Cancellation Identities and Renormalization
David Prinz
TL;DR
The work proposes a manifestly gauge-invariant renormalization framework by merging BRST symmetry with Connes-Kreimer Hopf-algebra renormalization and formal cancellation identities of tHooft-Veltman. It constructs a pBRST Feynman graph complex whose cohomology isolates transversal, gauge-invariant combinations of connected graphs, and then builds a derived renormalization Hopf algebra on these cohomology classes to realize multiplicative renormalization. The formalism is illustrated with Yang-Mills theory and (effective) Quantum General Relativity, and the authors discuss how gauge anomalies would obstruct the construction. The approach provides a mathematically rigid path to gauge-invariant renormalization and suggests future links to BV formalism and UV completions of 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.