On Kreimer's Hopf algebra structure of Feynman graphs

TL;DR

This paper revisits Kreimer's Hopf algebra framework for Feynman graphs, focusing on overlapping divergences, and proposes a formulation that directly defines Hopf algebra operations on all divergence types.

Contribution

It introduces a new formulation that directly applies Hopf algebra operations to any divergence type, clarifying the structure of primitive elements and their relation to Kreimer's original work.

Findings

Directly defines Hopf algebra operations on divergences

Characterizes primitive elements in the Hopf algebra

Clarifies the relation to Kreimer's original formulation

Abstract

We reinvestigate Kreimer's Hopf algebra structure of perturbative quantum field theories with a special emphasis on overlapping divergences. Kreimer first disentangles overlapping divergences into a linear combination of disjoint and nested ones and then tackles that linear combination by the Hopf algebra operations. We present a formulation where the Hopf algebra operations are directly defined on any type of divergence. We explain the precise relation to Kreimer's Hopf algebra and obtain thereby a characterization of their primitive elements.