From d538310c4644878778c5cd88deae5783995dde5d Mon Sep 17 00:00:00 2001 From: Mao Zeng Date: Fri, 25 Oct 2024 00:41:16 +0100 Subject: [PATCH] Add tadpole example and bubble systematic solution to IBP notes. --- _css/franklin.css | 6 + notes/bubble_ibp_grid.svg | 391 ++++++++++++++++++++++++++++++++++++++ notes/ibp.md | 67 ++++++- notes/tadpole.svg | 124 ++++++++++++ 4 files changed, 579 insertions(+), 9 deletions(-) create mode 100644 notes/bubble_ibp_grid.svg create mode 100644 notes/tadpole.svg diff --git a/_css/franklin.css b/_css/franklin.css index 651ac0d..52ece63 100644 --- a/_css/franklin.css +++ b/_css/franklin.css @@ -258,6 +258,12 @@ td { padding-left: 30%; } +.franklin-content .img-moretiny img { + width: 14%; + text-align: center; + padding-left: 38%; +} + /* ================================================================== KATEX ================================================================== */ 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.md b/notes/ibp.md index ee64830..502b809 100644 --- a/notes/ibp.md +++ b/notes/ibp.md @@ -1,14 +1,41 @@ # Scattering Amplitudes Lecture on 07 Nov: Integration-by-Parts Reduction ## Toy example: IBP for tadpole integrals -To be completed... Compare with analytic formula. +Let's look at the simplest toy example, the family of one-loop tadpole integrals with internal mass $m$, +@@img-moretiny ![one-loop bubble](/notes/tadpole.svg) @@ +\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 +\begin{equation} +\rho = q^2 - m^2 \, . +\end{equation} +The IBP identity is +\begin{align} +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{align} +We use +\begin{equation} +q^\mu \frac{\partial}{\partial q^\mu} \rho = 2q^2 = 2 (\rho + m^2) \, . +\end{equation} +The IBP identity becomes +\begin{equation} +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} \, . +\end{equation} +The exact result with $d=4-2\epsilon$ is +\begin{equation} +I_\nu = (-1)^\nu \frac {\Gamma(\nu-d/2)} {\Gamma(\nu)} \frac 1 {(m^2)^{\nu-d/2}} \, . +\end{equation} +The IBP identity agrees with the exact result using the gamma function relations $\Gamma(\nu+1) = \nu \Gamma(\nu)$, $\Gamma(\nu+1-d/2) = (\nu-d/2) \Gamma(\nu-d/2)$. ## Basic setup for bubble example Let's look at the bubble family of integrals, @@img-tiny ![one-loop bubble](/notes/bubble.svg) @@ \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}} \, , +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 \begin{equation} @@ -20,13 +47,13 @@ q^2 = \rho_1 + m^2, \quad p \cdot q = \frac 1 2 (\rho_2 - \rho_1 - p^2) \, . \end{equation} We can write down the IBP identity \begin{align} -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}} +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{align} -Using Eqs. (2) and (3), we have +Using Eqs. (8) and (9), we have \begin{align} 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 \, . @@ -46,20 +73,42 @@ Substituting $\nu_1=\nu_2=1$ and using symmetry relations $I_{1,2}=I_{2,1}, \, I \begin{equation} 0 = (d-3) I_{1,1} - (4m^2-p^2) I_{1,2} - I_{0,2} \, . \end{equation} -Combined with Eq. (7), we obtain +Combined with Eq. (13), we obtain \begin{equation} 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} \, . \end{equation} -By directly differentiating w.r.t. $m^2$ under the integral sign in Eq. (1), we obtain differential equations for the unknown integrals $(I_{1,0}, I_{1,1})$, +By directly differentiating w.r.t. $m^2$ under the integral sign in Eq. (7), we obtain differential equations for the unknown integrals $(I_{1,0}, I_{1,1})$, \begin{align} \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{align} +The differential equations can be solved with appropriate boundary conditions, as will be covered by future lectures. ## Systematic reduction of all possible bubble integrals -To be completed... +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 $I_{\nu_1, \nu_2}$ with any integer indices $\nu_1, \nu_2$ without relying on symmetry relations. All we need is one more IBP identity besides Eq. (10), by changing $q^\mu$ in the numerator to $p^\mu$. (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), +\begin{equation} +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} \, . +\end{equation} + +Eq. (12) and Eq. (17) involve a total of 5 integrals, for every choice of integers $(\nu_1, \nu_2)$. The integrals involved in the two linear relations are visualized on the $(\nu_1, \nu_2)$ lattice, +@@img ![bubble_ibp_grid](/notes/bubble_ibp_grid.svg) @@ +For the sake of solving the system, we define an auxiliary quantity, the *complexity* of an integral $I_{\nu_1, \nu_2}$, to be $\nu_1+\nu_2$. 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 $2\times 2$ linear system is non-degenerate if $\nu_1 \geq 1, \nu_2 \geq 1$. The results are +\begin{align} +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{align} +Applying these reduction formulas iteratively, we reduce the complexity of the integrals until all integral are reduced to $I_{1,1}$, $I_{2,0}$ and $I_{0,2}$. ## Using FIRE software to automate IBP reduction 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 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +