From b00af2a1bea68a332bf663ba462fc8757e1792c4 Mon Sep 17 00:00:00 2001 From: zengmao Date: Thu, 24 Oct 2024 23:43:05 +0000 Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20=20@=20d5383?= =?UTF-8?q?10c4644878778c5cd88deae5783995dde5d=20=F0=9F=9A=80?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 404.html | 2 +- amps/index.html | 2 +- css/franklin.css | 2 +- index.html | 2 +- notes/bubble_ibp_grid.svg | 391 ++++++++++++++++++++++++++++++++++++++ notes/ibp/index.html | 4 +- notes/tadpole.svg | 124 ++++++++++++ sitemap.xml | 6 +- 8 files changed, 524 insertions(+), 9 deletions(-) create mode 100644 notes/bubble_ibp_grid.svg create mode 100644 notes/tadpole.svg diff --git a/404.html b/404.html index a62ea9b..7e19678 100644 --- a/404.html +++ b/404.html @@ -1 +1 @@ - 404
Homepage

404


The requested page was not found



Click here to go back to the homepage.
Last modified: October 21, 2024. Mao Zeng.
\ No newline at end of file + 404
Homepage

404


The requested page was not found



Click here to go back to the homepage.
Last modified: October 24, 2024. Mao Zeng.
\ No newline at end of file diff --git a/amps/index.html b/amps/index.html index b9d8117..c64609c 100644 --- a/amps/index.html +++ b/amps/index.html @@ -1 +1 @@ - Higgs Centre Amplitudes Meetings
Homepage

Higgs Centre Amplitudes Meetings

This is a weekly series of informal seminars at the Higgs Centre for Theoretical Physics, Edinburgh University, focusing on scattering amplitudes in quantum field theory and related topics, encouraging open discussions.

Autumn 2024

24 Oct: Xabi Feal

Efficient high-order computation of cusp anomalous dimensions in the string-inspired worldline formalism

17 Oct: Yuyu Mo

Mellin-moment amplitude and onshell bootstrap in AdS

10 Oct: Asaad Elkhidir

Supertranslations from Scattering Amplitudes

03 Oct: Silvia Nagy

Asymptotics for subleading soft theorems at all orders

26 Sep: Dogan Akpinar

Spinning Black Hole Scattering at O(G^3) Using Fixed Spin Theories

Past Meetings

Spring 2024

09 Aug: Zehao Zhu

Calculation of the soft anomalous dimensions with time-like and light-like Wilson lines

06 Aug: Yuyu Mo

Two ways of building the BFKL ladder

30 Jul: Kai Yan, Yang Zhang

(Remote) Two-Loop Spacelike Splitting Amplitude for N=4 Super-Yang-Mills Theory

23 Jul: Luke Lippsteu

Genealogical constraints on Feynman integrals, and infrared finite scattering amplitudes

14 May: Giulio Crisanti

Recent advancements and hidden structures in intersection numbers for Feynman Integrals

07 May: Vasily Sotnikov

Pentagon Functions for Five-Point One-Mass Scattering in QCD

30 Apr: Riccardo Gonzo

Scattering and bound observables for spinning particles in Kerr spacetime

16 Apr: Zeno Capatti

Contour-deforming collider cross-sections

02 Apr: Jiajie Mei

From on-shell Amplitudes to Cosmological Correlators

13 Feb: Carlo Heissenberg

An eikonal-inspired approach to the one-loop gravitational waveform

30 Jan: Sid Smith

Fourier Calculus from Intersection Theory

2022-2023

07 Dec: Giuseppe de Laurentis

Non-Planar Two-Loop Amplitudes for Five-Parton Scattering

23 Nov: Kanghoon Lee

Binary black holes and quantum off-shell recursions

09 Nov: Ratan Sarkar

Isolation of 'Regions' with ASPIRE

26 Oct: Massimiliano Riva

Balance Laws in Gravitational Two-Body Scattering

19 Oct: Sebastian Pogel

Geometry in Feynman Integrals

04 Aug 2023: Canxin Shi

Classical double copy of worldline quantum field theory

30 Jun 2023: Temple He

Classical double copy of worldline quantum field theory

05 May 2023: Miguel Correia

Nonperturbative Anomalous Thresholds

28 Apr 2023: Piotr Bargiela

Three-loop four-particle QCD amplitudes

21 Apr 2023: Sid Smith

Introduction to Intersection Theory for Scattering Amplitudes

07 Apr 2023: Gregor Kalin

Gravitational Scattering at 4th Post-Minkowskian Order

31 Mar 2023: Zehao Zhu

Soft Anomalous Dimension for Multi-leg Processes involving both timelike and lightlike Wilson lines at 3 Loops

03 Feb 2023: Asaad Elkhidir

Large Gauge Effects and the Structure of Amplitudes

01 Dec 2022, Riccardo Gonzo

Bethe-Salpeter equation for classical gravitational bound states

24 Nov 2022, Andrea Cristofoli

Strong field amplitudes and classical physics

17 Nov 2022, Pavel Kovtun

Relativistic dissipation: from hydrodynamics to effective field theory

27 Oct 2022, Yao Ma

A geometric approach to the method of regions

20 Oct 2022, Jae Goode and Sam Teale

Diagrammatic Tensor Reduction

13 Oct 2022, Ingrid Holm

Classical radiation at one loop

29 Sep 2022, Andrea Pelloni

Computing DIS coefficient functions at 4-loop

22 Sep 2022, Eduardo Casali

Celestial holography and twistor strings

2021-2022

15 Jun 2022, Willian Norledge

Lie Theory of (Adjoint) Reflection Hyperplane

08 Jun 2022, Andrew McLeod

A New Amplitude/Form Factor Duality

18 May 2022, Kelian Haring

Gravitational Regge bounds

04 May 2022, Laura Johnson

On-shell Amplitudes for Supersymmetric Massive Gravity

27 Apr 2022, Rashid Alawadhi

The holonomy group and the double copy

20 Apr 2022, Guanda Lin

Color-kinematics duality and double-copy for form factors

30 Mar 2022, Emmet Byrne

The one-loop central emission vertex for two gluons in N=4 super Yang-Mills theory

16 Mar 2022, Christopher White

The case of the missing solutions: a biadjoint mystery

02 Mar 2022, Alasdair Ross

Uncertainty and Classical Amplitudes

23 Feb 2022, Nathan Moynihan

The Cotton Double Copy

16 Feb 2022, Atul Sharma

Advances in twistors and perturbative gravity

09 Feb 2022, Calum Milloy

Disentangling the Regge cut and Regge pole in perturbative QCD

25 Nov 2021, William Emond

Charge Quantization through Wilson Loops and the Double Copy

11 Nov 2021, Ben Page

Ansätze, Amplitudes and Singular Geometry

28 Oct 2021, Gang Chen

HEFT kinematic algebra, heavy double copy and classical gravitational scattering

21 Oct 2021, Leonardo de Cruz

Yang-Mills observables: from KMOC to eikonal through EFT

30 Sep 2021 Zeno Capatti

Local Unitarity

Last modified: October 21, 2024. Mao Zeng.
\ No newline at end of file + Higgs Centre Amplitudes Meetings
Homepage

Higgs Centre Amplitudes Meetings

This is a weekly series of informal seminars at the Higgs Centre for Theoretical Physics, Edinburgh University, focusing on scattering amplitudes in quantum field theory and related topics, encouraging open discussions.

Autumn 2024

24 Oct: Xabi Feal

Efficient high-order computation of cusp anomalous dimensions in the string-inspired worldline formalism

17 Oct: Yuyu Mo

Mellin-moment amplitude and onshell bootstrap in AdS

10 Oct: Asaad Elkhidir

Supertranslations from Scattering Amplitudes

03 Oct: Silvia Nagy

Asymptotics for subleading soft theorems at all orders

26 Sep: Dogan Akpinar

Spinning Black Hole Scattering at O(G^3) Using Fixed Spin Theories

Past Meetings

Spring 2024

09 Aug: Zehao Zhu

Calculation of the soft anomalous dimensions with time-like and light-like Wilson lines

06 Aug: Yuyu Mo

Two ways of building the BFKL ladder

30 Jul: Kai Yan, Yang Zhang

(Remote) Two-Loop Spacelike Splitting Amplitude for N=4 Super-Yang-Mills Theory

23 Jul: Luke Lippsteu

Genealogical constraints on Feynman integrals, and infrared finite scattering amplitudes

14 May: Giulio Crisanti

Recent advancements and hidden structures in intersection numbers for Feynman Integrals

07 May: Vasily Sotnikov

Pentagon Functions for Five-Point One-Mass Scattering in QCD

30 Apr: Riccardo Gonzo

Scattering and bound observables for spinning particles in Kerr spacetime

16 Apr: Zeno Capatti

Contour-deforming collider cross-sections

02 Apr: Jiajie Mei

From on-shell Amplitudes to Cosmological Correlators

13 Feb: Carlo Heissenberg

An eikonal-inspired approach to the one-loop gravitational waveform

30 Jan: Sid Smith

Fourier Calculus from Intersection Theory

2022-2023

07 Dec: Giuseppe de Laurentis

Non-Planar Two-Loop Amplitudes for Five-Parton Scattering

23 Nov: Kanghoon Lee

Binary black holes and quantum off-shell recursions

09 Nov: Ratan Sarkar

Isolation of 'Regions' with ASPIRE

26 Oct: Massimiliano Riva

Balance Laws in Gravitational Two-Body Scattering

19 Oct: Sebastian Pogel

Geometry in Feynman Integrals

04 Aug 2023: Canxin Shi

Classical double copy of worldline quantum field theory

30 Jun 2023: Temple He

Classical double copy of worldline quantum field theory

05 May 2023: Miguel Correia

Nonperturbative Anomalous Thresholds

28 Apr 2023: Piotr Bargiela

Three-loop four-particle QCD amplitudes

21 Apr 2023: Sid Smith

Introduction to Intersection Theory for Scattering Amplitudes

07 Apr 2023: Gregor Kalin

Gravitational Scattering at 4th Post-Minkowskian Order

31 Mar 2023: Zehao Zhu

Soft Anomalous Dimension for Multi-leg Processes involving both timelike and lightlike Wilson lines at 3 Loops

03 Feb 2023: Asaad Elkhidir

Large Gauge Effects and the Structure of Amplitudes

01 Dec 2022, Riccardo Gonzo

Bethe-Salpeter equation for classical gravitational bound states

24 Nov 2022, Andrea Cristofoli

Strong field amplitudes and classical physics

17 Nov 2022, Pavel Kovtun

Relativistic dissipation: from hydrodynamics to effective field theory

27 Oct 2022, Yao Ma

A geometric approach to the method of regions

20 Oct 2022, Jae Goode and Sam Teale

Diagrammatic Tensor Reduction

13 Oct 2022, Ingrid Holm

Classical radiation at one loop

29 Sep 2022, Andrea Pelloni

Computing DIS coefficient functions at 4-loop

22 Sep 2022, Eduardo Casali

Celestial holography and twistor strings

2021-2022

15 Jun 2022, Willian Norledge

Lie Theory of (Adjoint) Reflection Hyperplane

08 Jun 2022, Andrew McLeod

A New Amplitude/Form Factor Duality

18 May 2022, Kelian Haring

Gravitational Regge bounds

04 May 2022, Laura Johnson

On-shell Amplitudes for Supersymmetric Massive Gravity

27 Apr 2022, Rashid Alawadhi

The holonomy group and the double copy

20 Apr 2022, Guanda Lin

Color-kinematics duality and double-copy for form factors

30 Mar 2022, Emmet Byrne

The one-loop central emission vertex for two gluons in N=4 super Yang-Mills theory

16 Mar 2022, Christopher White

The case of the missing solutions: a biadjoint mystery

02 Mar 2022, Alasdair Ross

Uncertainty and Classical Amplitudes

23 Feb 2022, Nathan Moynihan

The Cotton Double Copy

16 Feb 2022, Atul Sharma

Advances in twistors and perturbative gravity

09 Feb 2022, Calum Milloy

Disentangling the Regge cut and Regge pole in perturbative QCD

25 Nov 2021, William Emond

Charge Quantization through Wilson Loops and the Double Copy

11 Nov 2021, Ben Page

Ansätze, Amplitudes and Singular Geometry

28 Oct 2021, Gang Chen

HEFT kinematic algebra, heavy double copy and classical gravitational scattering

21 Oct 2021, Leonardo de Cruz

Yang-Mills observables: from KMOC to eikonal through EFT

30 Sep 2021 Zeno Capatti

Local Unitarity

Last modified: October 24, 2024. Mao Zeng.
\ No newline at end of file diff --git a/css/franklin.css b/css/franklin.css index 685297a..68f87af 100644 --- a/css/franklin.css +++ b/css/franklin.css @@ -1 +1 @@ -@charset "utf-8"; :root{--block-background:hsl(0,0%,94%);--output-background:hsl(0,0%,98%);--small:12px;--normal:15px;--text-color:hsv(0,0%,20%)}html{font-family:Helvetica,Arial,sans-serif;font-size:var(--normal);color:var(--text-color)}.franklin-content .row{display:block}.franklin-content .left{float:left;margin-right:15px}.franklin-content .right{float:right}.franklin-content .container img{width:auto;padding-left:0;border-radius:10px}.franklin-content .footnote{position:relative;top:-0.5em;font-size:70%}.franklin-content .page-foot a{text-decoration:none;color:#a6a2a0;text-decoration:underline}.page-foot{font-size:80%;font-family:Arial,serif;color:#a6a2a0;text-align:center;margin-top:6em;border-top:1px solid lightgrey;padding-top:2em;margin-bottom:4em}.franklin-toc li{margin:.6rem 0}.franklin-content{position:relative;padding-left:12.5%;padding-right:12.5%;line-height:1.35em}@media (min-width:940px){.franklin-content{width:705px;margin-left:auto;margin-right:auto}}@media (max-width:480px){.franklin-content{padding-left:6%;padding-right:6%}}.franklin-content h1{font-size:20px}.franklin-content h2{font-size:18px}.franklin-content h3{font-size:16px}.franklin-content h1,h2,h3,h4,h5,h6{text-align:left}.franklin-content h1{padding-bottom:.5em;border-bottom:3px double lightgrey;margin-top:1.5em;margin-bottom:1em}.franklin-content h2{padding-bottom:.3em;border-bottom:1px solid lightgrey;margin-top:2em;margin-bottom:1em}.franklin-content h1 a{color:inherit}.franklin-content h1 a:hover{text-decoration:none}.franklin-content h2 a{color:inherit}.franklin-content h2 a:hover{text-decoration:none}.franklin-content h3 a{color:inherit}.franklin-content h3 a:hover{text-decoration:none}.franklin-content h4 a{color:inherit}.franklin-content h4 a:hover{text-decoration:none}.franklin-content h5 a{color:inherit}.franklin-content h5 a:hover{text-decoration:none}.franklin-content h6 a{color:inherit}.franklin-content h6 a:hover{text-decoration:none}.franklin-content table{margin-left:auto;margin-right:auto;border-collapse:collapse;text-align:center}.franklin-toc ol ol{list-style-type:lower-alpha}.franklin-content th,td{font-size:var(--small);padding:10px;border:1px solid black}.franklin-content blockquote{background:var(--block-background);border-left:7px solid #a8a8a8;margin:1.5em 10px;padding:.5em 10px;font-style:italic}.franklin-content blockquote p{display:inline}.franklin-content li p{margin:10px 0}.franklin-content a{color:#004de6;text-decoration:none}.franklin-content a:hover{text-decoration:underline}.franklin-content .eqref a{color:green}.franklin-content .bibref a{color:green}.franklin-content sup{font-size:70%;vertical-align:super;line-height:0}.franklin-content table.fndef{margin:0;margin-bottom:10px}.franklin-content .fndef tr,td{padding:0;border:0;text-align:left}.franklin-content .fndef tr{border-left:2px solid lightgray}.franklin-content .fndef td.fndef-backref{vertical-align:top;font-size:70%;padding-left:5px}.franklin-content .fndef td.fndef-content{font-size:80%;padding-left:10px;width:100%}.franklin-content img{width:70%;text-align:center;padding-left:10%}.franklin-content .img-small img{width:50%;text-align:center;padding-left:20%}.franklin-content .img-tiny img{width:30%;text-align:center;padding-left:30%}body{counter-reset:eqnum}.katex{font-size:1.2em !important}.katex-display .katex{display:inline-block;white-space:normal}.katex-display::after{counter-increment:eqnum;content:"(" counter(eqnum) ")";position:relative;float:right;padding-right:5px}.nonumber .katex-display::after{counter-increment:nothing;content:""}code{background-color:var(--block-background);padding:.1em .2em;border-radius:2px;font-size:var(--small)}.hljs{font-size:var(--small);line-height:1.35em;border-radius:10px}.hljs-meta{font-weight:bold}.hljs-meta.shell_{color:crimson}.hljs-meta.prompt_{color:#19b333}.code-output{background:var(--output-background);border:1px dashed #dbdbdb}.franklin-content .colbox-blue{background-color:#eef3f5;padding-top:5px;padding-right:10px;padding-left:10px;padding-bottom:5px;margin-left:5px;margin-top:5px;margin-bottom:5px;border-radius:0 10px 10px 0;border-left:5px solid #4c9cf1} \ No newline at end of file +@charset "utf-8"; :root{--block-background:hsl(0,0%,94%);--output-background:hsl(0,0%,98%);--small:12px;--normal:15px;--text-color:hsv(0,0%,20%)}html{font-family:Helvetica,Arial,sans-serif;font-size:var(--normal);color:var(--text-color)}.franklin-content .row{display:block}.franklin-content .left{float:left;margin-right:15px}.franklin-content .right{float:right}.franklin-content .container img{width:auto;padding-left:0;border-radius:10px}.franklin-content .footnote{position:relative;top:-0.5em;font-size:70%}.franklin-content .page-foot a{text-decoration:none;color:#a6a2a0;text-decoration:underline}.page-foot{font-size:80%;font-family:Arial,serif;color:#a6a2a0;text-align:center;margin-top:6em;border-top:1px solid lightgrey;padding-top:2em;margin-bottom:4em}.franklin-toc li{margin:.6rem 0}.franklin-content{position:relative;padding-left:12.5%;padding-right:12.5%;line-height:1.35em}@media (min-width:940px){.franklin-content{width:705px;margin-left:auto;margin-right:auto}}@media (max-width:480px){.franklin-content{padding-left:6%;padding-right:6%}}.franklin-content h1{font-size:20px}.franklin-content h2{font-size:18px}.franklin-content h3{font-size:16px}.franklin-content h1,h2,h3,h4,h5,h6{text-align:left}.franklin-content h1{padding-bottom:.5em;border-bottom:3px double lightgrey;margin-top:1.5em;margin-bottom:1em}.franklin-content h2{padding-bottom:.3em;border-bottom:1px solid lightgrey;margin-top:2em;margin-bottom:1em}.franklin-content h1 a{color:inherit}.franklin-content h1 a:hover{text-decoration:none}.franklin-content h2 a{color:inherit}.franklin-content h2 a:hover{text-decoration:none}.franklin-content h3 a{color:inherit}.franklin-content h3 a:hover{text-decoration:none}.franklin-content h4 a{color:inherit}.franklin-content h4 a:hover{text-decoration:none}.franklin-content h5 a{color:inherit}.franklin-content h5 a:hover{text-decoration:none}.franklin-content h6 a{color:inherit}.franklin-content h6 a:hover{text-decoration:none}.franklin-content table{margin-left:auto;margin-right:auto;border-collapse:collapse;text-align:center}.franklin-toc ol ol{list-style-type:lower-alpha}.franklin-content th,td{font-size:var(--small);padding:10px;border:1px solid black}.franklin-content blockquote{background:var(--block-background);border-left:7px solid #a8a8a8;margin:1.5em 10px;padding:.5em 10px;font-style:italic}.franklin-content blockquote p{display:inline}.franklin-content li p{margin:10px 0}.franklin-content a{color:#004de6;text-decoration:none}.franklin-content a:hover{text-decoration:underline}.franklin-content .eqref a{color:green}.franklin-content .bibref a{color:green}.franklin-content sup{font-size:70%;vertical-align:super;line-height:0}.franklin-content table.fndef{margin:0;margin-bottom:10px}.franklin-content .fndef tr,td{padding:0;border:0;text-align:left}.franklin-content .fndef tr{border-left:2px solid lightgray}.franklin-content .fndef td.fndef-backref{vertical-align:top;font-size:70%;padding-left:5px}.franklin-content .fndef td.fndef-content{font-size:80%;padding-left:10px;width:100%}.franklin-content img{width:70%;text-align:center;padding-left:10%}.franklin-content .img-small img{width:50%;text-align:center;padding-left:20%}.franklin-content .img-tiny img{width:30%;text-align:center;padding-left:30%}.franklin-content .img-moretiny img{width:14%;text-align:center;padding-left:38%}body{counter-reset:eqnum}.katex{font-size:1.2em !important}.katex-display .katex{display:inline-block;white-space:normal}.katex-display::after{counter-increment:eqnum;content:"(" counter(eqnum) ")";position:relative;float:right;padding-right:5px}.nonumber .katex-display::after{counter-increment:nothing;content:""}code{background-color:var(--block-background);padding:.1em .2em;border-radius:2px;font-size:var(--small)}.hljs{font-size:var(--small);line-height:1.35em;border-radius:10px}.hljs-meta{font-weight:bold}.hljs-meta.shell_{color:crimson}.hljs-meta.prompt_{color:#19b333}.code-output{background:var(--output-background);border:1px dashed #dbdbdb}.franklin-content .colbox-blue{background-color:#eef3f5;padding-top:5px;padding-right:10px;padding-left:10px;padding-bottom:5px;margin-left:5px;margin-top:5px;margin-bottom:5px;border-radius:0 10px 10px 0;border-left:5px solid #4c9cf1} \ No newline at end of file diff --git a/index.html b/index.html index 34ba395..8100df9 100644 --- a/index.html +++ b/index.html @@ -1 +1 @@ - Mao Zeng, University of Edinburgh
Homepage

Mao Zeng - Theoretical Physics, University of Edinburgh

Portrait

Contact: mao.zeng <A.T.> ed.ac.uk

I am a Royal Society University Research Fellow at the Higgs Centre for Theoretical Physics, University of Edinburgh. I obtained my BA and MA from Cambridge University, U.K., and PhD from Stony Brook University, USA, under the advice of Prof. George Sterman. I previously held research positions at UC Los Angeles (2015-2018), ETH Zürich (2018-2020), and Oxford University (2021).

My research uses quantum field theory to make precise predictions for fundamental physics experiments. In the last few years, my collaborators and I developed a formalism for computing classical general relativity obesrvables, relevant for LIGO / VIRGO observations, by taking classical limits of scattering amplitudes in quantum field theory. I am also involved in precise calculations relevant for the Large Hadron Collider, in perturbative QCD and sometimes in simplifying supersymmetric theories.

My research brings together a wide array of theoretical developments in high energy physics, including modern methods for scattering amplitudes, state-of-the-art loop integration techniques, and effective field theories.

Full CV: PDF

Links to my publications: INSPIRE | arXiv

Recent Conferences Organized

I co-organized the Higgs Centre Workshop on Gravitational Self-Force and Scattering Amplitudes held in March 2024.

I'm co-organizing the 2024 MIAPbP workshop, EFT and Multi-loop Methods for Advancing Precision in Collider and Gravitational Wave Physics to be held in October 2024.

I'm co-organizing the weekly Higgs Centre Amplitudes Meetings.

Last modified: October 21, 2024. Mao Zeng.
\ No newline at end of file + Mao Zeng, University of Edinburgh
Homepage

Mao Zeng - Theoretical Physics, University of Edinburgh

Portrait

Contact: mao.zeng <A.T.> ed.ac.uk

I am a Royal Society University Research Fellow at the Higgs Centre for Theoretical Physics, University of Edinburgh. I obtained my BA and MA from Cambridge University, U.K., and PhD from Stony Brook University, USA, under the advice of Prof. George Sterman. I previously held research positions at UC Los Angeles (2015-2018), ETH Zürich (2018-2020), and Oxford University (2021).

My research uses quantum field theory to make precise predictions for fundamental physics experiments. In the last few years, my collaborators and I developed a formalism for computing classical general relativity obesrvables, relevant for LIGO / VIRGO observations, by taking classical limits of scattering amplitudes in quantum field theory. I am also involved in precise calculations relevant for the Large Hadron Collider, in perturbative QCD and sometimes in simplifying supersymmetric theories.

My research brings together a wide array of theoretical developments in high energy physics, including modern methods for scattering amplitudes, state-of-the-art loop integration techniques, and effective field theories.

Full CV: PDF

Links to my publications: INSPIRE | arXiv

Recent Conferences Organized

I co-organized the Higgs Centre Workshop on Gravitational Self-Force and Scattering Amplitudes held in March 2024.

I'm co-organizing the 2024 MIAPbP workshop, EFT and Multi-loop Methods for Advancing Precision in Collider and Gravitational Wave Physics to be held in October 2024.

I'm co-organizing the weekly Higgs Centre Amplitudes Meetings.

Last modified: October 24, 2024. Mao Zeng.
\ No newline at end of file diff --git a/notes/bubble_ibp_grid.svg b/notes/bubble_ibp_grid.svg new file mode 100644 index 0000000..d37fc3e --- /dev/null +++ b/notes/bubble_ibp_grid.svg @@ -0,0 +1,391 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + highercomplexity + lowercomplexity + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/notes/ibp/index.html b/notes/ibp/index.html index be98531..5c9a032 100644 --- a/notes/ibp/index.html +++ b/notes/ibp/index.html @@ -1,10 +1,10 @@ - Scattering Amplitudes Lecture on 07 Nov: Integration-by-Parts Reduction
Homepage

Scattering Amplitudes Lecture on 07 Nov: Integration-by-Parts Reduction

Toy example: IBP for tadpole integrals

To be completed... Compare with analytic formula.

Basic setup for bubble example

Let's look at the bubble family of integrals,

one-loop bubble
Iν1,ν2=∫ddq1(q2−m2)ν1[(p+q)2−m2]ν2=∫ddq1ρ1ν1 ρ2ν2 ,\begin{aligned} I_{\nu_1, \nu_2} &= \int d^d q \frac{1} {(q^2-m^2)^{\nu_1} [(p+q)^2-m^2]^{\nu_2}} \\ &= \int d^d q \frac 1 {\rho_1^{\nu_1} \, \rho_2^{\nu_2}} \, , \end{aligned}

where we defined

ρ1=q2−m2,ρ2=(p+q)2−m2 . \rho_1 = q^2 - m^2, \quad \rho_2 = (p+q)^2-m^2 \, .

We can invert the above equation to write any dot product involving the loop momentum qq in terms of inverse propagator variables,

q2=ρ1+m2,p⋅q=12(ρ2−ρ1−p2) . q^2 = \rho_1 + m^2, \quad p \cdot q = \frac 1 2 (\rho_2 - \rho_1 - p^2) \, .

We can write down the IBP identity

0=∫ddq∂∂qμqμρ1ν1ρ2ν2=∫ddq[dρ1ν1ρ2ν2−ν1ρ1ν1+1ρ2ν2 qμ∂∂qμρ1−ν2ρ1ν1ρ2ν2+1 qμ∂∂qμρ2]\begin{aligned} 0 &= \int d^d q \frac{\partial}{\partial q^\mu} {\frac{q^\mu}{\rho_1^{\nu_1} \rho_2^{\nu_2}}} \\ &= \int d^d q \left[ \frac d {\rho_1^{\nu_1} \rho_2^{\nu_2}} - \frac {\nu_1} {\rho_1^{\nu_1+1} \rho_2^{\nu_2}} \, q^\mu \frac{\partial}{\partial q^\mu} \rho_1 - \frac {\nu_2} {\rho_1^{\nu_1} \rho_2^{\nu_2+1}} \, q^\mu \frac{\partial}{\partial q^\mu} \rho_2 \right] \end{aligned}

Using Eqs. (2) and (3), we have

qμ∂∂qμρ1=2q2=2ρ1+2m2qμ∂∂qμρ2=2p⋅q+2q2=ρ1+ρ2+2m2−p2 .\begin{aligned} q^\mu \frac{\partial}{\partial q^\mu} \rho_1 &= 2 q^2 = 2 \rho_1 + 2m^2 \\ q^\mu \frac{\partial}{\partial q^\mu} \rho_2 &= 2 p \cdot q + 2 q^2 = \rho_1 + \rho_2 + 2m^2 - p^2 \, . \end{aligned}

The IBP identity in the 2nd line of Eq. (4) evaluates to

0=(d−2ν1−ν2)Iν1,ν2−2ν1m2Iν1+1,ν2−ν2Iν1−1,ν2+1−ν2(2m2−p2)Iν1,ν2+1\begin{aligned} 0 = (d-2\nu_1-\nu_2) I_{\nu_1, \nu_2} - 2\nu_1 m^2 I_{\nu_1+1, \nu_2} - \nu_2 I_{\nu_1-1, \nu_2+1} - \nu_2 (2m^2-p^2) I_{\nu_1, \nu_2+1} \end{aligned}

Simple application to differential equations

Substituting ν1=1,ν2=0\nu_1=1, \nu_2=0 in Eq. (6) gives

0=(d−2)I1,0−2m2I2,0 ⟹ I2,0=12m2(d−2)I1,0 0 = (d-2) I_{1,0} - 2m^2 I_{2,0} \implies I_{2,0} = \frac{1}{2m^2} (d-2) I_{1,0}

Substituting ν1=ν2=1\nu_1=\nu_2=1 and using symmetry relations I1,2=I2,1, I0,2=I2,0I_{1,2}=I_{2,1}, \, I_{0,2}=I_{2,0} gives

0=(d−3)I1,1−(4m2−p2)I1,2−I0,2 . 0 = (d-3) I_{1,1} - (4m^2-p^2) I_{1,2} - I_{0,2} \, .

Combined with Eq. (7), we obtain

I1,2=d−34m2−p2I1,1−d−22m2(4m2−p2)I1,0 . I_{1,2} = \frac{d-3}{4m^2-p^2} I_{1,1} - \frac{d-2}{2m^2(4m^2-p^2)} I_{1,0} \, .

By directly differentiating w.r.t. m2m^2 under the integral sign in Eq. (1), we obtain differential equations for the unknown integrals (I1,0,I1,1)(I_{1,0}, I_{1,1}),

∂∂m2(I1,1I1,0)=(2I1,2I2,0)=(2(d−3)4m2−p2−d−2m2(4m2−p2)0d−22m2)⋅(I1,1I1,0)\begin{aligned} \frac {\partial}{\partial m^2} \begin{pmatrix} I_{1,1} \\ I_{1,0} \end{pmatrix} = \begin{pmatrix} 2 I_{1,2} \\ I_{2,0} \end{pmatrix} = \begin{pmatrix} \frac{2(d-3)}{4m^2-p^2} & - \frac{d-2}{m^2(4m^2-p^2)} \\ 0 & \frac{d-2}{2m^2} \end{pmatrix} \cdot \begin{pmatrix} I_{1,1} \\ I_{1,0} \end{pmatrix} \end{aligned}

Systematic reduction of all possible bubble integrals

To be completed...

Using FIRE software to automate IBP reduction

You need to have Wolfram Mathematica installed.

On Linux or Mac, install FIRE 6.5 with the following command in the terminal:

git clone https://gitlab.com/feynmanIntegrals/fire.git
+ Scattering Amplitudes Lecture on 07 Nov: Integration-by-Parts Reduction
Homepage

Scattering Amplitudes Lecture on 07 Nov: Integration-by-Parts Reduction

Toy example: IBP for tadpole integrals

Let's look at the simplest toy example, the family of one-loop tadpole integrals with internal mass mm,

one-loop bubble
Iν=∫ddqiπd/21(q2−m2)ν=∫ddqiπd/21ρν ,\begin{aligned} I_{\nu} &= \int \frac {d^d q}{i \pi^{d/2}} \frac{1} {(q^2-m^2)^{\nu}} \\ &= \int \frac {d^d q}{i \pi^{d/2}} \frac 1 {\rho^{\nu}} \, , \end{aligned}

where we defined

ρ=q2−m2 . \rho = q^2 - m^2 \, .

The IBP identity is

0=∫ddqiπd/2∂∂qμqμρν=∫ddqiπd/2[dρν−νρν+1qμ∂∂qμρ] .\begin{aligned} 0 &= \int \frac {d^d q}{i \pi^{d/2}} \frac{\partial}{\partial q^\mu} \frac{q^\mu}{\rho^{\nu} } &= \int \frac {d^d q}{i \pi^{d/2}} \left[ \frac {d} {\rho^\nu} - \frac {\nu} {\rho^{\nu+1}} q^\mu \frac{\partial}{\partial q^\mu} \rho \right] \, . \end{aligned}

We use

qμ∂∂qμρ=2q2=2(ρ+m2) . q^\mu \frac{\partial}{\partial q^\mu} \rho = 2q^2 = 2 (\rho + m^2) \, .

The IBP identity becomes

0=∫ddqiπd/2[d−2νρν−2m2νρν+1]=(d−2ν)Iν−2m2νIν+1 . 0 = \int \frac {d^d q}{i \pi^{d/2}} \left[ \frac {d - 2 \nu} {\rho^\nu} - \frac{2 m^2 \nu}{\rho^{\nu+1}} \right] = (d-2\nu) I_\nu - 2m^2 \nu I_{\nu+1} \, .

The exact result with d=4−2ϵd=4-2\epsilon is

Iν=(−1)νΓ(ν−d/2)Γ(ν)1(m2)ν−d/2 . I_\nu = (-1)^\nu \frac {\Gamma(\nu-d/2)} {\Gamma(\nu)} \frac 1 {(m^2)^{\nu-d/2}} \, .

The IBP identity agrees with the exact result using the gamma function relations Γ(ν+1)=νΓ(ν)\Gamma(\nu+1) = \nu \Gamma(\nu), Γ(ν+1−d/2)=(ν−d/2)Γ(ν−d/2)\Gamma(\nu+1-d/2) = (\nu-d/2) \Gamma(\nu-d/2).

Basic setup for bubble example

Let's look at the bubble family of integrals,

one-loop bubble
Iν1,ν2=∫ddqiπd/21(q2−m2)ν1[(p+q)2−m2]ν2=∫ddqiπd/21ρ1ν1 ρ2ν2 ,\begin{aligned} I_{\nu_1, \nu_2} &= \int \frac {d^d q}{i \pi^{d/2}} \frac{1} {(q^2-m^2)^{\nu_1} [(p+q)^2-m^2]^{\nu_2}} \\ &= \int \frac {d^d q}{i \pi^{d/2}} \frac 1 {\rho_1^{\nu_1} \, \rho_2^{\nu_2}} \, , \end{aligned}

where we defined

ρ1=q2−m2,ρ2=(p+q)2−m2 . \rho_1 = q^2 - m^2, \quad \rho_2 = (p+q)^2-m^2 \, .

We can invert the above equation to write any dot product involving the loop momentum qq in terms of inverse propagator variables,

q2=ρ1+m2,p⋅q=12(ρ2−ρ1−p2) . q^2 = \rho_1 + m^2, \quad p \cdot q = \frac 1 2 (\rho_2 - \rho_1 - p^2) \, .

We can write down the IBP identity

0=∫ddqiπd/2∂∂qμqμρ1ν1ρ2ν2=∫ddqiπd/2[dρ1ν1ρ2ν2−ν1ρ1ν1+1ρ2ν2 qμ∂∂qμρ1−ν2ρ1ν1ρ2ν2+1 qμ∂∂qμρ2]\begin{aligned} 0 &= \int \frac {d^d q}{i \pi^{d/2}} \frac{\partial}{\partial q^\mu} {\frac{q^\mu}{\rho_1^{\nu_1} \rho_2^{\nu_2}}} \\ &= \int \frac {d^d q}{i \pi^{d/2}} \left[ \frac d {\rho_1^{\nu_1} \rho_2^{\nu_2}} - \frac {\nu_1} {\rho_1^{\nu_1+1} \rho_2^{\nu_2}} \, q^\mu \frac{\partial}{\partial q^\mu} \rho_1 - \frac {\nu_2} {\rho_1^{\nu_1} \rho_2^{\nu_2+1}} \, q^\mu \frac{\partial}{\partial q^\mu} \rho_2 \right] \end{aligned}

Using Eqs. (8) and (9), we have

qμ∂∂qμρ1=2q2=2ρ1+2m2qμ∂∂qμρ2=2p⋅q+2q2=ρ1+ρ2+2m2−p2 .\begin{aligned} q^\mu \frac{\partial}{\partial q^\mu} \rho_1 &= 2 q^2 = 2 \rho_1 + 2m^2 \\ q^\mu \frac{\partial}{\partial q^\mu} \rho_2 &= 2 p \cdot q + 2 q^2 = \rho_1 + \rho_2 + 2m^2 - p^2 \, . \end{aligned}

The IBP identity in the 2nd line of Eq. (4) evaluates to

0=(d−2ν1−ν2)Iν1,ν2−2ν1m2Iν1+1,ν2−ν2Iν1−1,ν2+1−ν2(2m2−p2)Iν1,ν2+1\begin{aligned} 0 = (d-2\nu_1-\nu_2) I_{\nu_1, \nu_2} - 2\nu_1 m^2 I_{\nu_1+1, \nu_2} - \nu_2 I_{\nu_1-1, \nu_2+1} - \nu_2 (2m^2-p^2) I_{\nu_1, \nu_2+1} \end{aligned}

Simple application to differential equations

Substituting ν1=1,ν2=0\nu_1=1, \nu_2=0 in Eq. (6) gives

0=(d−2)I1,0−2m2I2,0 ⟹ I2,0=12m2(d−2)I1,0 0 = (d-2) I_{1,0} - 2m^2 I_{2,0} \implies I_{2,0} = \frac{1}{2m^2} (d-2) I_{1,0}

Substituting ν1=ν2=1\nu_1=\nu_2=1 and using symmetry relations I1,2=I2,1, I0,2=I2,0I_{1,2}=I_{2,1}, \, I_{0,2}=I_{2,0} gives

0=(d−3)I1,1−(4m2−p2)I1,2−I0,2 . 0 = (d-3) I_{1,1} - (4m^2-p^2) I_{1,2} - I_{0,2} \, .

Combined with Eq. (13), we obtain

I1,2=d−34m2−p2I1,1−d−22m2(4m2−p2)I1,0 . I_{1,2} = \frac{d-3}{4m^2-p^2} I_{1,1} - \frac{d-2}{2m^2(4m^2-p^2)} I_{1,0} \, .

By directly differentiating w.r.t. m2m^2 under the integral sign in Eq. (7), we obtain differential equations for the unknown integrals (I1,0,I1,1)(I_{1,0}, I_{1,1}),

∂∂m2(I1,1I1,0)=(2I1,2I2,0)=(2(d−3)4m2−p2−d−2m2(4m2−p2)0d−22m2)⋅(I1,1I1,0)\begin{aligned} \frac {\partial}{\partial m^2} \begin{pmatrix} I_{1,1} \\ I_{1,0} \end{pmatrix} = \begin{pmatrix} 2 I_{1,2} \\ I_{2,0} \end{pmatrix} = \begin{pmatrix} \frac{2(d-3)}{4m^2-p^2} & - \frac{d-2}{m^2(4m^2-p^2)} \\ 0 & \frac{d-2}{2m^2} \end{pmatrix} \cdot \begin{pmatrix} I_{1,1} \\ I_{1,0} \end{pmatrix} \end{aligned}

The differential equations can be solved with appropriate boundary conditions, as will be covered by future lectures.

Systematic reduction of all possible bubble integrals

Here we present a systematic method for solving IBP relations bubble integral. The method will be applicable to many other Feynman integrals. We will reduce bubble integrals Iν1,ν2I_{\nu_1, \nu_2} with any integer indices ν1,ν2\nu_1, \nu_2 without relying on symmetry relations. All we need is one more IBP identity besides Eq. (10), by changing qμq^\mu in the numerator to pμp^\mu. (For a general integral family, we allow the numerator to be any of the independent external and loop momenta.) The calculation steps are exactly analogous to those leading to Eq. (12), and we just quote the final result which is the counterpart of Eq. (12),

0=(ν1−ν2)Iν1,ν2−ν1Iν1+1,ν2−1+ν2Iν1−1,ν2+1+ν1p2Iν1+1,ν2−ν2p2Iν1,ν2+1 . 0 = (\nu_1-\nu_2) I_{\nu_1, \nu_2} - \nu_1 I_{\nu_1+1, \nu_2-1} + \nu_2 I_{\nu_1-1, \nu_2+1} + \nu_1 p^2 I_{\nu_1+1, \nu_2} - \nu_2 p^2 I_{\nu_1, \nu_2+1} \, .

Eq. (12) and Eq. (17) involve a total of 5 integrals, for every choice of integers (ν1,ν2)(\nu_1, \nu_2). The integrals involved in the two linear relations are visualized on the (ν1,ν2)(\nu_1, \nu_2) lattice,

bubble_ibp_grid

For the sake of solving the system, we define an auxiliary quantity, the complexity of an integral Iν1,ν2I_{\nu_1, \nu_2}, to be ν1+ν2\nu_1+\nu_2. We want to solve more "complex" integrals in terms of less complex ones. Two of the integrals have the same complexity. Fortunately, we have two IBP identities Eq. (12) and (17) which allow us to solve both integrals in terms of the three less complex integrals - the 2×22\times 2 linear system is non-degenerate if ν1≥1,ν2≥1\nu_1 \geq 1, \nu_2 \geq 1. The results are

Iν1+1,ν2=1p2(4m2−p2)[(2m2−p2)Iν1+1,ν2−1−2ν2ν1m2Iν1−1,ν2+1+dp2−ν1(2m2+p2)+2ν2(m2−p2)ν1Iν1,ν2] ,Iν1,ν2+1=1p2(4m2−p2)[(2m2−p2)Iν1−1,ν2+1−2ν1ν2m2Iν1+1,ν2−1+dp2−ν2(2m2+p2)+2ν1(m2−p2)ν2Iν1,ν2] .\begin{aligned} I_{\nu_1 + 1, \nu_2} = \frac{1}{p^2 (4m^2-p^2)} & \bigg[ (2m^2-p^2) I_{\nu_1+1, \nu_2-1} - 2\frac{\nu_2}{\nu_1} m^2 I_{\nu_1-1, \nu_2+1} \\ & \quad + \frac {dp^2 - \nu_1(2m^2+p^2) + 2\nu_2(m^2-p^2)}{\nu_1} I_{\nu_1, \nu_2} \bigg] \, , \\ I_{\nu_1, \nu_2+1} = \frac{1}{p^2 (4m^2-p^2)} & \bigg[ (2m^2-p^2) I_{\nu_1-1, \nu_2+1} - 2\frac{\nu_1}{\nu_2} m^2 I_{\nu_1+1, \nu_2-1} \\ & \quad + \frac {dp^2 - \nu_2(2m^2+p^2) + 2\nu_1(m^2-p^2)}{\nu_2} I_{\nu_1, \nu_2} \bigg] \, . \end{aligned}

Applying these reduction formulas iteratively, we reduce the complexity of the integrals until all integral are reduced to I1,1I_{1,1}, I2,0I_{2,0} and I0,2I_{0,2}.

Using FIRE software to automate IBP reduction

You need to have Wolfram Mathematica installed.

On Linux or Mac, install FIRE 6.5 with the following command in the terminal:

git clone https://gitlab.com/feynmanIntegrals/fire.git

We'll skip the installation of the C++ version of FIRE, since we'll only use the Mathematica version here.

Navigate into the examples/ folder:

cd fire/FIRE6/examples

and open the example notebook box.nb. You only need to look at the first half of the notebook, leading to the reduction result for F[1, {2, 2, 2, 2}], i.e. the box integral with all four propagators raised to a 2nd power.

- Last modified: October 21, 2024. Mao Zeng. + Last modified: October 24, 2024. Mao Zeng.
diff --git a/notes/tadpole.svg b/notes/tadpole.svg new file mode 100644 index 0000000..9bca810 --- /dev/null +++ b/notes/tadpole.svg @@ -0,0 +1,124 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/sitemap.xml b/sitemap.xml index 0c022c4..3a14641 100644 --- a/sitemap.xml +++ b/sitemap.xml @@ -3,19 +3,19 @@ https://tlienart.github.io/FranklinTemplates.jl/amps/index.html - 2024-10-21 + 2024-10-24 monthly 0.5 https://tlienart.github.io/FranklinTemplates.jl/notes/ibp/index.html - 2024-10-21 + 2024-10-24 monthly 0.5 https://tlienart.github.io/FranklinTemplates.jl/index.html - 2024-10-21 + 2024-10-24 monthly 0.5