The signed permutation group on Feynman graphs

TL;DR

This paper explores the algebraic structure of Feynman graphs in scalar quantum field theory, revealing simplifications in the perturbation series through the use of the signed permutation group and Hopf algebraic methods.

Contribution

It introduces a novel application of the signed permutation group and Hopf algebra to simplify the first non-trivial term in Feynman graph perturbation series.

Findings

Identifies specific graph combinations leading to simplification

Provides a scheme-independent and angle-independent result

Derives a formula reducing computational effort for first-order terms

Abstract

The Feynman rules assign to every graph an integral which can be written as a function of a scaling parameter L. Assuming L for the process under consideration is very small, so that contributions to the renormalizaton group are small, we can expand the integral and only consider the lowest orders in the scaling. The aim of this article is to determine specific combinations of graphs in a scalar quantum field theory that lead to a remarkable simplification of the first non-trivial term in the perturbation series. It will be seen that the result is independent of the renormalization scheme and the scattering angles. To achieve that goal we will utilize the parametric representation of scalar Feynman integrals as well as the Hopf algebraic structure of the Feynman graphs under consideration. Moreover, we will present a formula which reduces the effort of determining the first-order term in the perturbation series for the specific combination of graphs to a minimum.