Simple Feynman Diagram Calculation Examples

A collection of simple textbook-based examples.

The trace formalism is widely used when teaching courses in particle physics.
At the same time, like many formalisms, it can become tedious, especially when dealing with longer expressions or multiple diagrams.

This project provides a simple workflow that integrates Jupyter notebooks and Python for further processing.
The algebraic manipulations are handled using FORM. FORM is open source and can be seen as the modern successor to Schoonschip, one of the earliest programs designed for high-energy physics calculations.

Textbooks

• Quarks & Leptons: An Introductory Course to Modern Particle Physics — Halzen & Martin
• Diagrammatica: The Path to Feynman Diagrams — Martinus Veltman

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About FORM

• FORM: J.A.M. Vermaseren, "New features of FORM" (math-ph/0010025)
• Repository: https://github.com/form-dev/form
• Releases: https://github.com/form-dev/form/releases
• Manual: https://form-dev.github.io/form-docs/stable/manual/

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Structure

• /Notebooks — Interactive Jupyter Notebook environments that run FORM
• /Notebooks/scripts — .h files containing FORM code, along with .frm and .txt files generated from the notebooks

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Requirements

• FORM installed (command form must be available)
• Python
  • jupyterlab
  • numpy
  • matplotlib
  • sympy

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Pipeline Overview

• We aim to run form directly from within Jupyter notebooks

• Assume a notebook name.ipynb:

  1. Import utilities in the first cell
  2. Start the second cell with:

     %%pyForm name

  3. Write the FORM code in that cell
  4. On execution:
     • A file scripts/name.frm is generated
     • The FORM script is executed
     • Output is written using Format C in .txt files.
  5. Python then parses the output file for further manipulation

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Conventions for Amplitude Calculations

Gamma Matrix Notation

Term	Internal Notation	Mathematical Equivalent
Unit Matrix	gamma(i1, i2)	$(I)^{i_1}{}_{i_2}$
Gamma Matrix	gamma(i1, i2, mu)	$(\gamma^\mu)^{i_1}{}_{i_2}$
Slash Momentum	gamma(i1, i2, p_i)	$(\slashed{p}i)^{i_1}{}{i_2}$
Gamma 5 ($\gamma^5$)	gamma(i1, i2, k5)	$(\gamma^5)^{i_1}{}_{i_2}$
Left-Handed Projector	1/2 * gamma(i1, i2, k7)	$\left(\frac{1 - \gamma^5}{2}\right)^{i_1}{}_{i_2}$
Right-Handed Projector	1/2 * gamma(i1, i2, k6)	$\left(\frac{1 + \gamma^5}{2}\right)^{i_1}{}_{i_2}$
V−A Coupling	1/2 * ((cv+ca)*g(i1,i2,k7) + (cv-ca)*g(i1,i2,k6))	$(c_V - c_A \gamma^5)^{i_1}{}_{i_2}$

External Lines

• Outgoing fermion ($\bar{u}$): UB(i1, p1, m) → $\bar{u}_a(p_1, m)$
• Incoming fermion ($u$): U(i1, p1, m) → $u_a(p_1, m)$
• Outgoing anti-fermion ($v$): V(i1, p1, m) → $v_a(p_1, m)$
• Incoming anti-fermion ($\bar{v}$): VB(i1, p1, m) → $\bar{v}_a(p_1, m)$
• Massless Vector polarization ($\epsilon$): esum(mu1, p1) → $\epsilon_\mu(p_1)$
• Massive Vector polarization ($\epsilon$): esumM(mu1, p1 , m) → $\epsilon_\mu(p_1, m)$

Polarization sums

U(i1?,p?,m?) * UB(i2?,p?,m?) =  gamma(i1,i2,p) + gamma(i1,i2)*m;

V(i1?,p?,m?) * VB(i2?,p?,m?) =  gamma(i1,i2,p) - gamma(i1,i2)*m;

esum(mu1?,p?) * esum(mu2?,p?) = -d_(mu1,mu2);

esumM(mu1?, p?, m?) * esumM(mu2?, p?, m?) = -d_(mu1,mu2) + p(mu1)*p(mu2)/(m^2);

Propagators

fprop(i1?,i2?,p?,m?) = (g(i1,i2,p) +  g(i1,i2)*m)  * prop(p.p - m^2)

phprop(mu1?,mu2?,q?) = -d_(mu1,mu2) * prop(q.q);

Zprop(mu1?,mu2?,q?,m?) = (-d_(mu1,mu2) + q(mu1) * q(mu2)/(m^2)) * prop(q.q - m^2);

Wprop(mu1?,mu2?,q?,m?) = (-d_(mu1,mu2) + q(mu1) * q(mu2)/(m^2)) * prop(q.q - m^2);

The prop function is kept symbolic in intermediate expressions
It must be resolved using kinematics and momentum conservation.

For example:

* Momentum conservation
id q = p1 + p2;
.sort

* Propagator handling
id prop(x?) = (x)^-1;
.sort

id (q.q)^-1 = (s)^-1;
id (-mZ^2 + q.q)^-1 = (s - mZ^2)^-1;
.sort

Index and Momentum Mapping

To translate a handwritten Feynman rule:

1. Index direction: Follow fermion flow (matrix multiplication order)
2. Spinor indices: i1, i2, i3, ... (up to i40)
3. Lorentz indices: mu1, mu2, mu3, ... (up to mu20)

Squared Amplitudes

After defining the amplitude:

squareamplitude(Amp, Mat)

Kinematics

All calculations are performed using the Minkowski metric

$g^{\mu\nu} = \mathrm{diag}(1, -1, -1, -1)$

On-shell condition:

$p_i^2 = m_i^2$

Mandelstam variables:

$s = (p_1 + p_2)^2$,

$t = (p_1 - p_3)^2$,

$u = (p_1 - p_4)^2$

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