Gauge Symmetries and Renormalization

TL;DR

This paper explores the algebraic structure of gauge symmetries in quantum field theories, extending the Hopf algebra framework to complex theories including gravity and matter, and establishing criteria for their consistent renormalization.

Contribution

It generalizes the Hopf algebra approach to gauge symmetries to super- and non-renormalizable theories with multiple couplings, and applies it to effective quantum gravity and related models.

Findings

Extended Hopf algebra framework to super- and non-renormalizable theories.

Provided criteria for compatibility of Hopf ideals with renormalized Feynman rules.

Demonstrated the well-definedness of the Corolla polynomial for Quantum Yang--Mills.

Abstract

We study the perturbative renormalization of quantum gauge theories in the Hopf algebra setup of Connes and Kreimer. It was shown by van Suijlekom (2007) that the quantum counterparts of gauge symmetries -- the so-called Ward--Takahashi and Slavnov--Taylor identities -- correspond to Hopf ideals in the respective renormalization Hopf algebra. We generalize this correspondence to super- and non-renormalizable Quantum Field Theories, extend it to theories with multiple coupling constants and add a discussion on transversality. In particular, this allows us to apply our results to (effective) Quantum General Relativity, possibly coupled to matter from the Standard Model, as was suggested by Kreimer (2008). To this end, we introduce different gradings on the renormalization Hopf algebra and study combinatorial properties of the superficial degree of divergence. Then we generalize known coproduct and antipode identities to the super- and non-renormalizable cases and to theories with multiple vertex residues. Building upon our main result, we provide criteria for the compatibility of these Hopf ideals with the corresponding renormalized Feynman rules. A direct consequence of our findings is the well-definedness of the Corolla polynomial for Quantum Yang--Mills theory without reference to a particular renormalization scheme.