Renormalization of gauge fields using Hopf algebras

TL;DR

This paper explores the algebraic structure underlying gauge theories, demonstrating how Hopf algebras can encode Feynman graphs and maintain gauge identities during renormalization.

Contribution

It establishes the Hopf algebraic framework for non-abelian gauge theories and proves the compatibility of Slavnov-Taylor identities with this structure.

Findings

Green's functions form a Hopf subalgebra

Slavnov-Taylor identities are compatible with the coproduct

Hopf algebraic structure encodes gauge invariance

Abstract

We describe the Hopf algebraic structure of Feynman graphs for non-abelian gauge theories, and prove compatibility of the so-called Slavnov-Taylor identities with the coproduct. When these identities are taken into account, the coproduct closes on the Green's functions, which thus generate a Hopf subalgebra.