The structure of renormalization Hopf algebras for gauge theories I: Representing Feynman graphs on BV-algebras

TL;DR

This paper explores the algebraic structures underlying gauge theory renormalization, identifying Hopf subalgebras related to wave function and coupling renormalization, and connects these to BV-algebras via the classical master equation.

Contribution

It introduces a novel framework linking renormalization Hopf algebras with BV-algebras, elucidating the algebraic origin of key identities in gauge theories.

Findings

Identification of Hopf subalgebras related to wave function and coupling renormalization

Establishment of Hopf ideals from Slavnov-Taylor identities

Application of the framework to Yang-Mills gauge theory

Abstract

We study the structure of renormalization Hopf algebras of gauge theories. We identify certain Hopf subalgebras in them, whose character groups are semidirect products of invertible formal power series with formal diffeomorphisms. This can be understood physically as wave function renormalization and renormalization of the coupling constants, respectively. After taking into account the Slavnov-Taylor identities for the couplings as generators of a Hopf ideal, we find Hopf subalgebras in the corresponding quotient as well. In the second part of the paper, we explain the origin of these Hopf ideals by considering a coaction of the renormalization Hopf algebras on the Batalin-Vilkovisky (BV) algebras generated by the fields and couplings constants. The so-called classical master equation satisfied by the action in the BV-algebra implies the existence of the above Hopf ideals in the renormalization Hopf algebra. Finally, we exemplify our construction by applying it to Yang-Mills gauge theory.