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

Abstract

In renormalizable theories, we define equal-time commutators (ETCs) in terms of the equal-time limit and investigate their convergence in perturbation theory. We find that the equal-time limit vanishes for amplitudes with the effective dimension deff ⩽ −2 and is finite for tree-like amplitudes with deff = −1. Otherwise we expect divergent equal-time limits. We also find that, if the ETCs involved in verifying a Jacobi identity exist, then 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 itself 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 both diverge starting at the two-loop level.