Example gallery · FeynCalc

Here we collect examples of interesting calculations from textbooks and papers that can be reproduced using FeynCalc. The HTML pages below were generated from example notebooks shipped with the current stable version of Feyncalc. If you have FeynCalc installed, these files are located in

FileNameJoin[{$UserBaseDirectory, "Applications", "FeynCalc", "Examples"}]

$\phi^3$ theory

Tree level	1-loop level
	Renormalization ($\textrm{MS}, \overline{\textrm{MS}}$)

Tree level

1-loop level

Renormalization ($\textrm{MS}, \overline{\textrm{MS}}$)

$\phi^4$ theory

Tree level	1-loop level
	$\phi \phi \to \phi \phi$
	Renormalization ($\textrm{MS}, \overline{\textrm{MS}}$)

Tree level

1-loop level

$\phi \phi \to \phi \phi$

Renormalization ($\textrm{MS}, \overline{\textrm{MS}}$)

Quantum electrodynamics (QED)

Tree level	1-loop level	2-loop level
$e^+ e^- \to e^+ e^-$	$e^+ e^- \to e^+ e^-$	$e^- \to e^-$
$e^+ e^- \to \gamma \gamma$	$e^- \to e^-$	$\gamma \to \gamma$
$e^+ e^- \to \mu^+ \mu^-$	$e^- \to \gamma e^-$
$e^- e^- \to e^- e^-$	$\gamma$
$e^- \gamma \to e^- \gamma$	$\gamma \to \gamma$
$e^- \mu^+ \to e^- \mu^+$	$\gamma \to \gamma \gamma$
$\gamma \to \mu^+ \mu^-$	$\gamma \to \gamma \gamma \gamma \gamma$
	$e^+ e^- \to \mu^+ \mu^-$
	$\pi \to \gamma \gamma$
	Renormalization ($\textrm{MS}, \overline{\textrm{MS}}$)

Tree level

1-loop level

2-loop level

$e^+ e^- \to e^+ e^-$

$e^- \to e^-$

$e^+ e^- \to \gamma \gamma$

$e^- \to e^-$

$\gamma \to \gamma$

$e^+ e^- \to \mu^+ \mu^-$

$e^- \to \gamma e^-$

$e^- e^- \to e^- e^-$

$\gamma$

$e^- \gamma \to e^- \gamma$

$\gamma \to \gamma$

$e^- \mu^+ \to e^- \mu^+$

$\gamma \to \gamma \gamma$

$\gamma \to \mu^+ \mu^-$

$\gamma \to \gamma \gamma \gamma \gamma$

$e^+ e^- \to \mu^+ \mu^-$

$\pi \to \gamma \gamma$

Renormalization ($\textrm{MS}, \overline{\textrm{MS}}$)

Quantum chromodynamics (QCD)

Tree level	1-loop level	2-loop level
$e^+ e^- \to q \bar{q}$	$u_g \to u_g$	$u_g g \to u_g$
$\gamma g \to q \bar{q}$	$u_g g \to u_g$	$g \to g$
$\gamma \to q \bar{q}$	$g \to g$
$\gamma \to q \bar{q} g$	$g \to g$ (background field gauge)
$g g \to g g$	$g \to g g$
$g g\to q \bar{q}$	$q \to q$
$\mu^+ \mu^- \to q \bar{q}$	Renormalization ($\textrm{MS}, \overline{\textrm{MS}}$)
$q \gamma \to g q$	Renormalization ($\textrm{MS}, \overline{\textrm{MS}}$), massless quarks
$q g \to q g$
$q_i \bar{q}_i \to q_i \bar{q}_i$
$q_i \bar{q}_i \to q_j \bar{q}_j$
$q_i q_i \to q_i q_i$
$q_i \bar{q}_j \to q_i \bar{q}_j$
$q_i q_j \to q_i q_j$
$q \bar{q} \to e^+ e^-$
$q \bar{q} \to \gamma \gamma$
$q \bar{q} \to \gamma g$
$q \bar{q} \to g g$
$q \bar{q} \to \mu^+ \mu^-$

Tree level

1-loop level

2-loop level

$e^+ e^- \to q \bar{q}$

$u_g \to u_g$

$u_g g \to u_g$

$\gamma g \to q \bar{q}$

$u_g g \to u_g$

$g \to g$

$\gamma \to q \bar{q}$

$g \to g$

$\gamma \to q \bar{q} g$

$g \to g$ (background field gauge)

$\mu^+ \mu^- \to q \bar{q}$

Renormalization ($\textrm{MS}, \overline{\textrm{MS}}$)

$q \gamma \to g q$

Renormalization ($\textrm{MS}, \overline{\textrm{MS}}$), massless quarks

$q g \to q g$

$q_i \bar{q}_i \to q_i \bar{q}_i$

$q_i \bar{q}_i \to q_j \bar{q}_j$

$q_i q_i \to q_i q_i$

$q_i \bar{q}_j \to q_i \bar{q}_j$

$q_i q_j \to q_i q_j$

$q \bar{q} \to e^+ e^-$

$q \bar{q} \to \gamma \gamma$

$q \bar{q} \to \gamma g$

$q \bar{q} \to g g$

$q \bar{q} \to \mu^+ \mu^-$

Electroweak theory (EW)

Tree level	1-loop level
\(\bar{\nu}_e e^- \to \bar{\nu}_{\mu} \mu\)	$H \to g g$
$\bar{\nu}_e e^- \to \bar{q}_u q_d$
$\bar{\nu}_e e^- \to W^+ W^-$
$\bar{\nu}_e e^- \to Z^0 Z^0$
$e^- \nu_{\mu} \to \mu \nu_{e}$
$H \to f \bar{f}$
$H \to W^+ W^-$
$H \to Z^0 Z^0$
\(\mu \to e^- \bar{\nu}_{e} \nu_{\mu}\)
$\nu_l q_d \to l q_u$
$q \bar{q} \to Z^0 Z^0$
$t \to W^+ b$
$\bar{q}_d q_u \to \bar{\nu}_e \nu_e$
$\bar{q}_u q_d \to \bar{\nu}_e \nu_e$
$W^- \to e^- \bar{\nu}_e $
$W \to q_i \bar{q}_j$
$Z \to f \bar{f}$

Tree level

1-loop level

$\bar{\nu}_e e^- \to \bar{\nu}_{\mu} \mu$

$H \to g g$

$\bar{\nu}_e e^- \to \bar{q}_u q_d$

$\bar{\nu}_e e^- \to W^+ W^-$

$\bar{\nu}_e e^- \to Z^0 Z^0$

$e^- \nu_{\mu} \to \mu \nu_{e}$

$H \to f \bar{f}$

$H \to W^+ W^-$

$H \to Z^0 Z^0$

$\mu \to e^- \bar{\nu}_{e} \nu_{\mu}$

$\nu_l q_d \to l q_u$

$q \bar{q} \to Z^0 Z^0$

$t \to W^+ b$

$\bar{q}_d q_u \to \bar{\nu}_e \nu_e$

$\bar{q}_u q_d \to \bar{\nu}_e \nu_e$

$W^- \to e^- \bar{\nu}_e $

$W \to q_i \bar{q}_j$

$Z \to f \bar{f}$

Minimal Supersymmetric Standard Model (MSSM)

Tree level	1-loop level
\(\tilde{\nu}_e e^- \to \tilde{\nu}_e e^-\)

Tree level

1-loop level

$\tilde{\nu}_e e^- \to \tilde{\nu}_e e^-$

Acknowledgement

To generate Markdown files out of FeynCalc notebooks we employed the M2MD package by Kuba Podkalicki. We thank the author for helpful advice on using the package in a proper way. Our scripts for running M2MD can be found here.