The regularized BRST Jacobian of pure Yang-Mills theory

TL;DR

This paper demonstrates through explicit computation that the regularized Jacobian of BRST transformations in pure Yang-Mills theory does not produce an anomaly, confirming the consistency of the theory at the quantum level.

Contribution

The paper provides a direct path integral proof that the regularized Jacobian's trace variation is a local counterterm, showing no BRST anomaly in pure Yang-Mills theory.

Findings

Regularized Jacobian's trace variation is a local counterterm.

Explicit computation confirms absence of BRST anomaly.

Method applies a general regulator to the Jacobian.

Abstract

The Jacobian for infinitesimal BRST transformations of path integrals for pure Yang-Mills theory, viewed as a matrix $\unity +\Delta J$ in the space of Yang-Mills fields and (anti)ghosts, contains off-diagonal terms. Naively, the trace of $\Delta J$ vanishes, being proportional to the trace of the structure constants. However, the consistent regulator $\cR$, constructed from a general method, also contains off-diagonal terms. An explicit computation demonstrates that the regularized Jacobian $Tr\ \Delta J\exp -\cR /M^2$ for $M^2\rightarrow \infty $ is the variation of a local counterterm, which we give. This is a direct proof at the level of path integrals that there is no BRST anomaly.