Renormalization Hopf algebras for gauge theories and BRST-symmetries

TL;DR

This paper explores the structure of the Connes-Kreimer renormalization Hopf algebra in gauge theories, focusing on BRST-symmetries within quantum chromodynamics, and reveals how BRST-invariance leads to specific algebraic identities.

Contribution

It introduces a detailed analysis of the Hopf algebra structure in gauge theories, highlighting the role of BRST-invariance and the resulting Hopf ideals in encoding Slavnov-Taylor identities.

Findings

Defined a coaction of the Hopf algebra on coupling constants and fields.

Established that BRST-invariance implies the existence of Hopf ideals.

Connected Hopf algebra structures to gauge symmetry identities.

Abstract

The structure of the Connes-Kreimer renormalization Hopf algebra is studied for gauge theories, with particular emphasis on the BRST-formalism. We work in the explicit example of quantum chromodynamics, the physical theory of quarks and gluons. A coaction of the renormalization Hopf algebra is defined on the coupling constants and the fields. In this context, BRST-invariance of the action implies the existence of certain Hopf ideals in the renormalization Hopf algebra, encoding the Slavnov-Taylor identities for the coupling constants.