Non-locally regularized antibracket-antifield formalism and anomalies in chiral W3 gravity

Abstract

The nonlocal regularization method, recently proposed in ref. [1,2,3], 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 theh p anomaly generated by the addition of O(h 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.

On the quantum Batalin-Vilkovisky formalism and the renormalization of non linear symmetries

“…We thus see that the consistency condition (3.46) on the local contribution of the two loop anomaly does not require it to be just a cocycle of the extended BRST differentials, because of the non vanishing right hand side. This is in agreement with the analysis of [28]. Nevertheless, in the extended antifield formalism, the general solution of (3.48) can be characterized: it is given by as boundary up to the term…”

“…This answers, at least in the present context of the extended antifield formalism and dimensional regularization, 11 the question raised in [28] on the cohomological interpretation of the two loop anomaly consistency condition. One also sees on this example how the discussion of the quantum Batalin-Vilkovisky formalism is shifted to ξ, ξ * space in the extended formalism.…”

“…Indeed, its expression as a second order functional differential operator with respect to fields and antifields, obtained from formal path integral considerations, does not make sense when applied to local functionals. In [25,26,27,28], the antifield formalism has been discussed in the context of explicit regularization and renormalization schemes and the related question of anomalies (assumed to be absent in [16,20,21]) has been adressed. In particular, well defined expressions for the regularized ∆ operator are proposed at one loop level in [25] in the context of Pauli-Villars regularization and at higher orders in [28] for non-local regularization.…”

“…In [25,28], explicit expression for the ∆ operator have been obtained in the context of Pauli-Villars and non local regularization respectively. The aim of this section is to get such an expression in the context of dimensional renormalization.…”

Superspace formulation of general massive gauge theories and geometric interpretation of mass-dependent Becchi–Rouet–Stora–Tyutin symmetries

“…At the classical level, the extended BRST invariance in the osp(1,2)-approach is expressed by the classical master equations 1 2 (S m ,S m ) a ϩV m a S m ϭ0, where S m is the lowest order approximation in ប of W m . 12 for nonlocal regularization. ͑I.1͒.…”