Cancellation Identities and Renormalization

TL;DR

This paper develops a gauge-invariant renormalization framework using BRST cohomology and Connes--Kreimer theory, formalizing cancellation identities to ensure transversality and applying it to Yang--Mills and Quantum Gravity.

Contribution

It introduces a novel cohomological approach to gauge-invariant renormalization, combining BRST graph complexes with Hopf algebra structures for the first time.

Findings

Cohomology groups are zero in odd degrees.

Cohomology is generated by connected Green's functions in even degrees.

Framework successfully applied to Yang--Mills and Quantum Gravity.

Abstract

We construct a manifest gauge invariant renormalization framework by first introducing a perturbative BRST Feynman graph complex and then combining it with Connes--Kreimer renormalization theory: To this end, we first formalize the cancellation identities of 't Hooft (1971), which were used to prove the absence of gauge anomalies in Quantum Yang--Mills theories. Specifically, we start with some reasonable axioms of (generalized) gauge theories and then present the most general version of cancellation identities ensuring transversality. Then, we construct a perturbative BRST Feynman graph complex, whose cohomology groups consist of transversal invariant linear combinations of Feynman graphs. We prove that the cohomology groups are zero in odd degree and generated by connected combinatorial Green's functions in even degree, with a corresponding number of external ghost edges. Ultimately, we then formulate the renormalization Hopf algebra on these cohomology groups, which directly links to Hopf subalgebras for multiplicative renormalization. Finally, we exemplify the developed theory with Quantum Yang--Mills theory and (effective) Quantum General Relativity.