On the Invariance of Residues of Feynman Graphs

TL;DR

This paper investigates the symmetries of residues of Feynman graphs in massless Yukawa theory and QED, revealing partial invariance under permutations of subdivergence insertions, with implications for gauge invariance and transcendental weight.

Contribution

It demonstrates that the highest weight transcendental part of residues remains invariant under permutations of subdivergence insertions, using Hopf algebra structures and computational tools.

Findings

Partial invariance of residues under permutation of insertions.

Gauge invariance of the invariance in QED.

Use of Hopf algebra and GiNaC for automated calculations.

Abstract

We use simple iterated one-loop graphs in massless Yukawa theory and QED to pose the following question: what are the symmetries of the residues of a graph under a permutation of places to insert subdivergences. The investigation confirms partial invariance of the residue under such permutations: the highest weight transcendental is invariant under such a permutation. For QED this result is gauge invariant, ie the permutation invariance holds for any gauge. Computations are done making use of the Hopf algebra structure of graphs and employing GiNaC to automate the calculations.