Nonlocally Regularized Antibracket-Antifield Formalism and Anomalies in Chiral $W_3$ Gravity

TL;DR

This paper extends nonlocal regularization to gauge theories within the antibracket-antifield formalism, analyzing anomalies in chiral W3 gravity and revealing limitations of the quantum master equation at higher loops.

Contribution

It develops a fully regularized FA formalism compatible with higher loop corrections and demonstrates its application to chiral W3 gravity anomalies.

Findings

The regularized one-loop quantum master equation accurately computes the one-loop anomaly.

Two-loop anomalies are only partially reproduced by the quantum master equation, modified by counterterms.

The universal two-loop anomaly is obtained from the BRST variation of the effective action, indicating limitations of the quantum master equation.

Abstract

The nonlocal regularization method, recently proposed in ref.\,\ct{emkw91,kw92,kw93}, is extended to general gauge theories by reformulating it along the ideas of the antibracket-antifield formalism. From the interplay of both frameworks a fully regularized version of the field-antifield (FA) formalism arises, being able to deal with higher order loop corrections and to describe higher order loop contributions to the BRST anomaly. The quantum master equation, considered in the FA framework as the quantity parametrizing BRST anomalies, is argued to be incomplete at two and higher order loops and conjectured to reproduce only the one-loop corrections to the $\hbar^p$ anomaly generated by the addition of $O(\hbar^{k})$, $k<p$, counterterms. Chiral $W_3$ gravity is used to exemplify the nonlocally regularized FA formalism. First, the regularized one-loop quantum master equation is used to compute the complete one-loop anomaly. Its two-loop order, however, is shown to reproduce only the modification to the two-loop anomaly produced by the addition of a suitable one-loop counterterm, thereby providing an explicit verification of the previous statement for $p=2$. The well-known universal two-loop anomaly, instead, is alternatively obtained from the BRST variation of the nonlocally regulated effective action. Incompleteness of the quantum master equation is thus concluded to be a consequence of a naive derivation of the FA BRST Ward identity.