Background field method and the cohomology of renormalization

TL;DR

This paper proves a cohomology theorem for local functionals in various gauge theories using the background field method and Batalin-Vilkovisky formalism, simplifying the understanding of counterterms and anomalies.

Contribution

It introduces a unified cohomology theorem applicable to both renormalizable and nonrenormalizable theories with complex gauge symmetries, advancing the theoretical framework of renormalization.

Findings

Closed functionals decompose into exact plus physical-dependent parts

The theorem applies to theories with general covariance, Lorentz symmetry, and gauge symmetries

Simplifies the analysis of counterterms and anomalies in quantum field theories

Abstract

Using the background field method and the Batalin-Vilkovisky formalism, we prove a key theorem on the cohomology of perturbatively local functionals of arbitrary ghost numbers, in renormalizable and nonrenormalizable quantum field theories whose gauge symmetries are general covariance, local Lorentz symmetry, non-Abelian Yang-Mills symmetries and Abelian gauge symmetries. Interpolating between the background field approach and the usual, nonbackground approach by means of a canonical transformation, we take advantage of the properties of both approaches and prove that a closed functional is the sum of an exact functional plus a functional that depends only on the physical fields and possibly the ghosts. The assumptions of the theorem are the mathematical versions of general properties that characterize the counterterms and the local contributions to the potential anomalies. This makes the outcome a theorem on the cohomology of renormalization, rather than the whole local cohomology. The result supersedes numerous involved arguments that are available in the literature.