S0370269322007833-5bf4b9d297f0544e6037addfb7689840"><ce:label>34</ce:label><ce:textfn>Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway</ce:textfn><sa:affiliation><sa:organization>Faculty of Engineering and Science</sa:organization><sa:organization>Western Norway University of Applied Sciences</sa:organization><sa:city>Bergen</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0170">Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff0350" affiliation-id="S0370269322007833-56d412e1c3fe114d9a18225fa76274c9"><ce:label>35</ce:label><ce:textfn>Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Faculty of Nuclear Sciences and Physical Engineering</sa:organization><sa:organization>Czech Technical University in Prague</sa:organization><sa:city>Prague</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0175">Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0360" affiliation-id="S0370269322007833-de312990ec5b7745ed2a9609074bb450"><ce:label>36</ce:label><ce:textfn>Faculty of Physics, Sofia University, Sofia, Bulgaria</ce:textfn><sa:affiliation><sa:organization>Faculty of Physics</sa:organization><sa:organization>Sofia University</sa:organization><sa:city>Sofia</sa:city><sa:country>Bulgaria</sa:country></sa:affiliation><ce:source-text id="srct0180">Faculty of Physics, Sofia University, Sofia, Bulgaria</ce:source-text></ce:affiliation><ce:affiliation id="aff0370" affiliation-id="S0370269322007833-0879c60c6550c16bd60686725f5c6938"><ce:label>37</ce:label><ce:textfn>Faculty of Science, P.J. Šafárik University, Košice, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Faculty of Science</sa:organization><sa:organization>P.J. Šafárik University</sa:organization><sa:city>Košice</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0185">Faculty of Science, P.J. Šafárik University, Košice, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0380" affiliation-id="S0370269322007833-a18716885f5bc83f5d1aee8d0d80c7af"><ce:label>38</ce:label><ce:textfn>Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Frankfurt Institute for Advanced Studies</sa:organization><sa:organization>Johann Wolfgang Goethe-Universität Frankfurt</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0190">Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0390" affiliation-id="S0370269322007833-f0b0a2b18fef5547dcd39253b5714404"><ce:label>39</ce:label><ce:textfn>Fudan University, Shanghai, China</ce:textfn><sa:affiliation><sa:organization>Fudan University</sa:organization><sa:city>Shanghai</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0195">Fudan University, Shanghai, China</ce:source-text></ce:affiliation><ce:affiliation id="aff0400" affiliation-id="S0370269322007833-a2398937a38c48ebf6ac3ff5e5d60b95"><ce:label>40</ce:label><ce:textfn>Gangneung-Wonju National University, Gangneung, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Gangneung-Wonju National University</sa:organization><sa:city>Gangneung</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0200">Gangneung-Wonju National University, Gangneung, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0410" affiliation-id="S0370269322007833-73bc80872cd116960a09bc477e035838"><ce:label>41</ce:label><ce:textfn>Gauhati University, Department of Physics, Guwahati, India</ce:textfn><sa:affiliation><sa:organization>Gauhati University</sa:organization><sa:organization>Department of Physics</sa:organization><sa:city>Guwahati</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0205">Gauhati University, Department of Physics, Guwahati, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0420" affiliation-id="S0370269322007833-9fd27eebf76465366fe59cb8d9620ae8"><ce:label>42</ce:label><ce:textfn>Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany</ce:textfn><sa:affiliation><sa:organization>Helmholtz-Institut für Strahlen- und Kernphysik</sa:organization><sa:organization>Rheinische Friedrich-Wilhelms-Universität Bonn</sa:organization><sa:city>Bonn</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0210">Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0430" affiliation-id="S0370269322007833-66b9f8889421b091fa908925e638d91e"><ce:label>43</ce:label><ce:textfn>Helsinki Institute of Physics (HIP), Helsinki, Finland</ce:textfn><sa:affiliation><sa:organization>Helsinki Institute of Physics (HIP)</sa:organization><sa:city>Helsinki</sa:city><sa:country>Finland</sa:country></sa:affiliation><ce:source-text id="srct0215">Helsinki Institute of Physics (HIP), Helsinki, Finland</ce:source-text></ce:affiliation><ce:affiliation id="aff0440" affiliation-id="S0370269322007833-6968018d6fe1bd12a4413918be70ab85"><ce:label>44</ce:label><ce:textfn>High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico</ce:textfn><sa:affiliation><sa:organization>High Energy Physics Group</sa:organization><sa:organization>Universidad Autónoma de Puebla</sa:organization><sa:city>Puebla</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0220">High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0450" affiliation-id="S0370269322007833-16bdf6e3577a480da99159f5d54452b1"><ce:label>45</ce:label><ce:textfn>Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>Horia Hulubei National Institute of Physics and Nuclear Engineering</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0225">Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff0460" affiliation-id="S0370269322007833-78df0ad3e050545612fd8d71a4776a19"><ce:label>46</ce:label><ce:textfn>Indian Institute of Technology Bombay (IIT), Mumbai, India</ce:textfn><sa:affiliation><sa:organization>Indian Institute of Technology Bombay (IIT)</sa:organization><sa:city>Mumbai</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0230">Indian Institute of Technology Bombay (IIT), Mumbai, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0470" affiliation-id="S0370269322007833-b8ad6c9375b7a89b768adb13f27427b4"><ce:label>47</ce:label><ce:textfn>Indian Institute of Technology Indore, Indore, India</ce:textfn><sa:affiliation><sa:organization>Indian Institute of Technology Indore</sa:organization><sa:city>Indore</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0235">Indian Institute of Technology Indore, Indore, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0480" affiliation-id="S0370269322007833-25658fba725d22058ae2a8649ceeb084"><ce:label>48</ce:label><ce:textfn>INFN, Laboratori Nazionali di Frascati, Frascati, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Laboratori Nazionali di Frascati</sa:organization><sa:city>Frascati</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0240">INFN, Laboratori Nazionali di Frascati, Frascati, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0490" affiliation-id="S0370269322007833-5d9dee68bdf16e34f2f3f01d930f367d"><ce:label>49</ce:label><ce:textfn>INFN, Sezione di Bari, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Bari</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0245">INFN, Sezione di Bari, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0500" affiliation-id="S0370269322007833-340d750afed4ff705cf1dd72bf688ca4"><ce:label>50</ce:label><ce:textfn>INFN, Sezione di Bologna, Bologna, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Bologna</sa:organization><sa:city>Bologna</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0250">INFN, Sezione di Bologna, Bologna, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0510" affiliation-id="S0370269322007833-436e8c989d6c10cae74944a81714a4e2"><ce:label>51</ce:label><ce:textfn>INFN, Sezione di Cagliari, Cagliari, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Cagliari</sa:organization><sa:city>Cagliari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0255">INFN, Sezione di Cagliari, Cagliari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0520" affiliation-id="S0370269322007833-f0607c03a8b381da2bb83a11f8d899c8"><ce:label>52</ce:label><ce:textfn>INFN, Sezione di Catania, Catania, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Catania</sa:organization><sa:city>Catania</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0260">INFN, Sezione di Catania, Catania, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0530" affiliation-id="S0370269322007833-cb33711bf32ecc9cc4697cbbea10fa88"><ce:label>53</ce:label><ce:textfn>INFN, Sezione di Padova, Padova, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Padova</sa:organization><sa:city>Padova</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0265">INFN, Sezione di Padova, Padova, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0540" affiliation-id="S0370269322007833-5c70f16aa0d389aa33e2589db6fcd5d6"><ce:label>54</ce:label><ce:textfn>INFN, Sezione di Pavia, Pavia, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Pavia</sa:organization><sa:city>Pavia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0270">INFN, Sezione di Pavia, Pavia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0550" affiliation-id="S0370269322007833-f1faa0de67f6d8d1ece0914257c7635c"><ce:label>55</ce:label><ce:textfn>INFN, Sezione di Torino, Turin, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Torino</sa:organization><sa:city>Turin</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0275">INFN, Sezione di Torino, Turin, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0560" affiliation-id="S0370269322007833-0035d662cbe7eef98bed21545fd04324"><ce:label>56</ce:label><ce:textfn>INFN, Sezione di Trieste, Trieste, Italy</ce:textfn><sa:affiliation><sa:organization>INFN, Sezione di Trieste</sa:organization><sa:city>Trieste</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0280">INFN, Sezione di Trieste, Trieste, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0570" affiliation-id="S0370269322007833-22e3a6d8462b39bb6c155bce0c9b21cd"><ce:label>57</ce:label><ce:textfn>Inha University, Incheon, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Inha University</sa:organization><sa:city>Incheon</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0285">Inha University, Incheon, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0580" affiliation-id="S0370269322007833-d4941ebd67ab9bbf96b1e6a906f4397c"><ce:label>58</ce:label><ce:textfn>Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands</ce:textfn><sa:affiliation><sa:organization>Institute for Gravitational and Subatomic Physics (GRASP)</sa:organization><sa:organization>Utrecht University/Nikhef</sa:organization><sa:city>Utrecht</sa:city><sa:country>Netherlands</sa:country></sa:affiliation><ce:source-text id="srct0290">Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands</ce:source-text></ce:affiliation><ce:affiliation id="aff0590" affiliation-id="S0370269322007833-f6d8becf2ef35885577b8164313401cb"><ce:label>59</ce:label><ce:textfn>Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Institute of Experimental Physics</sa:organization><sa:organization>Slovak Academy of Sciences</sa:organization><sa:city>Košice</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0295">Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0600" affiliation-id="S0370269322007833-6fa96ff1fcc4aaff09633c1217f06ff0"><ce:label>60</ce:label><ce:textfn>Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India</ce:textfn><sa:affiliation><sa:organization>Institute of Physics</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Bhubaneswar</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0300">Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0610" affiliation-id="S0370269322007833-9f77e70dcbde10c6ed3d37b3b0071107"><ce:label>61</ce:label><ce:textfn>Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Institute of Physics of the Czech Academy of Sciences</sa:organization><sa:city>Prague</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0305">Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0620" affiliation-id="S0370269322007833-3309ea65bfe3ae2f96b232545cb67043"><ce:label>62</ce:label><ce:textfn>Institute of Space Science (ISS), Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>Institute of Space Science (ISS)</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0310">Institute of Space Science (ISS), Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff0630" affiliation-id="S0370269322007833-2759820a41f8b0ece8c42aa319465ed5"><ce:label>63</ce:label><ce:textfn>Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Institut für Kernphysik</sa:organization><sa:organization>Johann Wolfgang Goethe-Universität Frankfurt</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0315">Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0640" affiliation-id="S0370269322007833-b18018b82a469dc4e4eb94f595cf4812"><ce:label>64</ce:label><ce:textfn>Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:textfn><sa:affiliation><sa:organization>Instituto de Ciencias Nucleares</sa:organization><sa:organization>Universidad Nacional Autónoma de México</sa:organization><sa:city>Mexico City</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0320">Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0650" affiliation-id="S0370269322007833-2aa194eda46d62198ea9b929240200b8"><ce:label>65</ce:label><ce:textfn>Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil</ce:textfn><sa:affiliation><sa:organization>Instituto de Física</sa:organization><sa:organization>Universidade Federal do Rio Grande do Sul (UFRGS)</sa:organization><sa:city>Porto Alegre</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0325">Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff0660" affiliation-id="S0370269322007833-9e9b95fa2c082308cb0efc3488541c67"><ce:label>66</ce:label><ce:textfn>Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:textfn><sa:affiliation><sa:organization>Instituto de Física</sa:organization><sa:organization>Universidad Nacional Autónoma de México</sa:organization><sa:city>Mexico City</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0330">Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff0670" affiliation-id="S0370269322007833-4ef9aeae6b2f74e366edd12aafbea4cf"><ce:label>67</ce:label><ce:textfn>iThemba LABS, National Research Foundation, Somerset West, South Africa</ce:textfn><sa:affiliation><sa:organization>iThemba LABS</sa:organization><sa:organization>National Research Foundation</sa:organization><sa:city>Somerset West</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0335">iThemba LABS, National Research Foundation, Somerset West, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff0680" affiliation-id="S0370269322007833-2f4c4db0447d27fd2e43117b90bb74d4"><ce:label>68</ce:label><ce:textfn>Jeonbuk National University, Jeonju, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Jeonbuk National University</sa:organization><sa:city>Jeonju</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0340">Jeonbuk National University, Jeonju, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0690" affiliation-id="S0370269322007833-81ec7d4c8c69486a57ef1ca6a567076c"><ce:label>69</ce:label><ce:textfn>Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany</ce:textfn><sa:affiliation><sa:organization>Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik</sa:organization><sa:organization>Fachbereich Informatik und Mathematik</sa:organization><sa:city>Frankfurt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0345">Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0700" affiliation-id="S0370269322007833-c1d849875c0de0db6476f9cc96b07dd2"><ce:label>70</ce:label><ce:textfn>Korea Institute of Science and Technology Information, Daejeon, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Korea Institute of Science and Technology Information</sa:organization><sa:city>Daejeon</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0350">Korea Institute of Science and Technology Information, Daejeon, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff0710" affiliation-id="S0370269322007833-c5cd420fa3d6b7c79dc3ab9e7c2dfb06"><ce:label>71</ce:label><ce:textfn>KTO Karatay University, Konya, Turkey</ce:textfn><sa:affiliation><sa:organization>KTO Karatay University</sa:organization><sa:city>Konya</sa:city><sa:country>Turkey</sa:country></sa:affiliation><ce:source-text id="srct0355">KTO Karatay University, Konya, Turkey</ce:source-text></ce:affiliation><ce:affiliation id="aff0720" affiliation-id="S0370269322007833-0bdd954451acb47f491c8c4996fa87da"><ce:label>72</ce:label><ce:textfn>Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France</ce:textfn><sa:affiliation><sa:organization>Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie</sa:organization><sa:city>Orsay</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0360">Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France</ce:source-text></ce:affiliation><ce:affiliation id="aff0730" affiliation-id="S0370269322007833-1487532e5bfe30325bc10c0583f7c38e"><ce:label>73</ce:label><ce:textfn>Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France</ce:textfn><sa:affiliation><sa:organization>Laboratoire de Physique Subatomique et de Cosmologie</sa:organization><sa:organization>Université Grenoble-Alpes</sa:organization><sa:organization>CNRS-IN2P3</sa:organization><sa:city>Grenoble</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0365">Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France</ce:source-text></ce:affiliation><ce:affiliation id="aff0740" affiliation-id="S0370269322007833-dd5512fbf2faf90b56635e0b411d44a2"><ce:label>74</ce:label><ce:textfn>Lawrence Berkeley National Laboratory, Berkeley, CA, United States</ce:textfn><sa:affiliation><sa:organization>Lawrence Berkeley National Laboratory</sa:organization><sa:city>Berkeley</sa:city><sa:state>CA</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0370">Lawrence Berkeley National Laboratory, Berkeley, California, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0750" affiliation-id="S0370269322007833-ed03eefd58007822249c697d882deffc"><ce:label>75</ce:label><ce:textfn>Lund University Department of Physics, Division of Particle Physics, Lund, Sweden</ce:textfn><sa:affiliation><sa:organization>Lund University Department of Physics</sa:organization><sa:organization>Division of Particle Physics</sa:organization><sa:city>Lund</sa:city><sa:country>Sweden</sa:country></sa:affiliation><ce:source-text id="srct0375">Lund University Department of Physics, Division of Particle Physics, Lund, Sweden</ce:source-text></ce:affiliation><ce:affiliation id="aff0760" affiliation-id="S0370269322007833-5b47ca06644b7f88e2267e9accbe867b"><ce:label>76</ce:label><ce:textfn>Nagasaki Institute of Applied Science, Nagasaki, Japan</ce:textfn><sa:affiliation><sa:organization>Nagasaki Institute of Applied Science</sa:organization><sa:city>Nagasaki</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0380">Nagasaki Institute of Applied Science, Nagasaki, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0770" affiliation-id="S0370269322007833-61cd55f0b96e9831d6063318d9967d43"><ce:label>77</ce:label><ce:textfn>Nara Women's University (NWU), Nara, Japan</ce:textfn><sa:affiliation><sa:organization>Nara Women's University (NWU)</sa:organization><sa:city>Nara</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0385">Nara Women's University (NWU), Nara, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0780" affiliation-id="S0370269322007833-7f50d99eb340e408fbc323cbce0c8905"><ce:label>78</ce:label><ce:textfn>National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece</ce:textfn><sa:affiliation><sa:organization>National and Kapodistrian University of Athens</sa:organization><sa:organization>School of Science</sa:organization><sa:organization>Department of Physics</sa:organization><sa:city>Athens</sa:city><sa:country>Greece</sa:country></sa:affiliation><ce:source-text id="srct0390">National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece</ce:source-text></ce:affiliation><ce:affiliation id="aff0790" affiliation-id="S0370269322007833-58192de9f95a93cb8bf485b9a5647bf5"><ce:label>79</ce:label><ce:textfn>National Centre for Nuclear Research, Warsaw, Poland</ce:textfn><sa:affiliation><sa:organization>National Centre for Nuclear Research</sa:organization><sa:city>Warsaw</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0395">National Centre for Nuclear Research, Warsaw, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff0800" affiliation-id="S0370269322007833-5c73c93f2c9aba1b7fcfa30741ce17ca"><ce:label>80</ce:label><ce:textfn>National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India</ce:textfn><sa:affiliation><sa:organization>National Institute of Science Education and Research</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Jatni</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0400">National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0810" affiliation-id="S0370269322007833-1e229a9758219f877a7f903c2dde9c2b"><ce:label>81</ce:label><ce:textfn>National Nuclear Research Center, Baku, Azerbaijan</ce:textfn><sa:affiliation><sa:organization>National Nuclear Research Center</sa:organization><sa:city>Baku</sa:city><sa:country>Azerbaijan</sa:country></sa:affiliation><ce:source-text id="srct0405">National Nuclear Research Center, Baku, Azerbaijan</ce:source-text></ce:affiliation><ce:affiliation id="aff0820" affiliation-id="S0370269322007833-ab3fe404fa49046cc1d6ed593a4d5eb8"><ce:label>82</ce:label><ce:textfn>National Research and Innovation Agency - BRIN, Jakarta, Indonesia</ce:textfn><sa:affiliation><sa:organization>National Research and Innovation Agency - BRIN</sa:organization><sa:city>Jakarta</sa:city><sa:country>Indonesia</sa:country></sa:affiliation><ce:source-text id="srct0410">National Research and Innovation Agency - BRIN, Jakarta, Indonesia</ce:source-text></ce:affiliation><ce:affiliation id="aff0830" affiliation-id="S0370269322007833-3e7cb9222cae4e75df0d93d2ea96329f"><ce:label>83</ce:label><ce:textfn>Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark</ce:textfn><sa:affiliation><sa:organization>Niels Bohr Institute</sa:organization><sa:organization>University of Copenhagen</sa:organization><sa:city>Copenhagen</sa:city><sa:country>Denmark</sa:country></sa:affiliation><ce:source-text id="srct0415">Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark</ce:source-text></ce:affiliation><ce:affiliation id="aff0840" affiliation-id="S0370269322007833-11b144edfb6d992e108d17991cc7681b"><ce:label>84</ce:label><ce:textfn>Nikhef, National institute for subatomic physics, Amsterdam, Netherlands</ce:textfn><sa:affiliation><sa:organization>Nikhef, National institute for subatomic physics</sa:organization><sa:city>Amsterdam</sa:city><sa:country>Netherlands</sa:country></sa:affiliation><ce:source-text id="srct0420">Nikhef, National institute for subatomic physics, Amsterdam, Netherlands</ce:source-text></ce:affiliation><ce:affiliation id="aff0850" affiliation-id="S0370269322007833-7ffc71d9a245ae3927bb4d373ff478ed"><ce:label>85</ce:label><ce:textfn>Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Group</sa:organization><sa:organization>STFC Daresbury Laboratory</sa:organization><sa:city>Daresbury</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0425">Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff0860" affiliation-id="S0370269322007833-4ea73349ab4e79a7b6570dd0fd42cd13"><ce:label>86</ce:label><ce:textfn>Nuclear Physics Institute of the Czech Academy of Sciences, Husinec-Řež, Czech Republic</ce:textfn><sa:affiliation><sa:organization>Nuclear Physics Institute of the Czech Academy of Sciences</sa:organization><sa:city>Husinec-Řež</sa:city><sa:country>Czech Republic</sa:country></sa:affiliation><ce:source-text id="srct0430">Nuclear Physics Institute of the Czech Academy of Sciences, Husinec-Řež, Czech Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff0870" affiliation-id="S0370269322007833-43c0b745ad096562d858417552718575"><ce:label>87</ce:label><ce:textfn>Oak Ridge National Laboratory, Oak Ridge, TN, United States</ce:textfn><sa:affiliation><sa:organization>Oak Ridge National Laboratory</sa:organization><sa:city>Oak Ridge</sa:city><sa:state>TN</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0435">Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0880" affiliation-id="S0370269322007833-e4e32a64c7e436b0f2a688687dedfe5e"><ce:label>88</ce:label><ce:textfn>Ohio State University, Columbus, OH, United States</ce:textfn><sa:affiliation><sa:organization>Ohio State University</sa:organization><sa:city>Columbus</sa:city><sa:state>OH</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0440">Ohio State University, Columbus, Ohio, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff0890" affiliation-id="S0370269322007833-3c4eb171ede7112c64f1bc2c164f4732"><ce:label>89</ce:label><ce:textfn>Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia</ce:textfn><sa:affiliation><sa:organization>Physics department</sa:organization><sa:organization>Faculty of science, University of Zagreb</sa:organization><sa:city>Zagreb</sa:city><sa:country>Croatia</sa:country></sa:affiliation><ce:source-text id="srct0445">Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia</ce:source-text></ce:affiliation><ce:affiliation id="aff0900" affiliation-id="S0370269322007833-30d6dbb96747c914840798f88bf250cb"><ce:label>90</ce:label><ce:textfn>Physics Department, Panjab University, Chandigarh, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>Panjab University</sa:organization><sa:city>Chandigarh</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0450">Physics Department, Panjab University, Chandigarh, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0910" affiliation-id="S0370269322007833-712bc80495e97050498604da603cb5cc"><ce:label>91</ce:label><ce:textfn>Physics Department, University of Jammu, Jammu, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>University of Jammu</sa:organization><sa:city>Jammu</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0455">Physics Department, University of Jammu, Jammu, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0920" affiliation-id="S0370269322007833-500771749243cadcbaeeac3f726e6a62"><ce:label>92</ce:label><ce:textfn>Physics Department, University of Rajasthan, Jaipur, India</ce:textfn><sa:affiliation><sa:organization>Physics Department</sa:organization><sa:organization>University of Rajasthan</sa:organization><sa:city>Jaipur</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0460">Physics Department, University of Rajasthan, Jaipur, India</ce:source-text></ce:affiliation><ce:affiliation id="aff0930" affiliation-id="S0370269322007833-87f0c1fe6b138b8300b7695dc3391666"><ce:label>93</ce:label><ce:textfn>Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan</ce:textfn><sa:affiliation><sa:organization>Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2)</sa:organization><sa:organization>Hiroshima University</sa:organization><sa:city>Hiroshima</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0465">Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff0940" affiliation-id="S0370269322007833-0886f70da565231fd0e43398021fd604"><ce:label>94</ce:label><ce:textfn>Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany</ce:textfn><sa:affiliation><sa:organization>Physikalisches Institut</sa:organization><sa:organization>Eberhard-Karls-Universität Tübingen</sa:organization><sa:city>Tübingen</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0470">Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0950" affiliation-id="S0370269322007833-1c67678e124982de924355cdd6d6b91b"><ce:label>95</ce:label><ce:textfn>Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany</ce:textfn><sa:affiliation><sa:organization>Physikalisches Institut</sa:organization><sa:organization>Ruprecht-Karls-Universität Heidelberg</sa:organization><sa:city>Heidelberg</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0475">Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0960" affiliation-id="S0370269322007833-7a5765cc3928149c8b77128cd52462ae"><ce:label>96</ce:label><ce:textfn>Physik Department, Technische Universität München, Munich, Germany</ce:textfn><sa:affiliation><sa:organization>Physik Department</sa:organization><sa:organization>Technische Universität München</sa:organization><sa:city>Munich</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0480">Physik Department, Technische Universität München, Munich, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0970" affiliation-id="S0370269322007833-0a06f4f5fb60a25c8f1936ed71c96cfe"><ce:label>97</ce:label><ce:textfn>Politecnico di Bari and Sezione INFN, Bari, Italy</ce:textfn><sa:affiliation><sa:organization>Politecnico di Bari</sa:organization><sa:organization>Sezione INFN</sa:organization><sa:city>Bari</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0485">Politecnico di Bari and Sezione INFN, Bari, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff0980" affiliation-id="S0370269322007833-3451d358e90b759225813c9b9a6f098c"><ce:label>98</ce:label><ce:textfn>Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany</ce:textfn><sa:affiliation><sa:organization>Research Division</sa:organization><sa:organization>ExtreMe Matter Institute EMMI</sa:organization><sa:organization>GSI Helmholtzzentrum für Schwerionenforschung GmbH</sa:organization><sa:city>Darmstadt</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0490">Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff0990" affiliation-id="S0370269322007833-2771a2ae295804be11a4a937b894a546"><ce:label>99</ce:label><ce:textfn>Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Saha Institute of Nuclear Physics</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0495">Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1000" affiliation-id="S0370269322007833-b602cde70213041ca40ec5720e4e3e75"><ce:label>100</ce:label><ce:textfn>School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom</ce:textfn><sa:affiliation><sa:organization>School of Physics and Astronomy</sa:organization><sa:organization>University of Birmingham</sa:organization><sa:city>Birmingham</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0500">School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff1010" affiliation-id="S0370269322007833-1a1f0f3ae33ba0323ad4da08a216f4c3"><ce:label>101</ce:label><ce:textfn>Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru</ce:textfn><sa:affiliation><sa:organization>Sección Física</sa:organization><sa:organization>Departamento de Ciencias</sa:organization><sa:organization>Pontificia Universidad Católica del Perú</sa:organization><sa:city>Lima</sa:city><sa:country>Peru</sa:country></sa:affiliation><ce:source-text id="srct0505">Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru</ce:source-text></ce:affiliation><ce:affiliation id="aff1020" affiliation-id="S0370269322007833-24347400090c9a1efb87e7537af7461b"><ce:label>102</ce:label><ce:textfn>Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria</ce:textfn><sa:affiliation><sa:organization>Stefan Meyer Institut für Subatomare Physik (SMI)</sa:organization><sa:city>Vienna</sa:city><sa:country>Austria</sa:country></sa:affiliation><ce:source-text id="srct0510">Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria</ce:source-text></ce:affiliation><ce:affiliation id="aff1030" affiliation-id="S0370269322007833-317714469d8c208c77efd6f026942633"><ce:label>103</ce:label><ce:textfn>SUBATECH, IMT Atlantique, Nantes Université, CNRS-IN2P3, Nantes, France</ce:textfn><sa:affiliation><sa:organization>SUBATECH</sa:organization><sa:organization>IMT Atlantique</sa:organization><sa:organization>Nantes Université</sa:organization><sa:organization>CNRS-IN2P3</sa:organization><sa:city>Nantes</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0515">SUBATECH, IMT Atlantique, Nantes Université, CNRS-IN2P3, Nantes, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1040" affiliation-id="S0370269322007833-c3972f6af24eef12cc2ae3a53d6a7623"><ce:label>104</ce:label><ce:textfn>Suranaree University of Technology, Nakhon Ratchasima, Thailand</ce:textfn><sa:affiliation><sa:organization>Suranaree University of Technology</sa:organization><sa:city>Nakhon Ratchasima</sa:city><sa:country>Thailand</sa:country></sa:affiliation><ce:source-text id="srct0520">Suranaree University of Technology, Nakhon Ratchasima, Thailand</ce:source-text></ce:affiliation><ce:affiliation id="aff1050" affiliation-id="S0370269322007833-9186de958ef0b51277f8f34872e5c71b"><ce:label>105</ce:label><ce:textfn>Technical University of Košice, Košice, Slovak Republic</ce:textfn><sa:affiliation><sa:organization>Technical University of Košice</sa:organization><sa:city>Košice</sa:city><sa:country>Slovak Republic</sa:country></sa:affiliation><ce:source-text id="srct0525">Technical University of Košice, Košice, Slovak Republic</ce:source-text></ce:affiliation><ce:affiliation id="aff1060" affiliation-id="S0370269322007833-425836ada61fe0d3ee4ebf5b87598919"><ce:label>106</ce:label><ce:textfn>The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland</ce:textfn><sa:affiliation><sa:organization>The Henryk Niewodniczanski Institute of Nuclear Physics</sa:organization><sa:organization>Polish Academy of Sciences</sa:organization><sa:city>Cracow</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0530">The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff1070" affiliation-id="S0370269322007833-302a66584ead3636c3e74b86b51cf474"><ce:label>107</ce:label><ce:textfn>The University of Texas at Austin, Austin, TX, United States</ce:textfn><sa:affiliation><sa:organization>The University of Texas at Austin</sa:organization><sa:city>Austin</sa:city><sa:state>TX</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0535">The University of Texas at Austin, Austin, Texas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1080" affiliation-id="S0370269322007833-48ea3700cc067946e6c3032832f0ce0a"><ce:label>108</ce:label><ce:textfn>Universidad Autónoma de Sinaloa, Culiacán, Mexico</ce:textfn><sa:affiliation><sa:organization>Universidad Autónoma de Sinaloa</sa:organization><sa:city>Culiacán</sa:city><sa:country>Mexico</sa:country></sa:affiliation><ce:source-text id="srct0540">Universidad Autónoma de Sinaloa, Culiacán, Mexico</ce:source-text></ce:affiliation><ce:affiliation id="aff1090" affiliation-id="S0370269322007833-310945a13cafb9845fe651fc7aa661b6"><ce:label>109</ce:label><ce:textfn>Universidade de São Paulo (USP), São Paulo, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade de São Paulo (USP)</sa:organization><sa:city>São Paulo</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0545">Universidade de São Paulo (USP), São Paulo, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1100" affiliation-id="S0370269322007833-226f6a475b7a1634e1530e2077d7590e"><ce:label>110</ce:label><ce:textfn>Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade Estadual de Campinas (UNICAMP)</sa:organization><sa:city>Campinas</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0550">Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1110" affiliation-id="S0370269322007833-7582533209ee7a368c4d249782e93c03"><ce:label>111</ce:label><ce:textfn>Universidade Federal do ABC, Santo Andre, Brazil</ce:textfn><sa:affiliation><sa:organization>Universidade Federal do ABC</sa:organization><sa:city>Santo Andre</sa:city><sa:country>Brazil</sa:country></sa:affiliation><ce:source-text id="srct0555">Universidade Federal do ABC, Santo Andre, Brazil</ce:source-text></ce:affiliation><ce:affiliation id="aff1120" affiliation-id="S0370269322007833-f37f0b36132124cce5539b1f9c518079"><ce:label>112</ce:label><ce:textfn>University of Cape Town, Cape Town, South Africa</ce:textfn><sa:affiliation><sa:organization>University of Cape Town</sa:organization><sa:city>Cape Town</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0560">University of Cape Town, Cape Town, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff1130" affiliation-id="S0370269322007833-58fe35a5cc9d91eea34fd6e19293599b"><ce:label>113</ce:label><ce:textfn>University of Houston, Houston, TX, United States</ce:textfn><sa:affiliation><sa:organization>University of Houston</sa:organization><sa:city>Houston</sa:city><sa:state>TX</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0565">University of Houston, Houston, Texas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1140" affiliation-id="S0370269322007833-e9be2b1885c8285b7cb7b6534ca93d98"><ce:label>114</ce:label><ce:textfn>University of Jyväskylä, Jyväskylä, Finland</ce:textfn><sa:affiliation><sa:organization>University of Jyväskylä</sa:organization><sa:city>Jyväskylä</sa:city><sa:country>Finland</sa:country></sa:affiliation><ce:source-text id="srct0570">University of Jyväskylä, Jyväskylä, Finland</ce:source-text></ce:affiliation><ce:affiliation id="aff1150" affiliation-id="S0370269322007833-6e2074f6fd0b88987501b22e5d74f9c9"><ce:label>115</ce:label><ce:textfn>University of Kansas, Lawrence, KS, United States</ce:textfn><sa:affiliation><sa:organization>University of Kansas</sa:organization><sa:city>Lawrence</sa:city><sa:state>KS</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0575">University of Kansas, Lawrence, Kansas, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1160" affiliation-id="S0370269322007833-f538fc5a59ec1d1c8f86d2a83cb4ced6"><ce:label>116</ce:label><ce:textfn>University of Liverpool, Liverpool, United Kingdom</ce:textfn><sa:affiliation><sa:organization>University of Liverpool</sa:organization><sa:city>Liverpool</sa:city><sa:country>United Kingdom</sa:country></sa:affiliation><ce:source-text id="srct0580">University of Liverpool, Liverpool, United Kingdom</ce:source-text></ce:affiliation><ce:affiliation id="aff1170" affiliation-id="S0370269322007833-5507e3344bf7b46ba698c2c045a8aba6"><ce:label>117</ce:label><ce:textfn>University of Science and Technology of China, Hefei, China</ce:textfn><sa:affiliation><sa:organization>University of Science and Technology of China</sa:organization><sa:city>Hefei</sa:city><sa:country>China</sa:country></sa:affiliation><ce:source-text id="srct0585">University of Science and Technology of China, Hefei, China</ce:source-text></ce:affiliation><ce:affiliation id="aff1180" affiliation-id="S0370269322007833-5825f1740b6b974441dd90d9c8ad0a53"><ce:label>118</ce:label><ce:textfn>University of South-Eastern Norway, Kongsberg, Norway</ce:textfn><sa:affiliation><sa:organization>University of South-Eastern Norway</sa:organization><sa:city>Kongsberg</sa:city><sa:country>Norway</sa:country></sa:affiliation><ce:source-text id="srct0590">University of South-Eastern Norway, Kongsberg, Norway</ce:source-text></ce:affiliation><ce:affiliation id="aff1190" affiliation-id="S0370269322007833-d6620449e5365c80224ff22fe1ca4e4a"><ce:label>119</ce:label><ce:textfn>University of Tennessee, Knoxville, TN, United States</ce:textfn><sa:affiliation><sa:organization>University of Tennessee</sa:organization><sa:city>Knoxville</sa:city><sa:state>TN</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0595">University of Tennessee, Knoxville, Tennessee, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1200" affiliation-id="S0370269322007833-6d43022152ccaa62119e4287bf3b5270"><ce:label>120</ce:label><ce:textfn>University of the Witwatersrand, Johannesburg, South Africa</ce:textfn><sa:affiliation><sa:organization>University of the Witwatersrand</sa:organization><sa:city>Johannesburg</sa:city><sa:country>South Africa</sa:country></sa:affiliation><ce:source-text id="srct0600">University of the Witwatersrand, Johannesburg, South Africa</ce:source-text></ce:affiliation><ce:affiliation id="aff1210" affiliation-id="S0370269322007833-9b9071fc23073da0e4317b3863c91b9e"><ce:label>121</ce:label><ce:textfn>University of Tokyo, Tokyo, Japan</ce:textfn><sa:affiliation><sa:organization>University of Tokyo</sa:organization><sa:city>Tokyo</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0605">University of Tokyo, Tokyo, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff1220" affiliation-id="S0370269322007833-da3fb88e8779358429638ee8f8dc4cb5"><ce:label>122</ce:label><ce:textfn>University of Tsukuba, Tsukuba, Japan</ce:textfn><sa:affiliation><sa:organization>University of Tsukuba</sa:organization><sa:city>Tsukuba</sa:city><sa:country>Japan</sa:country></sa:affiliation><ce:source-text id="srct0610">University of Tsukuba, Tsukuba, Japan</ce:source-text></ce:affiliation><ce:affiliation id="aff1230" affiliation-id="S0370269322007833-9fc1715eadb7b2ba79b1a87cef8015cd"><ce:label>123</ce:label><ce:textfn>University Politehnica of Bucharest, Bucharest, Romania</ce:textfn><sa:affiliation><sa:organization>University Politehnica of Bucharest</sa:organization><sa:city>Bucharest</sa:city><sa:country>Romania</sa:country></sa:affiliation><ce:source-text id="srct0615">University Politehnica of Bucharest, Bucharest, Romania</ce:source-text></ce:affiliation><ce:affiliation id="aff1240" affiliation-id="S0370269322007833-43e2ac82e11c4ca2fabbc7e457c74b37"><ce:label>124</ce:label><ce:textfn>Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France</ce:textfn><sa:affiliation><sa:organization>Université Clermont Auvergne</sa:organization><sa:organization>CNRS/IN2P3</sa:organization><sa:organization>LPC</sa:organization><sa:city>Clermont-Ferrand</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0620">Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1250" affiliation-id="S0370269322007833-08f13150d05b32d440951efec6f80d29"><ce:label>125</ce:label><ce:textfn>Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France</ce:textfn><sa:affiliation><sa:organization>Université de Lyon</sa:organization><sa:organization>CNRS/IN2P3</sa:organization><sa:organization>Institut de Physique des 2 Infinis de Lyon</sa:organization><sa:city>Lyon</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0625">Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1260" affiliation-id="S0370269322007833-766d0075165899c16752f5f7ff1b8771"><ce:label>126</ce:label><ce:textfn>Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France</ce:textfn><sa:affiliation><sa:organization>Université de Strasbourg</sa:organization><sa:organization>CNRS</sa:organization><sa:organization>IPHC UMR 7178</sa:organization><sa:city>Strasbourg</sa:city><sa:postal-code>F-67000</sa:postal-code><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0630">Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1270" affiliation-id="S0370269322007833-36117066550bb6faaf9708b1b0ea670e"><ce:label>127</ce:label><ce:textfn>Université Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France</ce:textfn><sa:affiliation><sa:organization>Université Paris-Saclay Centre d'Etudes de Saclay (CEA)</sa:organization><sa:organization>IRFU</sa:organization><sa:organization>Départment de Physique Nucléaire (DPhN)</sa:organization><sa:city>Saclay</sa:city><sa:country>France</sa:country></sa:affiliation><ce:source-text id="srct0635">Université Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France</ce:source-text></ce:affiliation><ce:affiliation id="aff1280" affiliation-id="S0370269322007833-47ef552cdaf90d05d791574b14a8dcf9"><ce:label>128</ce:label><ce:textfn>Università degli Studi di Foggia, Foggia, Italy</ce:textfn><sa:affiliation><sa:organization>Università degli Studi di Foggia</sa:organization><sa:city>Foggia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0640">Università degli Studi di Foggia, Foggia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1290" affiliation-id="S0370269322007833-79fbcbe2a42e9099094360b5bf7bd602"><ce:label>129</ce:label><ce:textfn>Università del Piemonte Orientale, Vercelli, Italy</ce:textfn><sa:affiliation><sa:organization>Università del Piemonte Orientale</sa:organization><sa:city>Vercelli</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0645">Università del Piemonte Orientale, Vercelli, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1300" affiliation-id="S0370269322007833-83cd0c0929f483f88e3d2cb59f064287"><ce:label>130</ce:label><ce:textfn>Università di Brescia, Brescia, Italy</ce:textfn><sa:affiliation><sa:organization>Università di Brescia</sa:organization><sa:city>Brescia</sa:city><sa:country>Italy</sa:country></sa:affiliation><ce:source-text id="srct0650">Università di Brescia, Brescia, Italy</ce:source-text></ce:affiliation><ce:affiliation id="aff1310" affiliation-id="S0370269322007833-f1ae52f852d4d7d99988b3e872f887e4"><ce:label>131</ce:label><ce:textfn>Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:textfn><sa:affiliation><sa:organization>Variable Energy Cyclotron Centre</sa:organization><sa:organization>Homi Bhabha National Institute</sa:organization><sa:city>Kolkata</sa:city><sa:country>India</sa:country></sa:affiliation><ce:source-text id="srct0655">Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India</ce:source-text></ce:affiliation><ce:affiliation id="aff1320" affiliation-id="S0370269322007833-bfd4fe0d3b8675ea55d96ce64033d817"><ce:label>132</ce:label><ce:textfn>Warsaw University of Technology, Warsaw, Poland</ce:textfn><sa:affiliation><sa:organization>Warsaw University of Technology</sa:organization><sa:city>Warsaw</sa:city><sa:country>Poland</sa:country></sa:affiliation><ce:source-text id="srct0660">Warsaw University of Technology, Warsaw, Poland</ce:source-text></ce:affiliation><ce:affiliation id="aff1330" affiliation-id="S0370269322007833-4e11d38f810a3540206ffb5151a35d3b"><ce:label>133</ce:label><ce:textfn>Wayne State University, Detroit, MI, United States</ce:textfn><sa:affiliation><sa:organization>Wayne State University</sa:organization><sa:city>Detroit</sa:city><sa:state>MI</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0665">Wayne State University, Detroit, Michigan, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1340" affiliation-id="S0370269322007833-0b76517a6de7ff0579ec14780f79d81b"><ce:label>134</ce:label><ce:textfn>Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany</ce:textfn><sa:affiliation><sa:organization>Westfälische Wilhelms-Universität Münster</sa:organization><sa:organization>Institut für Kernphysik</sa:organization><sa:city>Münster</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0670">Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1350" affiliation-id="S0370269322007833-6b1ae45228f1f040c55767dc107b589d"><ce:label>135</ce:label><ce:textfn>Wigner Research Centre for Physics, Budapest, Hungary</ce:textfn><sa:affiliation><sa:organization>Wigner Research Centre for Physics</sa:organization><sa:city>Budapest</sa:city><sa:country>Hungary</sa:country></sa:affiliation><ce:source-text id="srct0675">Wigner Research Centre for Physics, Budapest, Hungary</ce:source-text></ce:affiliation><ce:affiliation id="aff1360" affiliation-id="S0370269322007833-96e79e7381eab77664732c3553e247e8"><ce:label>136</ce:label><ce:textfn>Yale University, New Haven, CT, United States</ce:textfn><sa:affiliation><sa:organization>Yale University</sa:organization><sa:city>New Haven</sa:city><sa:state>CT</sa:state><sa:country>United States</sa:country></sa:affiliation><ce:source-text id="srct0680">Yale University, New Haven, Connecticut, United States</ce:source-text></ce:affiliation><ce:affiliation id="aff1370" affiliation-id="S0370269322007833-f3db17ceaf6ea4190e2176a6a129c96b"><ce:label>137</ce:label><ce:textfn>Yonsei University, Seoul, Republic of Korea</ce:textfn><sa:affiliation><sa:organization>Yonsei University</sa:organization><sa:city>Seoul</sa:city><sa:country>Republic of Korea</sa:country></sa:affiliation><ce:source-text id="srct0685">Yonsei University, Seoul, Republic of Korea</ce:source-text></ce:affiliation><ce:affiliation id="aff1380" affiliation-id="S0370269322007833-2e935afd39a812eb118b95cc53ef2ac3"><ce:label>138</ce:label><ce:textfn>Zentrum für Technologie und Transfer (ZTT), Worms, Germany</ce:textfn><sa:affiliation><sa:organization>Zentrum für Technologie und Transfer (ZTT)</sa:organization><sa:city>Worms</sa:city><sa:country>Germany</sa:country></sa:affiliation><ce:source-text id="srct0690">Zentrum für Technologie und Transfer (ZTT), Worms, Germany</ce:source-text></ce:affiliation><ce:affiliation id="aff1390" affiliation-id="S0370269322007833-7972cea8d4a6e5b7150baf4a74e03970"><ce:label>139</ce:label><ce:textfn>Affiliated with an institute covered by a cooperation agreement with CERN</ce:textfn><sa:affiliation><sa:address-line>Affiliated with an institute covered by a cooperation agreement with CERN</sa:address-line></sa:affiliation><ce:source-text id="srct0695">Affiliated with an institute covered by a cooperation agreement with CERN</ce:source-text></ce:affiliation><ce:affiliation id="aff1400" affiliation-id="S0370269322007833-3c2eaa2a494293a99583fccbf5d95e9f"><ce:label>140</ce:label><ce:textfn>Affiliated with an international laboratory covered by a cooperation agreement with CERN</ce:textfn><sa:affiliation><sa:address-line>Affiliated with an international laboratory covered by a cooperation agreement with CERN</sa:address-line></sa:affiliation><ce:source-text id="srct0700">Affiliated with an international laboratory covered by a cooperation agreement with CERN</ce:source-text></ce:affiliation><ce:footnote id="fn0010"><ce:label>I</ce:label><ce:note-para id="np0010">Deceased.</ce:note-para></ce:footnote><ce:footnote id="fn0020"><ce:label>II</ce:label><ce:note-para id="np0020">Also at: Max-Planck-Institut für Physik, Munich, Germany.</ce:note-para></ce:footnote><ce:footnote id="fn0030"><ce:label>III</ce:label><ce:note-para id="np0030">Also at: Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA), Bologna, Italy.</ce:note-para></ce:footnote><ce:footnote id="fn0040"><ce:label>IV</ce:label><ce:note-para id="np0040">Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy.</ce:note-para></ce:footnote><ce:footnote id="fn0050"><ce:label>V</ce:label><ce:note-para id="np0050">Also at: Department of Applied Physics, Aligarh Muslim University, Aligarh, India.</ce:note-para></ce:footnote><ce:footnote id="fn0060"><ce:label>VI</ce:label><ce:note-para id="np0060">Also at: Institute of Theoretical Physics, University of Wroclaw, Poland.</ce:note-para></ce:footnote><ce:footnote id="fn0070"><ce:label>VII</ce:label><ce:note-para id="np0070">Also at: An institution covered by a cooperation agreement with CERN.</ce:note-para></ce:footnote></ce:author-group></ce:collaboration><ce:footnote id="fn0080"><ce:label>⋆</ce:label><ce:note-para id="np0080"><ce:italic>E-mail address:</ce:italic> <ce:inter-ref xlink:role="http://www.elsevier.com/xml/linking-roles/text/html" xlink:href="mailto:alice-publications@cern.ch" id="inf0020">alice-publications@cern.ch</ce:inter-ref>.</ce:note-para></ce:footnote></ce:author-group><ce:date-received day="5" month="5" year="2022"/><ce:date-revised day="2" month="12" year="2022"/><ce:date-accepted day="23" month="12" year="2022"/><ce:miscellaneous id="ms0010">Editor: M. Pierini</ce:miscellaneous><ce:abstract id="ab0010"><ce:section-title id="st0010">Abstract</ce:section-title><ce:abstract-sec id="as0010"><ce:simple-para id="sp0070">This letter reports measurements which characterize the underlying event associated with hard scatterings at mid-pseudorapidity (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math>) in pp, p–Pb and Pb–Pb collisions at centre-of-mass energy per nucleon pair, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The measurements are performed with ALICE at the LHC. Different multiplicity classes are defined based on the event activity measured at forward rapidities. The hard scatterings are identified by the leading particle defined as the charged particle with the largest transverse momentum (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) in the collision and having 8 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mo linebreak="badbreak" linebreakstyle="after"><</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra of associated particles (0.5 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) are measured in different azimuthal regions defined with respect to the leading particle direction: toward, transverse, and away. The associated charged particle yields in the transverse region are subtracted from those of the away and toward regions. The remaining jet-like yields are reported as a function of the multiplicity measured in the transverse region. The measurements show a suppression of the jet-like yield in the away region and an enhancement of high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> associated particles in the toward region in central Pb–Pb collisions, as compared to minimum-bias pp collisions. These observations are consistent with previous measurements that used two-particle correlations, and with an interpretation in terms of parton energy loss in a high-density quark gluon plasma. These yield modifications vanish in peripheral Pb–Pb collisions and are not observed in either high-multiplicity pp or p–Pb collisions.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:data-availability id="dav0001"><ce:section-title id="st0080">Data availability</ce:section-title><ce:para id="pr0200">This manuscript has associated data in a HEPData repository at <ce:inter-ref xlink:href="https://www.hepdata.net/" xlink:role="http://www.elsevier.com/xml/linking-roles/research-data" id="inf0550">https://www.hepdata.net/</ce:inter-ref>.</ce:para></ce:data-availability></head><body><ce:sections><ce:section id="se0010" role="introduction"><ce:label>1</ce:label><ce:section-title id="st0020">Introduction</ce:section-title><ce:para id="pr0010">In proton-proton (pp) collisions, jets, originating from partonic scatterings with large momentum transfer, are accompanied by particles produced by initial- and final-state radiation (ISR and FSR, respectively), as well as, by a plethora of other mechanisms. These include proton break-up, and, in a scenario incorporating multi-parton interactions (MPI) <ce:cross-refs refid="br0010 br0020" id="crs0010">[1,2]</ce:cross-refs>, several semi-hard parton-parton scatterings in a single pp collision. These jet-accompanying particles experimentally make up the underlying event (UE) and are commonly studied via azimuthal separations from the jets to minimise the influence of hard scatterings. The present study follows the strategy originally introduced by the CDF collaboration <ce:cross-ref refid="br0030" id="crf10910">[3]</ce:cross-ref>. First, the leading charged particle in the event is found, i.e., the charged particle with the highest transverse momentum in the collision (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math>). Secondly, the associated particles (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math>) are measured in three topological regions depending on their azimuthal angle relative to the leading particle, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>φ</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msup><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">assoc</mml:mi></mml:mrow></mml:msup><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:msup><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">|</mml:mo></mml:math>, see <ce:cross-ref refid="fg0010" id="crf10920">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/>.</ce:para><ce:para id="pr0020">The toward region contains the primary jet within the acceptance of the detector, while the away region contains the back-scattered particles of the recoil jet <ce:cross-ref refid="br0040" id="crf10930">[4]</ce:cross-ref>. In contrast, the transverse region is dominated by the underlying-event dynamics, but it also includes contributions from ISR and FSR <ce:cross-ref refid="br0050" id="crf10940">[5]</ce:cross-ref>.</ce:para><ce:para id="pr0030">The measurements performed at RHIC and LHC in small systems (pp, p–A, and d–A collisions) have shown for high particle multiplicities similar phenomena as were originally observed only in A–A collisions and have been attributed there to the formation of the strongly interacting quark gluon plasma <ce:cross-refs refid="br0060 br0070" id="crs0020">[6,7]</ce:cross-refs>, namely, long range angular correlations and collectivity <ce:cross-ref refid="br0080" id="crf10950">[8]</ce:cross-ref>. The origin of these effects in small systems is still an open question; on one hand, hydrodynamical calculations describe some aspects of the data <ce:cross-ref refid="br0090" id="crf10960">[9]</ce:cross-ref>; on the other hand, mechanisms like colour reconnection <ce:cross-ref refid="br0100" id="crf10970">[10]</ce:cross-ref>, rope hadronisation <ce:cross-ref refid="br0110" id="crf10980">[11]</ce:cross-ref>, and string shoving <ce:cross-ref refid="br0120" id="crf10990">[12]</ce:cross-ref> can produce collective-like effects in Monte Carlo event generators such as <ce:small-caps>PYTHIA</ce:small-caps> 8 <ce:cross-ref refid="br0130" id="crf11000">[13]</ce:cross-ref>. Thus, investigating pp collisions as a function of the charged particle multiplicity has become ever more pertinent <ce:cross-refs refid="br0090 br0140 br0150 br0160 br0170 br0180" id="crs0030">[9,14–18]</ce:cross-refs>. The interpretation of the results from the analysis of high-multiplicity pp collisions is challenging due to the selection biases of the sample towards events in which partonic scatterings with large momentum transfer (hard scatterings) occurred. To mitigate this inherent bias, Martin et al. <ce:cross-ref refid="br0190" id="crf11010">[19]</ce:cross-ref> suggested to use the charged-particle multiplicity in the transverse region (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>) as a classifier of the activity in the collisions, since the correlation between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math> and the hardest scattering in the collision is small. The ALICE collaboration has reported the first <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math> spectra measured in pp collisions at centre-of-mass energy, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"><mml:msqrt><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>13</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV <ce:cross-ref refid="br0200" id="crf11020">[20]</ce:cross-ref>. Event generators, such as <ce:small-caps>PYTHIA</ce:small-caps> 8 <ce:cross-ref refid="br0130" id="crf11030">[13]</ce:cross-ref> and EPOS-LHC <ce:cross-ref refid="br0210" id="crf11040">[21]</ce:cross-ref>, do not provide a good description of the measured distribution of the ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>/<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> is the event-averaged charged-particle multiplicity in the transverse region, underestimating in particular the number of collisions with large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">></mml:mo><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo></mml:math>. In the framework of MPI-based models, like those implemented in <ce:small-caps>PYTHIA</ce:small-caps> 8 and <ce:small-caps>HERWIG</ce:small-caps> 7 <ce:cross-ref refid="br0220" id="crf11050">[22]</ce:cross-ref>, the probability for a hard scattering in the collision increases with decreasing impact parameter<ce:cross-ref refid="fn0090" id="crf11060"><ce:sup>VIII</ce:sup></ce:cross-ref><ce:footnote id="fn0090"><ce:label>VIII</ce:label><ce:note-para id="np0090">In event generators like <ce:small-caps>PYTHIA</ce:small-caps> 8 the impact parameter profile is described by an overlap matter distribution of the two incoming hadrons.</ce:note-para></ce:footnote> between the colliding protons. Thus, requiring a high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> particle (e.g., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>8</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) in a given pp collision biases the selection of collisions towards those with a smaller impact parameter <ce:cross-ref refid="br0230" id="crf11070">[23]</ce:cross-ref>, which in turn biases the selection towards pp collisions with more MPI <ce:cross-ref refid="br0200" id="crf11080">[20]</ce:cross-ref>. This feature of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>-based analysis is important for the isolation of potential MPI and colour reconnection effects, which according to <ce:small-caps>PYTHIA</ce:small-caps> 8, produce effects resembling collective behaviour <ce:cross-ref refid="br0100" id="crf11090">[10]</ce:cross-ref>. By construction, MPI and colour reconnection effects are expected to be more relevant in the transverse region than in the away and toward regions <ce:cross-ref refid="br0240" id="crf11100">[24]</ce:cross-ref>. It is worth mentioning that the MPI picture has been used to explain the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in p–Pb collisions and peripheral Pb–Pb collisions <ce:cross-refs refid="br0250 br0260 br0270" id="crs0040">[25–27]</ce:cross-refs>. Studies, as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup></mml:math>, are therefore important to the understanding of the effects observed in high-multiplicity pp collisions. Last but not least, measurements of UE observables are also important to tune event generators <ce:cross-ref refid="br0280" id="crf11110">[28]</ce:cross-ref> that include hard partonic scatterings and MPI.</ce:para><ce:para id="pr0040">This letter reports the inclusive charged-particle transverse momentum spectra in pp, p–Pb and Pb–Pb collisions at centre-of-mass energy per nucleon pair <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV containing a high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> leading particle within the kinematic intervals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>8</mml:mn><mml:mo>≤</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math>. This guarantees the selection of collisions in which the average activity in the transverse region is roughly flat as a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math> <ce:cross-ref refid="br0200" id="crf11120">[20]</ce:cross-ref>, and therefore, any additional selection on the charged particle multiplicity will only modulate the UE activity. The measurements are performed considering different event classes defined in terms of the multiplicity registered in the forward detectors. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra of associated charged particles (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mn>0.5</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math>) are measured in the toward, away, and transverse regions as a function of the average charged particle multiplicity in the transverse region. To further investigate the possible modification of the particles produced in the hard scattering in pp, p–Pb, and Pb–Pb collisions, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> distributions in the toward (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) and away (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) regions obtained after the subtraction of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in the transverse region (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) are also reported. The subtracted yields (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>) are further normalised to those measured in minimum-bias (MB) pp collisions,<ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pp</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">MB</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">st</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">sa</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">pp</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">MB</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <ce:italic>X</ce:italic> indicates the collision system and the event multiplicity class. In this way, the hard process <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in the toward and away regions are isolated, and thus allowing us to study possible modifications to the produced particles due to medium effects in high-multiplicity pp, p–Pb, and Pb–Pb collisions. In heavy-ion collisions, this ratio is sensitive to the same effects which were studied using the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math> quantity <ce:cross-refs refid="br0290 br0300 br0310" id="crs0050">[29–31]</ce:cross-refs>, where jets produced in the early stage of the collision propagate through the hot and dense quark–gluon plasma. Their interaction with the coloured medium lead to parton-energy loss (jet quenching) <ce:cross-ref refid="br0320" id="crf11130">[32]</ce:cross-ref> which, for example, results in the suppression of the charged-particle yield at high <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> <ce:cross-ref refid="br0330" id="crf11140">[33]</ce:cross-ref>, and the suppression of the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> yield in the away region <ce:cross-refs refid="br0290 br0300" id="crs0060">[29,30]</ce:cross-refs>. It is worth mentioning that jet quenching effects have not been observed so far in small systems <ce:cross-refs refid="br0330 br0340" id="crs0070">[33,34]</ce:cross-refs>.</ce:para></ce:section><ce:section id="se0020"><ce:label>2</ce:label><ce:section-title id="st0030">Experiment and data analysis</ce:section-title><ce:para id="pr0050">This analysis is based on the data recorded by the ALICE apparatus during the pp and Pb–Pb runs at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV in 2015, and the p–Pb run at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV in 2016. The present study uses the V0 detector, and the Silicon Pixel Detector (SPD) for triggering and background rejection. The V0 consists of two arrays of scintillating tiles placed on each side of the interaction point covering the full azimuthal acceptance and the pseudorapidity intervals of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"><mml:mn>2.8</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>5.1</mml:mn></mml:math> (V0A) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>3.7</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>η</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1.7</mml:mn></mml:math> (V0C). The SPD is the innermost part of the Inner Tracking System (ITS) and it is the closest detector to the interaction point. It consists of two cylindrical silicon pixel layers at radial distances of 3.9 and 7.6 cm from the beam line and the pseudorapidity coverages of the two layers are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1.4</mml:mn></mml:math>, respectively. The data were collected using a minimum-bias trigger, which required a signal in both V0A and V0C detectors. The offline event selection was optimised to reject beam-induced background in all collision systems by utilising the timing signals in the two V0 detectors. In Pb–Pb collisions, the beam-induced background is further suppressed by correlating the timing signals of the neutron zero degree calorimeters, which are positioned on both sides of the interaction point at 112.5<ce:hsp sp="0.20"/>m distance along the beam axis <ce:cross-ref refid="br0350" id="crf11150">[35]</ce:cross-ref>. The signals from the zero degree calorimeters are also used to suppress the contamination from electromagnetic interactions. This is performed by requesting the coincidence of the signals coming from both side zero degree calorimeters by which the background due to single nucleus electromagnetic dissociation processes is excluded. A criterion based on the offline reconstruction of multiple primary vertices in the SPD is applied to reduce the pileup caused by multiple interactions in the same bunch crossing <ce:cross-ref refid="br0360" id="crf11160">[36]</ce:cross-ref>. The results presented in this letter are for minimum-bias triggered pp collisions having at least one charged particle in the pseudorapidity interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1</mml:mn></mml:math> (INEL>0). The INEL>0 event class corresponds to about 75% of the total inelastic cross section <ce:cross-ref refid="br0370" id="crf11170">[37]</ce:cross-ref>. For pp and Pb–Pb collisions, the sample is subdivided into different multiplicity classes based on the total charge deposited in both V0 sub-detectors, which is termed as V0M amplitude <ce:cross-ref refid="br0380" id="crf11180">[38]</ce:cross-ref>. For p–Pb collisions, the sample is subdivided based on the total charge deposited in V0A sub-detector (V0A amplitude) <ce:cross-ref refid="br0390" id="crf11190">[39]</ce:cross-ref>, which is located in the Pb-going direction. The V0A estimator has been implemented in previous measurements that used p–Pb data (see e.g. <ce:cross-ref refid="br0400" id="crf11200">[40]</ce:cross-ref>). This allows for comparisons with other observables for similar V0A multiplicity classes. To ensure that a hard scattering took place in the collision, events are required to have a trigger particle within <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>8</mml:mn><mml:mo>≤</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math> GeV/<ce:italic>c</ce:italic>. In this <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math> interval, the momentum resolution effects are negligible on the extracted yields, and therefore, no <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup></mml:math> resolution correction is applied. The total number of analysed collisions before the trigger particle selection are about 10<ce:sup>8</ce:sup>, 10<ce:sup>8</ce:sup>, and 10<ce:sup>7</ce:sup> for pp, p–Pb, and Pb–Pb collisions, respectively.</ce:para><ce:para id="pr0060">The transverse momentum of particles is determined from measurements in the central barrel with the ITS and the Time Projection Chamber (TPC). The ITS is a tracking detector which consists of six cylindrical layers of silicon detectors. The TPC is a cylindrical drift detector which covers a radial distance of 85-247<ce:hsp sp="0.20"/>cm from the beam axis and it has longitudinal dimension extending from about -250<ce:hsp sp="0.20"/>cm to +250<ce:hsp sp="0.20"/>cm around the nominal interaction point. Primary charged particles are measured in the pseudorapidity range of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math> and with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>0.5</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, where <ce:italic>η</ce:italic> is measured in the laboratory frame for the three collision systems. The configuration for p–Pb collisions with protons at 4<ce:hsp sp="0.20"/>TeV energy colliding with Pb ions that have per-nucleon energies of (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"><mml:mi>Z</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>A</mml:mi></mml:math>) × 4<ce:hsp sp="0.20"/>TeV ∼ 1.58<ce:hsp sp="0.20"/>TeV results in a shift in the rapidity of the nucleon–nucleon centre-of-mass system by 0.465 in the direction of the proton beam (negative z-direction). Here <ce:italic>Z</ce:italic> and <ce:italic>A</ce:italic> are the atomic and mass numbers of the Pb ion, respectively. Therefore, the detector coverage <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"><mml:mo stretchy="false">|</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.8</mml:mn></mml:math> corresponds to roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si30.svg"><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>0.3</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">cms</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>1.3</mml:mn></mml:math> for p–Pb collisions. The particles with mean proper lifetime larger than 1<ce:hsp sp="0.20"/>cm/<ce:italic>c</ce:italic>, which are either produced directly in the interaction or from decays of particles with mean proper lifetime smaller than 1<ce:hsp sp="0.20"/>cm/<ce:italic>c</ce:italic> are termed as primary particles <ce:cross-ref refid="br0410" id="crf11210">[41]</ce:cross-ref>. The track selection follows a procedure similar to the one described in Ref. <ce:cross-ref refid="br0420" id="crf11220">[42]</ce:cross-ref> and only few specific details are reported here. Tracks (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">tracks</mml:mi></mml:mrow></mml:msub></mml:math>) are required to have two hits in the ITS, out of which at least one should be in either of the two innermost layers. The geometrical track length <ce:italic>L</ce:italic> is calculated in the TPC readout plane, excluding the information from the pads at the sector boundaries (≈3<ce:hsp sp="0.20"/>cm from the sector edges). The trajectory lengths built from radial segments, i.e. the crossed TPC pad rows, traversed in the TPC by a particle are required to be larger than 85% of the geometrical track length. The pad rows are made of at least 3 neighbouring individual observations (clusters), and their height varies from 7.5<ce:hsp sp="0.20"/>mm to 15<ce:hsp sp="0.20"/>mm <ce:cross-ref refid="br0430" id="crf11230">[43]</ce:cross-ref>. The trajectory lengths built from clusters (one cluster per pad row) is required to be larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.svg"><mml:mn>0.7</mml:mn><mml:mo>×</mml:mo><mml:mi>L</mml:mi></mml:math>. The fraction of TPC clusters shared with another track is required to be lower than 0.4. The fit quality for the ITS and TPC track points must satisfy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">hits</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>36</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">clusters</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>4</mml:mn></mml:math>, respectively, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">hits</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">clusters</mml:mi></mml:mrow></mml:msub></mml:math> are the numbers of hits in the ITS and the number of clusters in the TPC, respectively. Only tracks with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>36</mml:mn></mml:math> are included in the analysis, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> is calculated comparing the track parameters from the combined ITS and TPC track reconstruction to that derived only from the TPC and constrained to the interaction point. The definition of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math> can be found in Ref. <ce:cross-ref refid="br0440" id="crf11240">[44]</ce:cross-ref>. To reduce the contamination from secondary particles, tracks are accepted if their distance-of-closest-approach (DCA) to the reconstructed primary interaction vertex satisfies in the longitudinal (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow></mml:msub></mml:math>) and transverse (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">xy</mml:mi></mml:mrow></mml:msub></mml:math>) directions the conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math><ce:hsp sp="0.20"/>cm and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">xy</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>0.018</mml:mn></mml:math><ce:hsp sp="0.20"/>cm + 0.035<ce:hsp sp="0.20"/>(cm×GeV/<ce:italic>c</ce:italic>)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.svg"><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>.</ce:para><ce:para id="pr0070">The measurement of the transverse momentum spectra of charged particles follows the standard procedure of the ALICE collaboration <ce:cross-refs refid="br0420 br0450" id="crs0080">[42,45]</ce:cross-refs>. The raw yields are corrected for efficiency and contamination from secondary particles. The efficiency correction is calculated from Monte Carlo simulations with GEANT3 <ce:cross-ref refid="br0460" id="crf11250">[46]</ce:cross-ref> transport code, which made use of PYTHIA 8 (Monash) <ce:cross-ref refid="br0280" id="crf11260">[28]</ce:cross-ref>, EPOS-LHC <ce:cross-ref refid="br0210" id="crf11270">[21]</ce:cross-ref> and HIJING <ce:cross-ref refid="br0470" id="crf11280">[47]</ce:cross-ref> event generators for pp, p–Pb and Pb–Pb collisions, respectively and incorporated a detailed description of the detector material, geometry and response. Since the event generators do not reproduce the relative abundances of different particle species in the real data, the efficiency obtained from the simulations is re-weighted considering the particle composition from data as outlined in <ce:cross-ref refid="br0420" id="crf11290">[42]</ce:cross-ref>. A multi-component template fit based on the DCA distributions from the simulation is used for the estimation of secondary contamination <ce:cross-ref refid="br0420" id="crf11300">[42]</ce:cross-ref>.</ce:para><ce:para id="pr0080">The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra for the toward and away regions include contributions from the jet fragmentation, ISR, and FSR, as well as, the contribution from the underlying event. In order to increase the sensitivity to the hardest process of the event, the particle yields measured in the transverse region are subtracted from the corresponding yields in both the toward and away regions: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.svg"><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>. This approach assumes that the background (UE, ISR, and FSR) in the toward and away regions is similar to the activity in the transverse region. However, one has to keep in mind that in Pb–Pb collisions two-particle correlations are affected by anisotropic transverse flow. In particular, the main contribution is due to the elliptic flow, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>, which is the second order coefficient in the Fourier expansion of the azimuthal distribution of the particle momenta <ce:cross-ref refid="br0480" id="crf11310">[48]</ce:cross-ref>. This elliptic azimuthal anisotropy modulates the background according to:<ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si46.svg"><mml:mi>B</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>φ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="badbreak" linebreakstyle="after">=</mml:mo><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">(</mml:mo><mml:mn>1</mml:mn><mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant="normal">cos</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>φ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="true" maxsize="2.4ex" minsize="2.4ex">)</mml:mo><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si47.svg"><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> is approximately given by the product of anisotropic flow coefficients for trigger and associated particles at their respective momenta i.e. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant="normal">assoc</mml:mi></mml:mrow></mml:msubsup></mml:math>. The existing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> measurements over a broad transverse momentum range <ce:cross-ref refid="br0490" id="crf11320">[49]</ce:cross-ref> suggest that the effect of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulation of background should be more relevant in semi-central Pb–Pb collisions. The effect is expected to be important at low and intermediate transverse momenta and decreases for high transverse momentum particles <ce:cross-ref refid="br0300" id="crf11330">[30]</ce:cross-ref>. In the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> region of interest for the jet quenching studies, namely <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>4</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, the effect of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulation is estimated to be small (about 5%) for Pb–Pb collisions. Given that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> effect is larger in Pb–Pb collisions than in pp and p–Pb collisions, no correction for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulation is applied for pp and p–Pb collisions since its effect is smaller than the other sources of systematic uncertainty.</ce:para><ce:para id="pr0090">The results are shown as a function of the average number of charged particles in the transverse region <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math>. The values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> are extracted in each multiplicity class from the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si31.svg"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">tracks</mml:mi></mml:mrow></mml:msub></mml:math> distributions in the transverse region that are corrected for detector effects using a Bayesian unfolding <ce:cross-ref refid="br0500" id="crf11340">[50]</ce:cross-ref>. The Bayesian unfolding requires the multiplicity response matrix, which is built from the correlation between the measured multiplicity and the multiplicity at generator level (without detector effects) in the transverse region. This has been obtained from MC simulations which include the propagation of particles through the detector using GEANT 3. As a crosscheck, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> values are also calculated by integrating the transverse momentum distributions in the interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si50.svg"><mml:mn>0.5</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>8</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>. The difference between the results from the two strategies is assigned as the systematic uncertainty on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math>, where the effects related to the discrepancy between data and MC in the particle composition and secondary contamination are considered. This uncertainty amounts up to 3.5%, 4% and 6.5% for pp, p–Pb and Pb–Pb collisions, respectively.</ce:para><ce:para id="pr0100">The systematic uncertainties related to the track selection criteria were studied by repeating the analysis varying one-by-one the track selection criteria <ce:cross-refs refid="br0420 br0450" id="crs0090">[42,45]</ce:cross-refs>. In particular, the upper limits of the track fit quality parameters in the ITS (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si51.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ITS</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">hits</mml:mi></mml:mrow></mml:msub></mml:math>) and in the TPC (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.svg"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">TPC</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">clusters</mml:mi></mml:mrow></mml:msub></mml:math>) were varied in the ranges of 25–49 and 3–5, respectively. The maximum fraction of shared TPC clusters was varied between 0.2 to 1 and the maximum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si39.svg"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow></mml:msub></mml:math> was varied between 1 and 5<ce:hsp sp="0.20"/>cm <ce:cross-ref refid="br0420" id="crf11350">[42]</ce:cross-ref>. We have also quantified the impact of not including the ITS hit requirement in the track selection. The systematic uncertainty on the primary particle composition was estimated using a procedure similar to the one described in <ce:cross-ref refid="br0420" id="crf11360">[42]</ce:cross-ref>. To quantify the uncertainty due to the imperfect simulation of the detector response, the track matching between the TPC and the ITS information in the data and in the simulation were compared. To achieve this, the fraction of secondary particles was rescaled according to fits to the measured DCA distributions. After this rescaling, the agreement between data and model was found to be within 3% for all collision systems. This value was assigned as an additional systematic uncertainty <ce:cross-ref refid="br0420" id="crf11370">[42]</ce:cross-ref>. The systematic uncertainty on the secondary particle contamination considers the imperfection of the method (multi-component template fit) used to extract the correction. The fit ranges were varied and the fit was repeated using templates with two (primaries, secondaries) or three (primaries, secondaries from material, secondaries from weak decays) components. The maximum spread among these variations was assigned as the systematic uncertainty on the secondary contamination. This contribution dominates at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>. The density of materials used in simulations of the experimental setup was varied by ± 4.5% <ce:cross-ref refid="br0350" id="crf11380">[35]</ce:cross-ref>, resulting in a negligible systematic uncertainty in the considered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> range of 0.5 to 6.0 GeV/<ce:italic>c</ce:italic>. For the estimation of total systematic uncertainty, all the above listed contributions were summed in quadrature. The systematic uncertainties are independent of the difference between the azimuthal angle of the associated particle and that of the trigger particle. The estimated systematic uncertainties on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra significantly depend on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>, while the dependence on the multiplicity classes is mild. The ranges of systematic uncertainties in the three considered collision systems are reported in <ce:cross-ref refid="tbl0010" id="crf11390">Table 1</ce:cross-ref><ce:float-anchor refid="tbl0010"/> for the various sources described above.</ce:para></ce:section><ce:section id="se0030" role="results"><ce:label>3</ce:label><ce:section-title id="st0040">Results and discussion</ce:section-title><ce:para id="pr0110">The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra measured in the transverse region for pp, p–Pb, and Pb–Pb collisions are shown in <ce:cross-ref refid="fg0020" id="crf11400">Fig. 2</ce:cross-ref><ce:float-anchor refid="fg0020"/> (top panel). Results are presented for different multiplicity classes. The ratios between the spectra in the individual multiplicity classes and the MB (0−100%) one are shown in the bottom panel. In the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si53.svg"><mml:mn>0.5</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, the ratios for the highest multiplicity class (0−5%) are larger than unity and show an increasing trend with increasing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si54.svg"><mml:mo linebreak="badbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) followed at higher <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> by a slow decrease. Instead, for the lowest multiplicity classes (40−60% and 60−90%) the ratios are lower than unity and follow an opposite trend with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math>, decreasing at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> and increasing for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si55.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>3</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>. The behaviour of the ratios as a function of the event activity is reminiscent of analogous ratios as a function of the number of MPI in pp collisions simulated with <ce:small-caps>PYTHIA</ce:small-caps> 8, including colour reconnection <ce:cross-ref refid="br0510" id="crf11410">[51]</ce:cross-ref>. In particular, at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mn>2</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>3</mml:mn></mml:math> GeV/<ce:italic>c</ce:italic> the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectrum of pp collisions with large MPI activity exhibits an enhancement with respect to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectrum of MB pp collisions. The effect was not observed before in data because, in contrast to the present analysis, the jet contribution was included in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra <ce:cross-ref refid="br0450" id="crf11420">[45]</ce:cross-ref>.</ce:para><ce:para id="pr0120">The top (bottom) panel of <ce:cross-ref refid="fg0030" id="crf11430">Fig. 3</ce:cross-ref><ce:float-anchor refid="fg0030"/> shows the charged particle yields for the toward (away) region after the subtraction of the yields measured in the transverse region in pp, p–Pb and Pb–Pb collisions. Results are compared with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra measured for MB pp collisions (0−100% V0M pp event class) quantified with the ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math>, as defined in Eq. <ce:cross-ref refid="fm0010" id="crf11440">(1)</ce:cross-ref>. At low transverse momenta, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>4</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is close to unity in pp and p–Pb collisions. In contrast, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> in Pb–Pb collisions exhibits a strong multiplicity dependence over the whole measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> interval. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> magnitude is larger for semi-peripheral Pb–Pb collisions, the maximum is observed for 20−40% Pb–Pb collisions, and is smaller for the most central and most peripheral classes. Given that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> contribution is not subtracted from the jet-like yields reported in <ce:cross-ref refid="fg0030" id="crf11450">Fig. 3</ce:cross-ref>, the centrality dependence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> follows the behaviour of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> as a function of collision centrality and particle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> in Pb–Pb collisions at LHC energies <ce:cross-ref refid="br0520" id="crf11460">[52]</ce:cross-ref>.</ce:para><ce:para id="pr0130"><ce:cross-ref refid="fg0040" id="crf11660">Fig. 4</ce:cross-ref><ce:float-anchor refid="fg0040"/> shows the measured values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> in the transverse momentum interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mn>4</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> as a function of the average multiplicity in the transverse region for all the multiplicity classes considered in pp, p–Pb and Pb–Pb collisions. The figure shows that, within uncertainties, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are close to unity for all the multiplicity classes measured in pp and p–Pb collisions. This indicates that effects induced by possible energy loss in these systems are not observed within uncertainties. This result is consistent with previous studies of nuclear modification factor <ce:cross-ref refid="br0330" id="crf11480">[33]</ce:cross-ref> and hadron-jet recoil measurements <ce:cross-ref refid="br0340" id="crf11490">[34]</ce:cross-ref>. By contrast, for Pb–Pb collisions the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are compatible to unity for peripheral collisions, and show a gradual enhancement (reduction) with the increase in multiplicity for the toward (away) region. The behaviour is the same for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si63.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values measured either assuming a flat background or a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>-modulated background. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>-modulated background was estimated following the approach depicted in Eq. <ce:cross-ref refid="fm0020" id="crf11500">(2)</ce:cross-ref> and using the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> data reported in <ce:cross-ref refid="br0490" id="crf11510">[49]</ce:cross-ref>. This behaviour is qualitatively similar to the di-hadron correlation results reported by the STAR and ALICE collaborations <ce:cross-refs refid="br0290 br0300" id="crs0100">[29,30]</ce:cross-refs>. In Pb–Pb collisions, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> provides information about the fragmenting jet leaving the medium, while on the away side, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> reflects the survival probability of the recoiling parton during passage through the medium. Thus a suppression of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si66.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> would indicate that fewer partons survive the passage through the medium and is expected from the strong in-medium energy loss. On the other hand, the enhancement observed in the toward region is also subject to medium effects. The ratio is sensitive to a) a possible change of the fragmentation functions, b) a possible modification of the quark to gluon jet ratio in the final state due to different coupling with medium, and c) a possible bias on the parton spectrum due to trigger particle selection. Moreover, given that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is sensitive to the same effects as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math>, the interpretation of the results is similar to that reported in <ce:cross-ref refid="br0300" id="crf11520">[30]</ce:cross-ref>. It is likely that all three effects play a role <ce:cross-ref refid="br0300" id="crf11530">[30]</ce:cross-ref>. A detailed quantification of the contribution of each effect is beyond the scope of the present paper.</ce:para><ce:para id="pr0140">In order to get further insight into the effect, the measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are compared in <ce:cross-ref refid="fg0050" id="crf11540">Fig. 5</ce:cross-ref><ce:float-anchor refid="fg0050"/> with model predictions. Following the similar treatment of the experimental data, for the models, the total sample is subdivided into different V0M classes and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> is calculated for each class. For high-multiplicity pp collisions, although <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is close to unity, a small trend with multiplicity is visible, which is not seen at similar multiplicities (20−90% V0A) in p–Pb data. To understand the source of these slight deviations from unity, the data are compared with the predictions from the <ce:small-caps>PYTHIA</ce:small-caps> 8 (Monash tune <ce:cross-ref refid="br0280" id="crf11550">[28]</ce:cross-ref>) and EPOS-LHC <ce:cross-ref refid="br0210" id="crf11560">[21]</ce:cross-ref> event generators. In PYTHIA, the hadronization of quarks is simulated using the Lund string fragmentation model <ce:cross-ref refid="br0530" id="crf11570">[53]</ce:cross-ref>. Various PYTHIA tunes have been developed through extensive comparison of Monte Carlo distributions with the minimum-bias data from different experiments. The Monash tune of <ce:small-caps>PYTHIA</ce:small-caps> 8 is tuned to LHC data and uses an updated set of hadronization parameters compared to the previous tunes <ce:cross-ref refid="br0280" id="crf11580">[28]</ce:cross-ref>. EPOS-LHC is built on the Parton-Based Gribov Regge Theory. Utilising the colour exchange mechanism of string excitation, the model is tuned to LHC data <ce:cross-ref refid="br0210" id="crf11590">[21]</ce:cross-ref>. In this model, a part of the collision system which has high parton densities becomes a “core” region that evolves hydrodynamically as a quark–gluon plasma and it is surrounded by a more dilute “corona” for which fragmentation occurs in the vacuum. The upper panel of <ce:cross-ref refid="fg0050" id="crf11600">Fig. 5</ce:cross-ref> shows <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> for different multiplicity classes. The observed deviations from unity are reproduced by <ce:small-caps>PYTHIA</ce:small-caps> 8 for both the toward and away regions. Given that <ce:small-caps>PYTHIA</ce:small-caps> 8 does not incorporate any jet quenching mechanism, the origin of the effect in high <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> collisions is related to a remaining bias towards harder fragmentation and more activity from initial and final state radiation <ce:cross-ref refid="br0540" id="crf11610">[54]</ce:cross-ref>. These effects enhance the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> yield in the toward region, and produce a broadening in the away region <ce:cross-ref refid="br0550" id="crf11620">[55]</ce:cross-ref>. The EPOS-LHC results in the away region are similar to both data and <ce:small-caps>PYTHIA</ce:small-caps> 8. However, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> EPOS-LHC exhibits a trend with a maximum at intermediate multiplicity and a reduction toward low and high multiplicities, which is not consistent with the measurements.</ce:para><ce:para id="pr0150">The middle and bottom panels of <ce:cross-ref refid="fg0050" id="crf11630">Fig. 5</ce:cross-ref> show <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> measured for p–Pb and Pb–Pb collisions, respectively. The data are compared to <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr <ce:cross-ref refid="br0560" id="crf11640">[56]</ce:cross-ref> and EPOS-LHC predictions. The Angantyr model in <ce:small-caps>PYTHIA</ce:small-caps> 8 extrapolates the dynamics from pp collisions to p–Pb and Pb–Pb collisions, generalising the formalism adopted for pp collisions by including a description of the nucleon positions within the colliding nuclei and utilising the Glauber model to calculate the number of interacting nucleons and binary nucleon–nucleon collisions. <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr, which does not include jet quenching effects, predicts <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values consistent with unity for all the multiplicity classes in Pb–Pb collisions. Whereas for p–Pb collisions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> is consistent with unity, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> is slightly below unity. In EPOS-LHC, a certain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> cutoff is defined in such a way that, above this cutoff, a particle loses part of its momentum in the core but survives as an independent particle produced by a flux tube. Soft particles, which are below the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> cutoff, get completely absorbed and form the core. This sort of energy loss mechanism implemented in EPOS-LHC depends on the system size <ce:cross-refs refid="br0210 br0570 br0580" id="crs0110">[21,57,58]</ce:cross-refs>. <ce:cross-ref refid="fg0050" id="crf11650">Fig. 5</ce:cross-ref> (middle) shows that for p–Pb collisions, EPOS-LHC does not describe either the magnitude or the trend of the multiplicity dependence of the measured ratio in the toward region, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math>. However, the model is in reasonable agreement with data in the away region. For Pb–Pb collisions, EPOS-LHC predicts a significant enhancement of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> for low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"><mml:mo stretchy="false">〈</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ch</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">〉</mml:mo></mml:math> ranges and deviates significantly from the experimental results.</ce:para><ce:para id="pr0160">In summary, while the data from Pb–Pb collisions are in qualitative agreement with expectations from parton energy loss due to the presence of a hot and dense medium, pp and p–Pb data do not show any hint of medium effects in the multiplicity range which is reported.</ce:para></ce:section><ce:section id="se0040"><ce:label>4</ce:label><ce:section-title id="st0050">Summary</ce:section-title><ce:para id="pr0170">The transverse momentum spectra (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mn>0.5</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) of primary charged particles in three azimuthal regions (toward, away and transverse) defined with respect to the direction of the particle with the highest transverse momentum in the event (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"><mml:mn>8</mml:mn><mml:mo>≤</mml:mo><mml:msubsup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">trig</mml:mi></mml:mrow></mml:msubsup><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>15</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>) are reported. The spectra are studied in intervals of the multiplicity measured at forward pseudorapidities for pp, p–Pb, and Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>5.02</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:math> spectra in the transverse region are subtracted from those of the away and toward regions. This is based on the assumption that the transverse side provides a good estimation of the underlying event contribution in both the toward and away regions. However, for the interpretation of the results one has to keep in mind that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si45.svg"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> modulates the background and this effect is important for semi-central Pb–Pb collisions and for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after">></mml:mo><mml:mn>4</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic> the effect is less than 5% in central and peripheral Pb–Pb collisions. Ratios to MB pp (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math>), i.e., the multiplicity dependent yields normalised to the yield measured in MB pp collisions, are reported. At low transverse momentum (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si70.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>2</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>), within 20%, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are multiplicity independent for both the toward and away regions in pp and p–Pb collisions. In contrast, in Pb–Pb collisions for both toward and away regions the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values exhibit a centrality dependence which is expected given the residual presence of elliptic flow. In the highest transverse momentum interval (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si64.svg"><mml:mn>4</mml:mn><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mn>6</mml:mn></mml:math><ce:hsp sp="0.20"/>GeV/<ce:italic>c</ce:italic>), the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values in pp collisions are closer to unity but they exhibit a small reduction (increase) towards high V0 activity in pp collisions. This trend is well reproduced by <ce:small-caps>PYTHIA</ce:small-caps> 8. In the model, it is due to a selection bias towards pp collisions with harder fragmentation and larger activity from initial and final state radiation. For p–Pb collisions, within uncertainties, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are consistent with unity and do not show a multiplicity dependence. <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr fairly describes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si62.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math>, but it underestimates by about 10% the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msubsup></mml:math> values in the low multiplicity classes (40−90% V0A event class). For Pb–Pb collisions, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> values are close to unity for peripheral collisions, and show a gradual increase (reduction) in the toward (away) region with increasing multiplicity. A similar observable, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math>, based on the per-trigger yield of associated particles in di-hadron correlation has been studied for central and peripheral Pb–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:msqrt><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt><mml:mo linebreak="goodbreak" linebreakstyle="after">=</mml:mo><mml:mn>2.76</mml:mn></mml:math><ce:hsp sp="0.20"/>TeV. The behaviour of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> exhibits the same features as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.svg"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">AA</mml:mi></mml:mrow></mml:msub></mml:math>: in central collisions, on the away-side, a suppression is observed as expected from strong in-medium energy loss. In the toward region, an enhancement is observed. <ce:small-caps>PYTHIA</ce:small-caps> 8/Angantyr predicts <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si73.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:math> for all multiplicity intervals, and it does not reproduce the observed away-side suppression or toward-side enhancement. Generally, EPOS-LHC does not describe the measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">a</mml:mi></mml:mrow></mml:msubsup></mml:math> ratios.</ce:para><ce:para id="pr0180">In summary, within the multiplicity reach reported in this paper, no jet quenching effects are observed in pp and p–Pb collisions within uncertainties. Further studies are required to extend the present analysis to higher multiplicities, which are currently limited by the event selection based on the forward V0 detector. The analysis of future pp and p–Pb collisions with much larger integrated luminosity may remove this limitation.</ce:para> </ce:section></ce:sections><ce:conflict-of-interest id="coi0001"><ce:section-title id="st0090">Declaration of Competing Interest</ce:section-title><ce:para id="pr0210">The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.</ce:para></ce:conflict-of-interest><ce:acknowledgment id="ac0010"><ce:section-title id="st0060">Acknowledgements</ce:section-title><ce:para id="pr0190">The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: <ce:grant-sponsor id="gsp0010">A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation (ANSL)</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0020" sponsor-id="https://doi.org/10.13039/501100007029">State Committee of Science</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0030">World Federation of Scientists (WFS)</ce:grant-sponsor>, Armenia; <ce:grant-sponsor id="gsp0040" sponsor-id="https://doi.org/10.13039/501100001822">Austrian Academy of Sciences</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0050" sponsor-id="https://doi.org/10.13039/501100002428">Austrian Science Fund</ce:grant-sponsor> (FWF): [<ce:grant-number refid="gsp0050">M 2467-N36</ce:grant-number>] and <ce:grant-sponsor id="gsp0060">Nationalstiftung für Forschung, Technologie und Entwicklung</ce:grant-sponsor>, Austria; <ce:grant-sponsor id="gsp0070">Ministry of Communications and High Technologies, National Nuclear Research Center</ce:grant-sponsor>, Azerbaijan; Conselho Nacional de Desenvolvimento Científico e Tecnológico (<ce:grant-sponsor id="gsp0080" sponsor-id="https://doi.org/10.13039/501100003593">CNPq</ce:grant-sponsor>), <ce:grant-sponsor id="gsp0090" sponsor-id="https://doi.org/10.13039/501100004809">Financiadora de Estudos e Projetos</ce:grant-sponsor> (Finep), <ce:grant-sponsor id="gsp0100" sponsor-id="https://doi.org/10.13039/501100001807">Fundação de Amparo à Pesquisa do Estado de São Paulo</ce:grant-sponsor> (<ce:grant-sponsor id="gsp0110" sponsor-id="https://doi.org/10.13039/501100001807">FAPESP</ce:grant-sponsor>) and <ce:grant-sponsor id="gsp0120" sponsor-id="https://doi.org/10.13039/501100004909">Universidade Federal do Rio Grande do Sul</ce:grant-sponsor> (<ce:grant-sponsor id="gsp0130" sponsor-id="https://doi.org/10.13039/501100004909">UFRGS</ce:grant-sponsor>), Brazil; Bulgarian <ce:grant-sponsor id="gsp0140" sponsor-id="https://doi.org/10.13039/501100005992">Ministry of Education and Science</ce:grant-sponsor>, within the National Roadmap for Research Infrastructures 2020–2027 (object CERN), Bulgaria; <ce:grant-sponsor id="gsp0150" sponsor-id="https://doi.org/10.13039/501100002338">Ministry of Education of China</ce:grant-sponsor> (MOEC), <ce:grant-sponsor id="gsp0160">Ministry of Science & Technology of China</ce:grant-sponsor> (MSTC) and <ce:grant-sponsor id="gsp0170" sponsor-id="https://doi.org/10.13039/501100001809">National Natural Science Foundation of China</ce:grant-sponsor> (NSFC), China; <ce:grant-sponsor id="gsp0180" sponsor-id="https://doi.org/10.13039/100015526">Ministry of Science and Education</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0190" sponsor-id="https://doi.org/10.13039/501100004488">Croatian Science Foundation</ce:grant-sponsor>, Croatia; <ce:grant-sponsor id="gsp0200" sponsor-id="https://doi.org/10.13039/501100019929">Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear</ce:grant-sponsor> (CEADEN), <ce:grant-sponsor id="gsp0210">Cubaenergía</ce:grant-sponsor>, Cuba; <ce:grant-sponsor id="gsp0220">Ministry of Education, Youth and Sports of the Czech Republic</ce:grant-sponsor>, Czech Republic; The <ce:grant-sponsor id="gsp0230">Danish Council for Independent Research | Natural Sciences</ce:grant-sponsor>, the <ce:grant-sponsor id="gsp0240" sponsor-id="https://doi.org/10.13039/100008398">Villum Fonden</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0250" sponsor-id="https://doi.org/10.13039/501100001732">Danish National Research Foundation</ce:grant-sponsor> (DNRF), Denmark; <ce:grant-sponsor id="gsp0260">Helsinki Institute of Physics</ce:grant-sponsor> (HIP), Finland; Commissariat à l'Energie Atomique (<ce:grant-sponsor id="gsp0270" sponsor-id="https://doi.org/10.13039/501100006489">CEA</ce:grant-sponsor>) and <ce:grant-sponsor id="gsp0280" sponsor-id="https://doi.org/10.13039/501100012441">Institut National de Physique Nucléaire et de Physique des Particules</ce:grant-sponsor> (IN2P3) and <ce:grant-sponsor id="gsp0290" sponsor-id="https://doi.org/10.13039/501100004794">Centre National de la Recherche Scientifique</ce:grant-sponsor> (CNRS), France; Bundesministerium für Bildung und Forschung (<ce:grant-sponsor id="gsp0300" sponsor-id="https://doi.org/10.13039/501100002347">BMBF</ce:grant-sponsor>) and <ce:grant-sponsor id="gsp0310" sponsor-id="https://doi.org/10.13039/501100010958">GSI Helmholtzzentrum für Schwerionenforschung GmbH</ce:grant-sponsor>, Germany; <ce:grant-sponsor id="gsp0320" sponsor-id="https://doi.org/10.13039/501100003448">General Secretariat for Research and Technology</ce:grant-sponsor>, Ministry of Education, Research and Religions, Greece; <ce:grant-sponsor id="gsp0330" sponsor-id="https://doi.org/10.13039/501100018818">National Research, Development and Innovation Office</ce:grant-sponsor>, Hungary; Department of Atomic Energy Government of India (<ce:grant-sponsor id="gsp0340" sponsor-id="https://doi.org/10.13039/501100001502">DAE</ce:grant-sponsor>), Department of Science and Technology, Government of India (<ce:grant-sponsor id="gsp0350" sponsor-id="https://doi.org/10.13039/501100001409">DST</ce:grant-sponsor>), <ce:grant-sponsor id="gsp0360" sponsor-id="https://doi.org/10.13039/501100001501">University Grants Commission</ce:grant-sponsor>, Government of India (UGC) and <ce:grant-sponsor id="gsp0370" sponsor-id="https://doi.org/10.13039/501100001412">Council of Scientific and Industrial Research</ce:grant-sponsor> (CSIR), India; National Research and Innovation Agency - <ce:grant-sponsor id="gsp0380" sponsor-id="https://doi.org/10.13039/100020473">BRIN</ce:grant-sponsor>, Indonesia; Istituto Nazionale di Fisica Nucleare (<ce:grant-sponsor id="gsp0390" sponsor-id="https://doi.org/10.13039/501100004007">INFN</ce:grant-sponsor>), Italy; Japanese <ce:grant-sponsor id="gsp0400" sponsor-id="https://doi.org/10.13039/501100001700">Ministry of Education, Culture, Sports, Science and Technology</ce:grant-sponsor> (MEXT) and <ce:grant-sponsor id="gsp0410" sponsor-id="https://doi.org/10.13039/501100001691">Japan Society for the Promotion of Science</ce:grant-sponsor> (JSPS) KAKENHI, Japan; Consejo Nacional de Ciencia (<ce:grant-sponsor id="gsp0420" sponsor-id="https://doi.org/10.13039/501100003141">CONACYT</ce:grant-sponsor>) y Tecnología, through <ce:grant-sponsor id="gsp0430" sponsor-id="https://doi.org/10.13039/501100007709">Fondo de Cooperación Internacional en Ciencia y Tecnología</ce:grant-sponsor> (FONCICYT) and <ce:grant-sponsor id="gsp0440" sponsor-id="https://doi.org/10.13039/501100006087">Dirección General de Asuntos del Personal Académico</ce:grant-sponsor> (DGAPA), Mexico; <ce:grant-sponsor id="gsp0450" sponsor-id="https://doi.org/10.13039/501100003246">Nederlandse Organisatie voor Wetenschappelijk Onderzoek</ce:grant-sponsor> (NWO), Netherlands; The <ce:grant-sponsor id="gsp0460" sponsor-id="https://doi.org/10.13039/501100005416">Research Council of Norway</ce:grant-sponsor>, Norway; <ce:grant-sponsor id="gsp0470">Commission on Science and Technology for Sustainable Development in the South</ce:grant-sponsor> (COMSATS), Pakistan; <ce:grant-sponsor id="gsp0480" sponsor-id="https://doi.org/10.13039/501100011871">Pontificia Universidad Católica del Perú</ce:grant-sponsor>, Peru; <ce:grant-sponsor id="gsp0490">Ministry of Education and Science</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0500" sponsor-id="https://doi.org/10.13039/501100004281">National Science Centre</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0510">WUT ID-UB</ce:grant-sponsor>, Poland; <ce:grant-sponsor id="gsp0520" sponsor-id="https://doi.org/10.13039/501100003708">Korea Institute of Science and Technology Information</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0530" sponsor-id="https://doi.org/10.13039/501100003725">National Research Foundation of Korea</ce:grant-sponsor> (NRF), Republic of Korea; <ce:grant-sponsor id="gsp0540">Ministry of Education and Scientific Research</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0550" sponsor-id="https://doi.org/10.13039/501100019278">Institute of Atomic Physics</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0560" sponsor-id="https://doi.org/10.13039/501100015622">Ministry of Research and Innovation</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0570" sponsor-id="https://doi.org/10.13039/501100019278">Institute of Atomic Physics</ce:grant-sponsor> and <ce:grant-sponsor id="gsp0580">University Politehnica of Bucharest</ce:grant-sponsor>, Romania; <ce:grant-sponsor id="gsp0590" sponsor-id="https://doi.org/10.13039/501100003193">Ministry of Education, Science, Research and Sport of the Slovak Republic</ce:grant-sponsor>, Slovakia; <ce:grant-sponsor id="gsp0600">National Research Foundation of South Africa</ce:grant-sponsor>, South Africa; <ce:grant-sponsor id="gsp0610" sponsor-id="https://doi.org/10.13039/501100004359">Swedish Research Council</ce:grant-sponsor> (VR) and <ce:grant-sponsor id="gsp0620">Knut & Alice Wallenberg Foundation</ce:grant-sponsor> (KAW), Sweden; <ce:grant-sponsor id="gsp0630" sponsor-id="https://doi.org/10.13039/100012470">European Organization for Nuclear Research</ce:grant-sponsor>, Switzerland; <ce:grant-sponsor id="gsp0640" sponsor-id="https://doi.org/10.13039/501100004352">Suranaree University of Technology</ce:grant-sponsor> (SUT), <ce:grant-sponsor id="gsp0650" sponsor-id="https://doi.org/10.13039/501100004192">National Science and Technology Development Agency</ce:grant-sponsor> (NSTDA), <ce:grant-sponsor id="gsp0660" sponsor-id="https://doi.org/10.13039/501100017170">Thailand Science Research and Innovation</ce:grant-sponsor> (TSRI) and <ce:grant-sponsor id="gsp0670">National Science, Research and Innovation Fund</ce:grant-sponsor> (NSRF), Thailand; <ce:grant-sponsor id="gsp0680" sponsor-id="https://doi.org/10.13039/100020381">Turkish Energy, Nuclear and Mineral Research Agency</ce:grant-sponsor> (TENMAK), Turkey; <ce:grant-sponsor id="gsp0690" sponsor-id="https://doi.org/10.13039/501100004742">National Academy of Sciences of Ukraine</ce:grant-sponsor>, Ukraine; <ce:grant-sponsor id="gsp0700" sponsor-id="https://doi.org/10.13039/501100000271">Science and Technology Facilities Council</ce:grant-sponsor> (STFC), United Kingdom; National Science Foundation of the United States of America (<ce:grant-sponsor id="gsp0710" sponsor-id="https://doi.org/10.13039/100000001">NSF</ce:grant-sponsor>) and <ce:grant-sponsor id="gsp0720" sponsor-id="https://doi.org/10.13039/100000015">United States Department of Energy</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0730" sponsor-id="https://doi.org/10.13039/100006209">Office of Nuclear Physics</ce:grant-sponsor> (DOE NP), United States of America. In addition, individual groups or members have received support from: Marie Skłodowska Curie, Strong 2020 - <ce:grant-sponsor id="gsp0740" sponsor-id="https://doi.org/10.13039/100010661">Horizon 2020</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0750" sponsor-id="https://doi.org/10.13039/501100000781">European Research Council</ce:grant-sponsor> (grant nos. <ce:grant-number refid="gsp0750">824093</ce:grant-number>, <ce:grant-number refid="gsp0750">896850</ce:grant-number>, <ce:grant-number refid="gsp0750">950692</ce:grant-number>), <ce:grant-sponsor id="gsp0760" sponsor-id="https://doi.org/10.13039/501100000780">European Union</ce:grant-sponsor>; <ce:grant-sponsor id="gsp0770" sponsor-id="https://doi.org/10.13039/501100002341">Academy of Finland</ce:grant-sponsor> (Center of Excellence in Quark Matter) (grant nos. <ce:grant-number refid="gsp0770">346327</ce:grant-number>, <ce:grant-number refid="gsp0770">346328</ce:grant-number>), Finland; <ce:grant-sponsor id="gsp0780">Programa de Apoyos para la Superación del Personal Académico</ce:grant-sponsor>, <ce:grant-sponsor id="gsp0790" sponsor-id="https://doi.org/10.13039/501100005739">UNAM</ce:grant-sponsor>, Mexico.</ce:para></ce:acknowledgment></body><tail><ce:bibliography id="bl0010"><ce:section-title id="st0070">References</ce:section-title><ce:bibliography-sec id="bs0010"><ce:bib-reference id="br0010"><ce:label>[1]</ce:label><sb:reference id="bibD17CEA27EFFA6FC144ED8C3DC8A6A4C2s1"><sb:contribution><sb:authors><sb:author><ce:given-name>T.</ce:given-name><ce:surname>Sjöstrand</ce:surname></sb:author><sb:author><ce:given-name>M.</ce:given-name><ce:surname>van Zijl</ce:surname></sb:author></sb:authors><sb:title><sb:maintitle>A multiple interaction model for the event structure in hadron collisions</sb:maintitle></sb:title></sb:contribution><sb:host><sb:issue><sb:series><sb:title><sb:maintitle>Phys. 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Proc. 1038 no. 1, (2008) 139–148, arXiv:0806.0242 [hep-ph].</ce:source-text></ce:bib-reference></ce:bibliography-sec></ce:bibliography></tail></article> diff --git a/tests/units/hindawi/data/example1.xml b/tests/units/hindawi/data/example1.xml index 79a771c4..d69e5f08 100644 --- a/tests/units/hindawi/data/example1.xml +++ b/tests/units/hindawi/data/example1.xml @@ -66,4 +66,3 @@ </ns1:record> </ns0:metadata> </ns0:record> - diff --git a/tests/units/hindawi/data/example2.xml b/tests/units/hindawi/data/example2.xml index 5c2bfacf..0adc5a8c 100644 --- a/tests/units/hindawi/data/example2.xml +++ b/tests/units/hindawi/data/example2.xml @@ -182,4 +182,3 @@ </ns1:record> </ns0:metadata> </ns0:record> - diff --git a/tests/units/hindawi/data/example3.xml b/tests/units/hindawi/data/example3.xml index 40b8645b..95511b1f 100644 --- a/tests/units/hindawi/data/example3.xml +++ b/tests/units/hindawi/data/example3.xml @@ -55,4 +55,3 @@ </ns1:record> </ns0:metadata> </ns0:record> - diff --git a/tests/units/hindawi/data/without_doi.xml b/tests/units/hindawi/data/without_doi.xml index 40b8645b..95511b1f 100644 --- a/tests/units/hindawi/data/without_doi.xml +++ b/tests/units/hindawi/data/without_doi.xml @@ -55,4 +55,3 @@ </ns1:record> </ns0:metadata> </ns0:record> - diff --git a/tests/units/hindawi/data/without_page_nr.xml b/tests/units/hindawi/data/without_page_nr.xml index 717e6940..b31202ac 100644 --- a/tests/units/hindawi/data/without_page_nr.xml +++ b/tests/units/hindawi/data/without_page_nr.xml @@ -181,4 +181,3 @@ </ns1:record> </ns0:metadata> </ns0:record> - diff --git a/tests/units/hindawi/test_hindawi_enhance.py b/tests/units/hindawi/test_hindawi_enhance.py index 623925a5..f8e71fb2 100644 --- a/tests/units/hindawi/test_hindawi_enhance.py +++ b/tests/units/hindawi/test_hindawi_enhance.py @@ -1,10 +1,7 @@ import xml.etree.ElementTree as ET import pytest -from common.parsing.xml_extractors import RequiredFieldNotFoundExtractionError -from hindawi.hindawi_file_processing import ( - enhance_hindawi, -) +from hindawi.hindawi_file_processing import enhance_hindawi from hindawi.parser import HindawiParser diff --git a/tests/units/iop/data/aca95c.xml b/tests/units/iop/data/aca95c.xml index 39528652..e6179bb5 100644 --- a/tests/units/iop/data/aca95c.xml +++ b/tests/units/iop/data/aca95c.xml @@ -24,14 +24,14 @@ <tex-math><?CDATA $ U(1)_R $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M3.jpg" xlink:type="simple"/> </inline-formula> symmetry, we show that the Kahler potential and superpotential in the RNMSSM change as follows:</p><p> - <disp-formula> - <label>1</label> + <disp-formula> + <label>1</label> <tex-math id="cpc_47_4_043105_E1"> <?CDATA $ K=K_{{\rm{MSSM}}} + N^{\dagger}N+\delta K, $?> </tex-math> <graphic orientation="portrait" position="float" xlink:href="cpc_47_4_043105_E1.jpg" xlink:type="simple"/> </disp-formula> </p><p> - <disp-formula> - <label>2</label> + <disp-formula> + <label>2</label> <tex-math id="cpc_47_4_043105_E2"> <?CDATA $ W=W_{{\rm{MSSM}}}+\lambda NH_{u}H_{d}+\delta W, $?> </tex-math> <graphic orientation="portrait" position="float" xlink:href="cpc_47_4_043105_E2.jpg" xlink:type="simple"/> </disp-formula> @@ -102,24 +102,24 @@ <inline-formula> <tex-math><?CDATA $ {\rm{Field}} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M175.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </th><th align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $({S U}(3)_{c}, {S U}(2)_{L})_{U(1)_{Y}}$?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M176.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </th><th align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA ${U}(1)_{R}$?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M177.jpg" xlink:type="simple"/> </inline-formula> </th></tr></thead><tbody><tr><td align="center" colspan="1" rowspan="1" valign="middle"> - <italic toggle="yes">Q</italic> + <italic toggle="yes">Q</italic> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (3,2)_{1/6} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M178.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ 1 $?></tex-math> @@ -129,12 +129,12 @@ <inline-formula> <tex-math><?CDATA $ \bar{u} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M180.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (\bar{3},1)_{-2/3} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M181.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ 1 $?></tex-math> @@ -144,24 +144,24 @@ <inline-formula> <tex-math><?CDATA $ \bar{d} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M183.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (\bar{3},1)_{1/3} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M184.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ 1 $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M185.jpg" xlink:type="simple"/> </inline-formula> </td></tr><tr><td align="center" colspan="1" rowspan="1" valign="middle"> - <italic toggle="yes">L</italic> + <italic toggle="yes">L</italic> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (1,2)_{-1/2} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M186.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ 1 $?></tex-math> @@ -171,12 +171,12 @@ <inline-formula> <tex-math><?CDATA $ \bar{e} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M188.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (1,1)_{1} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M189.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ 1 $?></tex-math> @@ -186,12 +186,12 @@ <inline-formula> <tex-math><?CDATA $ H_{u} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M191.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (1,2)_{1/2} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M192.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ 0 $?></tex-math> @@ -201,24 +201,24 @@ <inline-formula> <tex-math><?CDATA $ H_{d} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M194.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (1,2)_{-1/2} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M195.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ 0 $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M196.jpg" xlink:type="simple"/> </inline-formula> </td></tr><tr><td align="center" colspan="1" rowspan="1" valign="middle"> - <italic toggle="yes">N</italic> + <italic toggle="yes">N</italic> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (1,1)_{0} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M197.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ 2 $?></tex-math> @@ -228,12 +228,12 @@ <inline-formula> <tex-math><?CDATA $ \sigma_{1} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M199.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (1, 1)_{0} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M200.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ 0 $?></tex-math> @@ -243,24 +243,24 @@ <inline-formula> <tex-math><?CDATA $ \sigma_{3} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M202.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (1, 3)_{0} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M203.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ 0 $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M204.jpg" xlink:type="simple"/> </inline-formula> </td></tr><tr><td align="center" colspan="1" rowspan="1" valign="middle"> - <italic toggle="yes">T</italic> + <italic toggle="yes">T</italic> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (1,3)_{0} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M205.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ 2 $?></tex-math> @@ -270,12 +270,12 @@ <inline-formula> <tex-math><?CDATA $ \sigma_{8} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M207.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (8,1)_{0} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M208.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ 0 $?></tex-math> @@ -297,8 +297,8 @@ <tex-math><?CDATA $ \sigma_i $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M22.jpg" xlink:type="simple"/> </inline-formula> as</p><p> - <disp-formula> - <label>3</label> + <disp-formula> + <label>3</label> <tex-math id="cpc_47_4_043105_E3"> <?CDATA $ \begin{array}{*{20}{l}} { } \mathcal{L}_{{\rm{soft}}} \supset m_{1}\tilde{\sigma}_{1}\tilde{B}+ m_{2}\tilde{\sigma}_{3}\tilde{W}+m_{3}\tilde{\sigma}_{8}\tilde{g}. \end{array} $?> </tex-math> <graphic orientation="portrait" position="float" xlink:href="cpc_47_4_043105_E3.jpg" xlink:type="simple"/> </disp-formula> @@ -306,8 +306,8 @@ <tex-math><?CDATA $ m_i $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M23.jpg" xlink:type="simple"/> </inline-formula> in Eq. (3) are read from soft SUSY-breaking operators such as</p><p> - <disp-formula> - <label>4</label> + <disp-formula> + <label>4</label> <tex-math id="cpc_47_4_043105_E4"> <?CDATA $ \int {\rm d}^{2}\theta \frac{D_{\alpha}X}{M} W^{\alpha}_{i}\sigma^{i}+\int {\rm d}^{4}\theta \frac{X^{\dagger}X}{M^{2}}\sigma^{\dagger}_{i}\sigma_{i}, $?> </tex-math> <graphic orientation="portrait" position="float" xlink:href="cpc_47_4_043105_E4.jpg" xlink:type="simple"/> </disp-formula> @@ -329,8 +329,8 @@ <tex-math><?CDATA $S U(2)_L$?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M28.jpg" xlink:type="simple"/> </inline-formula> triplet <italic toggle="yes">T</italic> shown in <xref ref-type="table" rid="cpc_47_4_043105_t1">Table 1</xref>, the determinant of the charged chargino mass matrix vanishes owing to the unbroken <italic toggle="yes">R</italic> symmetry. This mass issue can be resolved by an extension of the matter content. The simplest way to achieve this is via the addition of the triplet <italic toggle="yes">T</italic> [<xref ref-type="bibr" rid="cpc_47_4_043105_bib16">16</xref>], which couples to the Higgs doublets as</p><p> - <disp-formula> - <label>5</label> + <disp-formula> + <label>5</label> <tex-math id="cpc_47_4_043105_E5"> <?CDATA $ \delta W=\int {\rm d}^{2}\theta \; y_{T}H_{d}TH_{u}. $?> </tex-math> <graphic orientation="portrait" position="float" xlink:href="cpc_47_4_043105_E5.jpg" xlink:type="simple"/> </disp-formula> @@ -338,8 +338,8 @@ <tex-math><?CDATA $ +2 $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M29.jpg" xlink:type="simple"/> </inline-formula>. With the quantum numbers of <italic toggle="yes">T</italic>, one can infer its soft masses from the SUSY-breaking operators,</p><p> - <disp-formula> - <label>6</label> + <disp-formula> + <label>6</label> <tex-math id="cpc_47_4_043105_E6"> <?CDATA $ \int {\rm d}^{4}\theta \frac{X^{\dagger}X}{M^{2}} T^{\dagger}T+ \int {\rm d}^{2}\theta \frac{D^{2}X}{M}{\rm{tr}}\left(T\sigma_{3}\right), $?> </tex-math> <graphic orientation="portrait" position="float" xlink:href="cpc_47_4_043105_E6.jpg" xlink:type="simple"/> </disp-formula> @@ -368,14 +368,14 @@ <tex-math><?CDATA $ T^{\pm} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M43.jpg" xlink:type="simple"/> </inline-formula>, respectively. With the new superpotential term in Eq. (5) taken into account, the chargino and neutralino mass matrices are now given as</p><p> - <disp-formula> - <label>7</label> + <disp-formula> + <label>7</label> <tex-math id="cpc_47_4_043105_E7"> <?CDATA $ \begin{array}{*{20}{l}} \left(\begin{array}{cccc} \tilde{\sigma}^-_{3} & \tilde{T}^- & \tilde{W}^- & \tilde{H}_{d}^-\end{array}\right) \left( \begin{array}{cccc} 0 & m'_{2} & m_{2} & 0 \\ m'_{2} & 0 & 0 & y_{T}\upsilon_{d} \\ m_{2} & 0 & 0 & \sqrt{2}m_{W}\sin\beta \\ 0 & -y_{T}\upsilon_{u} & \sqrt{2}m_{W}\cos\beta & 0 \\ \end{array}\right) \left(\begin{array}{c} \tilde{\sigma}^+_{3}\\ \tilde{T}^+\\ \tilde{W}^+ \\ \tilde{H}_{u}^+ \\ \end{array}\right), \end{array} $?> </tex-math> <graphic orientation="portrait" position="float" xlink:href="cpc_47_4_043105_E7.jpg" xlink:type="simple"/> </disp-formula> </p><p> - <disp-formula> - <label>8</label> + <disp-formula> + <label>8</label> <tex-math id="cpc_47_4_043105_E8"> <?CDATA $ \begin{array}{*{20}{l}} \left(\begin{array}{cccc} \tilde{\sigma}^{0}_{1} & \tilde{\sigma}^{0}_{3} & \tilde{H}_{u}^{0} & \tilde{H}_{d}^{0}\end{array}\right) \left( \begin{array}{cccc} m_{1 } & 0 & 0 & 0\\ 0 & m_{2} & 0 & m'_{2} \\ m_{Z}\sin\theta_{W} \cos\beta & -m_{Z}\cos\theta_{W} \sin\beta & -\lambda\upsilon_{d} & -y_{T}\upsilon_{d}\\ -m_{Z}\sin\theta_{W} \cos\beta & m_{Z}\cos\theta_{W} \cos\beta & -\lambda\upsilon_{u} & -y_{T}\upsilon_{u} \\ \end{array}\right) \left(\begin{array}{c} \tilde{B}^{0}\\ \tilde{W}^{0} \\ \tilde{N}^{0} \\ \tilde{T}^{0} \\ \end{array}\right), \end{array} $?> </tex-math> <graphic orientation="portrait" position="float" xlink:href="cpc_47_4_043105_E8.jpg" xlink:type="simple"/> </disp-formula> @@ -389,8 +389,8 @@ <tex-math><?CDATA $ \tan\beta=\upsilon_{u}/\upsilon_{d} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M46.jpg" xlink:type="simple"/> </inline-formula>, and we simply assume the singlet vevs</p><p> - <disp-formula> - <label>9</label> + <disp-formula> + <label>9</label> <tex-math id="cpc_47_4_043105_E9"> <?CDATA $ \begin{array}{*{20}{l}} { } \left \lt \sigma^{0}_{1}\right \gt =\left \lt \sigma^{0}_{3}\right \gt =0. \end{array} $?> </tex-math> <graphic orientation="portrait" position="float" xlink:href="cpc_47_4_043105_E9.jpg" xlink:type="simple"/> </disp-formula> @@ -542,7 +542,7 @@ <inline-formula> <tex-math><?CDATA ${\rm{Parameter} }$?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M211.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </th><th align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA ${\rm{Range} }$?></tex-math> @@ -552,7 +552,7 @@ <inline-formula> <tex-math><?CDATA $ m_1 $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M213.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ [-490.8, 490.3] $?></tex-math> @@ -562,7 +562,7 @@ <inline-formula> <tex-math><?CDATA $ m_2 $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M215.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ [-137.6, 157.1] $?></tex-math> @@ -572,7 +572,7 @@ <inline-formula> <tex-math><?CDATA $ m'_2 $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M217.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ [-430, 409] $?></tex-math> @@ -582,14 +582,14 @@ <inline-formula> <tex-math><?CDATA $ \tan\beta $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M219.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ (1, 10] $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M220.jpg" xlink:type="simple"/> </inline-formula> </td></tr><tr><td align="center" colspan="1" rowspan="1" valign="middle"> - <italic toggle="yes">λ</italic> + <italic toggle="yes">λ</italic> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ [0.5, 0.7] $?></tex-math> @@ -599,7 +599,7 @@ <inline-formula> <tex-math><?CDATA $ y_T $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M222.jpg" xlink:type="simple"/> - </inline-formula> + </inline-formula> </td><td align="center" colspan="1" rowspan="1" valign="middle"> <inline-formula> <tex-math><?CDATA $ [0.49, 1.0] $?></tex-math> @@ -642,8 +642,8 @@ <tex-math><?CDATA $ \chi^{0}_1 $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M99.jpg" xlink:type="simple"/> </inline-formula> pair is dominated by lepton final states [<xref ref-type="bibr" rid="cpc_47_4_043105_bib36">36</xref>]</p><p> - <disp-formula> - <label>10</label> + <disp-formula> + <label>10</label> <tex-math id="cpc_47_4_043105_E10"> <?CDATA $ \sigma_{\ell\bar{\ell}}\upsilon_{{\rm{rel}}}\approx \frac{g'^{4}m^{2}_{\chi^{0}_{1}}}{8\pi}\left[\frac{1}{16(m^{2}_{\chi^{0}_{1}}+m^{2}_{\tilde{\ell}_{L}})^{2}}+ \frac{1}{(m^{2}_{\chi^{0}_{1}}+m^{2}_{\tilde{\ell}_{R}})^{2}}\right],\\ $?> </tex-math> <graphic orientation="portrait" position="float" xlink:href="cpc_47_4_043105_E10.jpg" xlink:type="simple"/> </disp-formula> @@ -738,14 +738,14 @@ <tex-math><?CDATA $ \sigma_{\tau^{+}\tau^{-}}\upsilon_{{\rm{rel}}} $?></tex-math> <inline-graphic xlink:href="cpc_47_4_043105_M128.jpg" xlink:type="simple"/> </inline-formula> (in purple).</p></caption><graphic content-type="print" id="cpc_47_4_043105_f2_eps" orientation="portrait" position="float" xlink:href="cpc_47_4_043105_f2.eps" xlink:type="simple"/><graphic content-type="online" id="cpc_47_4_043105_f2_online" orientation="portrait" position="float" xlink:href="cpc_47_4_043105_f2.jpg" xlink:type="simple"/></fig></sec><sec id="cpc_47_4_043105_s03-03"><label>C.</label><title>Higgs

Compared to the R-symmetric MSSM (RMSSM) [43–47] or conventional NMSSM, the scalar mass spectrum of the Higgs sector in the minimal RNMSSM is different. Let us begin with relevant soft masses in our model.

- - + +

where the first three terms arise from the following soft SUSY-breaking operators:

- - + + @@ -759,14 +759,14 @@ are given by

- - + +

and

- - + + @@ -777,14 +777,14 @@ are eliminated by the b term because of the conditions of electroweak symmetry breaking.

Second, the mass squared matrices for the CP-odd scalars are

- - + +

which contain a massless Goldstone mode, and

- - + + @@ -792,8 +792,8 @@ is given by

- - + + @@ -893,14 +893,14 @@ - + - b + b @@ -910,14 +910,14 @@ - + - λ + λ @@ -927,7 +927,7 @@ - + @@ -937,8 +937,8 @@ , with

- - + + @@ -949,14 +949,14 @@ .

Sfermions

Apart from Majorana gaugino masses, R symmetry also prohibits holomorphic soft masses, such as A terms related to sfermions. It only allows scalar soft masses

- - + +

which arise from soft SUSY-breaking operators such as

- - + + @@ -975,7 +975,7 @@ - + @@ -1000,7 +1000,7 @@ - + @@ -1025,7 +1025,7 @@ - + @@ -1050,7 +1050,7 @@ - + diff --git a/tests/units/iop/data/title_and_abstract_with_cdata.xml b/tests/units/iop/data/title_and_abstract_with_cdata.xml index 8608149c..b935d886 100644 --- a/tests/units/iop/data/title_and_abstract_with_cdata.xml +++ b/tests/units/iop/data/title_and_abstract_with_cdata.xml @@ -4766,4 +4766,4 @@ PTEP202020208083C10.1093/ptep/ptaa104 L. WorkmanR. et al - PTEP20222022083C10.1093/ptep/ptac097 \ No newline at end of file + PTEP20222022083C10.1093/ptep/ptac097 diff --git a/tests/units/iop/test_iop_parser.py b/tests/units/iop/test_iop_parser.py index 8f00bd09..89eb98ef 100644 --- a/tests/units/iop/test_iop_parser.py +++ b/tests/units/iop/test_iop_parser.py @@ -1,13 +1,13 @@ import xml.etree.ElementTree as ET +from common.cleanup import replace_cdata_format from common.constants import ARXIV_EXTRACTION_PATTERN from common.enhancer import Enhancer from common.exceptions import UnknownLicense from common.parsing.xml_extractors import RequiredFieldNotFoundExtractionError from common.utils import parse_element_text, parse_to_ET_element -from common.cleanup import replace_cdata_format -from iop.parser import IOPParser from iop.iop_process_file import process_xml +from iop.parser import IOPParser from pytest import fixture, mark, param, raises @@ -541,7 +541,6 @@ def test_no_authors(shared_datadir, parser): parser._publisher_specific_parsing(article) - def test_title(shared_datadir, parser): content = (shared_datadir / "title_and_abstract_with_cdata.xml").read_text() content = replace_cdata_format(content) @@ -805,7 +804,7 @@ def test_no_collaborations_value(shared_datadir, parser): assert "collaborations" not in parsed_article -def test_title(shared_datadir, parser): +def test_title(shared_datadir, parser): # noqa content = (shared_datadir / "just_required_fields.xml").read_text() article = ET.fromstring(content) parsed_article = parser._publisher_specific_parsing(article) @@ -894,10 +893,10 @@ def test_cdata_abstract_title(shared_datadir): content = replace_cdata_format(content) ET_article = parse_to_ET_element(content) - abstract_element = ET_article.find("front/article-meta/abstract/p") abstract_text = parse_element_text(abstract_element) - assert (abstract_text + assert ( + abstract_text == "Recently, the experimental measurements of the branching ratios and different" " polarization asymmetries for processes occurring through flavor-changing-charged" " current $ (b\\rightarrow c\\tau\\overline{\\nu}_{\\tau}) $ transitions by BABAR," @@ -931,12 +930,14 @@ def test_cdata_abstract_title(shared_datadir): ) title_element = ET_article.find("front/article-meta/title-group/article-title") title_text = parse_element_text(title_element) - assert (title_text - == "Analysis of ${{\\boldsymbol b}{\\bf\\rightarrow} {\\boldsymbol c}{\\boldsymbol\\tau}\\bar" - "{\\boldsymbol\\nu}_{\\boldsymbol\\tau}}$ anomalies using weak effective Hamiltonian with " - "complex couplings and their impact on various physical observables" + assert ( + title_text + == "Analysis of ${{\\boldsymbol b}{\\bf\\rightarrow} {\\boldsymbol c}{\\boldsymbol\\tau}\\bar" + "{\\boldsymbol\\nu}_{\\boldsymbol\\tau}}$ anomalies using weak effective Hamiltonian with " + "complex couplings and their impact on various physical observables" ) + def test_cdata_without_regex(): paseudo_aricle = """

diff --git a/tests/units/oup/data/oup_orcid.xml b/tests/units/oup/data/oup_orcid.xml index 12c3643c..81365ed5 100644 --- a/tests/units/oup/data/oup_orcid.xml +++ b/tests/units/oup/data/oup_orcid.xml @@ -3085,4 +3085,4 @@ m_{H^+_1},\; m_{H^+_2}, \; m_{H^{++}_2} &\in& [1,\; 20]\;\mathrm{ TeV},\; v_R \i -
\ No newline at end of file + diff --git a/tests/units/oup/data/ptab170_no_journal_title_and_pubmed_value.xml b/tests/units/oup/data/ptab170_no_journal_title_and_pubmed_value.xml index b61bac9b..0b5fd1b4 100644 --- a/tests/units/oup/data/ptab170_no_journal_title_and_pubmed_value.xml +++ b/tests/units/oup/data/ptab170_no_journal_title_and_pubmed_value.xml @@ -37,7 +37,7 @@ - Detectable electric current induced by the dark matter axion in a + Detectable electric current induced by the dark matter axion in a conductor diff --git a/tests/units/oup/test_oup_parser.py b/tests/units/oup/test_oup_parser.py index 7d6d9656..d20f9898 100644 --- a/tests/units/oup/test_oup_parser.py +++ b/tests/units/oup/test_oup_parser.py @@ -635,54 +635,36 @@ def test_authors_parsing_with_orcid(article_with_orcid): "surname": "Hong", "given_names": "T T", "affiliations": [ - { - "organization": "An Giang University", - "country": "Vietnam" - }, - { - "organization": "Vietnam National University", - "country": "Vietnam" - } + {"organization": "An Giang University", "country": "Vietnam"}, + {"organization": "Vietnam National University", "country": "Vietnam"}, ], "orcid": "0000-0002-7719-4160", - "full_name": "Hong, T T" + "full_name": "Hong, T T", }, { "surname": "Le", "given_names": "V K", "affiliations": [ - { - "organization": "An Giang University", - "country": "Vietnam" - }, - { - "organization": "Binh Thuy Junior High School", - "country": "Vietnam" - } + {"organization": "An Giang University", "country": "Vietnam"}, + {"organization": "Binh Thuy Junior High School", "country": "Vietnam"}, ], - "full_name": "Le, V K" + "full_name": "Le, V K", }, { "surname": "Phuong", "given_names": "L T T", "affiliations": [ - { - "organization": "An Giang University", - "country": "Vietnam" - } + {"organization": "An Giang University", "country": "Vietnam"} ], - "full_name": "Phuong, L T T" + "full_name": "Phuong, L T T", }, { "surname": "Hoi", "given_names": "N C", "affiliations": [ - { - "organization": "An Giang University", - "country": "Vietnam" - } + {"organization": "An Giang University", "country": "Vietnam"} ], - "full_name": "Hoi, N C" + "full_name": "Hoi, N C", }, { "surname": "Ngan", @@ -690,10 +672,10 @@ def test_authors_parsing_with_orcid(article_with_orcid): "affiliations": [ { "organization": "Department of Physics, Can Tho University", - "country": "Vietnam" + "country": "Vietnam", } ], - "full_name": "Ngan, N T K" + "full_name": "Ngan, N T K", }, { "surname": "Nha", @@ -702,15 +684,15 @@ def test_authors_parsing_with_orcid(article_with_orcid): "affiliations": [ { "organization": "Subatomic Physics Research Group, Science and Technology Advanced Institute, Van Lang University", - "country": "Vietnam" + "country": "Vietnam", }, { "organization": "Faculty of Applied Technology, School of Engineering and Technology, Van Lang University", - "country": "Vietnam" - } + "country": "Vietnam", + }, ], "orcid": "0009-0005-5993-6895", - "full_name": "Nha, N H T" - } + "full_name": "Nha, N H T", + }, ] assert article_with_orcid["authors"] == expected_output diff --git a/tests/units/springer/test_parser.py b/tests/units/springer/test_parser.py index 77146de0..18bf50ef 100644 --- a/tests/units/springer/test_parser.py +++ b/tests/units/springer/test_parser.py @@ -1,9 +1,9 @@ import xml.etree.ElementTree as ET from os import listdir +from common.enhancer import Enhancer from pytest import fixture from springer.parser import SpringerParser -from common.enhancer import Enhancer @fixture(scope="module") @@ -333,90 +333,94 @@ def test_abstract(parsed_articles): @fixture def article_with_orcid(parser, datadir): with open(datadir / "s10052-024-12692-y.xml") as file: - yield parser._generic_parsing(parser._publisher_specific_parsing(ET.fromstring(file.read()))) + yield parser._generic_parsing( + parser._publisher_specific_parsing(ET.fromstring(file.read())) + ) def test_article_with_cleaned_orcid(article_with_orcid): - expected_output = [{ - "surname": "Hong", - "given_names": "T.", - "email": "tthong@agu.edu.vn", - "affiliations": [ - { - "value": "An Giang University, Long Xuyen, 880000, Vietnam", - "organization": "An Giang University", - "country": "Vietnam" - }, - { - "value": "Vietnam National University, Ho Chi Minh City, 700000, Vietnam", - "organization": "Vietnam National University", - "country": "Vietnam" - } - ], - "full_name": "Hong, T." - }, + expected_output = [ { - "surname": "Tran", - "given_names": "Q.", - "email": "tqduyet@agu.edu.vn", - "affiliations": [ - { - "value": "An Giang University, Long Xuyen, 880000, Vietnam", - "organization": "An Giang University", - "country": "Vietnam" - }, - { - "value": "Vietnam National University, Ho Chi Minh City, 700000, Vietnam", - "organization": "Vietnam National University", - "country": "Vietnam" - } - ], - "full_name": "Tran, Q." - }, + "surname": "Hong", + "given_names": "T.", + "email": "tthong@agu.edu.vn", + "affiliations": [ + { + "value": "An Giang University, Long Xuyen, 880000, Vietnam", + "organization": "An Giang University", + "country": "Vietnam", + }, + { + "value": "Vietnam National University, Ho Chi Minh City, 700000, Vietnam", + "organization": "Vietnam National University", + "country": "Vietnam", + }, + ], + "full_name": "Hong, T.", + }, { - "surname": "Nguyen", - "given_names": "T.", - "email": "thanhphong@ctu.edu.vn", - "affiliations": [ - { - "value": "Department of Physics, Can Tho University, 3/2 Street, Can Tho, Vietnam", - "organization": "Can Tho University", - "country": "Vietnam" - } - ], - "full_name": "Nguyen, T." - }, + "surname": "Tran", + "given_names": "Q.", + "email": "tqduyet@agu.edu.vn", + "affiliations": [ + { + "value": "An Giang University, Long Xuyen, 880000, Vietnam", + "organization": "An Giang University", + "country": "Vietnam", + }, + { + "value": "Vietnam National University, Ho Chi Minh City, 700000, Vietnam", + "organization": "Vietnam National University", + "country": "Vietnam", + }, + ], + "full_name": "Tran, Q.", + }, { - "surname": "Hue", - "given_names": "L.", - "email": "lethohue@vlu.edu.vn", - "affiliations": [ - { - "value": "Subatomic Physics Research Group, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam", - "organization": "Van Lang University", - "country": "Vietnam" - } - ], - "full_name": "Hue, L." - }, + "surname": "Nguyen", + "given_names": "T.", + "email": "thanhphong@ctu.edu.vn", + "affiliations": [ + { + "value": "Department of Physics, Can Tho University, 3/2 Street, Can Tho, Vietnam", + "organization": "Can Tho University", + "country": "Vietnam", + } + ], + "full_name": "Nguyen, T.", + }, { - "orcid": "0009-0005-5993-6895", - "surname": "Nha", - "given_names": "N.", - "email": "nguyenhuathanhnha@vlu.edu.vn", - "affiliations": [ - { - "value": "Subatomic Physics Research Group, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam", - "organization": "Van Lang University", - "country": "Vietnam" - }, - { - "value": "Faculty of Applied Technology, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam", - "organization": "Van Lang University", - "country": "Vietnam" - } - ], - "full_name": "Nha, N." - }] + "surname": "Hue", + "given_names": "L.", + "email": "lethohue@vlu.edu.vn", + "affiliations": [ + { + "value": "Subatomic Physics Research Group, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam", + "organization": "Van Lang University", + "country": "Vietnam", + } + ], + "full_name": "Hue, L.", + }, + { + "orcid": "0009-0005-5993-6895", + "surname": "Nha", + "given_names": "N.", + "email": "nguyenhuathanhnha@vlu.edu.vn", + "affiliations": [ + { + "value": "Subatomic Physics Research Group, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam", + "organization": "Van Lang University", + "country": "Vietnam", + }, + { + "value": "Faculty of Applied Technology, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam", + "organization": "Van Lang University", + "country": "Vietnam", + }, + ], + "full_name": "Nha, N.", + }, + ] assert expected_output == article_with_orcid["authors"] diff --git a/tests/units/springer/test_parser/s10052-024-12692-y.xml b/tests/units/springer/test_parser/s10052-024-12692-y.xml index c99a7c13..15af4e09 100644 --- a/tests/units/springer/test_parser/s10052-024-12692-y.xml +++ b/tests/units/springer/test_parser/s10052-024-12692-y.xml @@ -490,4 +490,4 @@ - \ No newline at end of file +