An analysis on the convergence of equal-time commutators and the closure of the BRST algebra in Yang-Mills theories

TL;DR

This paper investigates the convergence of equal-time commutators in Yang-Mills theories and demonstrates that the BRST charge commutes with itself in anomaly-free cases, with potential divergences arising at higher loops if anomalies are present.

Contribution

It provides a detailed analysis of the convergence properties of equal-time commutators and establishes conditions under which the BRST algebra closes in Yang-Mills theories.

Findings

Equal-time commutators vanish for amplitudes with effective dimension ≤ -2.

BRST charge commutes with itself if the theory is anomaly-free.

Divergences in commutators appear at two-loop level if anomalies are not canceled.

Abstract

In renormalizable theories, we define equal-time commutators (ETC'S) in terms of the equal-time limit and investigate its convergence in perturbation theory. We find that the equal-time limit vanishes for amplitudes with the effective dimension $d_{\em eff} \leq -2$ and is finite for those with $d_{\em eff} =-1$ but without nontrivial discontinuity. Otherwise we expect divergent equal-time limits. We also find that, if the ETC's involved in verifying an Jacobi identity exist, the identity is satisfied. Under these circumstances, we show in the Yang-Mills theory that the ETC of the $0$ component of the BRST current with each other vanishes to all orders in perturbation theory if the theory is free from the chiral anomaly, from which we conclude that $[\, Q\,,\,Q\,]=0$, where $Q$ is the BRST charge. For the case that the chiral anomaly is not canceled, we use various broken Ward identities to show that $[\, Q\,,\,Q\,]$ is finite and $[\,Q\,,\,[\, Q\,,\,Q]\,]$ vanishes at the one-loop level and that they start to diverge at the two-loop level unless there is some unexpected cancellation mechanism that improves the degree of convergence.