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

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

1994-01-21

arxiv.org/abs/hep-th/9401106v1

Nucl.Phys. B427 (1994) 398-424

Physics

High Energy Physics

High Energy Physics - Theory

35 pages

Scientific paper

10.1016/0550-3213(94)90282-8

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.

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