diff --git a/notes/ibp/index.html b/notes/ibp/index.html index 5bf3b47..622a606 100644 --- a/notes/ibp/index.html +++ b/notes/ibp/index.html @@ -1,4 +1,4 @@ - Scattering Amplitudes Lecture on 07 Nov: Integration-by-Parts Reduction
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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−aρ1ν1+1ρ2ν2 qμ∂∂qμρ1−bρ1ν1ρ2ν2+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 a {\rho_1^{\nu_1+1} \rho_2^{\nu_2}} \, q^\mu \frac{\partial}{\partial q^\mu} \rho_1 - \frac b {\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

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−aρ1ν1+1ρ2ν2 qμ∂∂qμρ1−bρ1ν1ρ2ν2+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 a {\rho_1^{\nu_1+1} \rho_2^{\nu_2}} \, q^\mu \frac{\partial}{\partial q^\mu} \rho_1 - \frac b {\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

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