Renormalization of gauge fields: A Hopf algebra approach

TL;DR

This paper demonstrates that the algebraic structure underlying renormalization in gauge theories naturally incorporates Ward and Slavnov-Taylor identities, ensuring their compatibility through a Hopf algebra framework.

Contribution

It introduces a Hopf algebra approach that rigorously proves the compatibility of gauge identities with the renormalization process.

Findings

Ward and Slavnov-Taylor identities form a Hopf ideal

The quotient Hopf algebra incorporates these identities

Provides a combinatorial proof of their compatibility

Abstract

We study the Connes-Kreimer Hopf algebra of renormalization in the case of gauge theories. We show that the Ward identities and the Slavnov-Taylor identities (in the abelian and non-abelian case respectively) are compatible with the Hopf algebra structure, in that they generate a Hopf ideal. Consequently, the quotient Hopf algebra is well-defined and has those identities built in. This provides a purely combinatorial and rigorous proof of compatibility of the Slavnov-Taylor identities with renormalization.