Computations and generation of elements on the Hopf algebra of Feynman graphs

TL;DR

This paper introduces two software tools, feyngen and feyncop, for generating high-order Feynman graphs and computing their coproducts within the Hopf algebra framework, validated through combinatorics and algebraic identities.

Contribution

The paper presents new programs for generating Feynman graphs and calculating their Hopf algebra coproducts, enabling advanced computational analysis in quantum field theory.

Findings

feyngen efficiently generates high loop order graphs.

feyncop accurately computes coproducts of Feynman graphs.

Programs validated using combinatorics and algebraic identities.

Abstract

Two programs, feyngen and feyncop, were developed. feyngen is designed to generate high loop order Feynman graphs for Yang-Mills, QED and $\phi^k$ theories. feyncop can compute the coproduct of these graphs on the underlying Hopf algebra of Feynman graphs. The programs can be validated by exploiting zero dimensional field theory combinatorics and identities on the Hopf algebra which follow from the renormalizability of the theories. A benchmark for both programs was made.