The BPHZ renormalised BV master equation and two-loop anomalies in chiral gravities

Abstract

Anomalies and BRST invariance are governed, in the context of Lagrangian Batalin-Vilkovisky quantization, by the master equation, whose classical limit is (S, S) = 0. Using Zimmerman's normal products and the BPHZ renormalisation method, we obtain a corresponding local quantum operator equation, which is valid to all orders in perturbation theory. The formulation implies a calculational method for anomalies that is useful also outside this context and that remains completely within regularised perturbation theory. It makes no difference in principle whether the anomaly appears at one loop or at higher loops. The method is illustrated by computing the one-and two-loop anomalies in chiral W 3 gravity.

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

“…As discussed for instance in section 4 of [27] in the context of the BPHZ renormalized antifield formalism, even though there is a well defined expression for the anomaly, there is no room for the formal Batalin-Vilkovisky ∆ operator in the final renormalized theory. Contact with the quantum Batalin-Vilkovisky formalism in the present set-up has thus to be done on the renormalized theory before the regulator τ is removed.…”

“…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.…”

“…The renormalized one loop action is 27) whereΣ 1 remains to be determined. Using the remark after theorem 4, we get…”

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

“…10 The reason is the following: The only quantity that remains undefined in the above-mentioned approaches of quantizing general gauge theories is the right-hand side of the quantum master equations ͑that problem already occurs in the Batalin-Vilkovisky field-antifield formalism͒. 10 The reason is the following: The only quantity that remains undefined in the above-mentioned approaches of quantizing general gauge theories is the right-hand side of the quantum master equations ͑that problem already occurs in the Batalin-Vilkovisky field-antifield formalism͒.…”

Superspace Formulation for the BRST Quantization of the Chiral Schwinger Model

“…The application of this formalism to anomalous gauge theories was first discussed by Troost, van Nieuwenhuizen and Van Proeyen [4], that succeeded in using the Pauli Villars regularization in order to give a regularized meaning to the master equation at one loop order. The possibility of more general regularizations that enable the quantization of more complex gauge theories where the higher loop order terms play an important role is presently under study [5,6]. For the case of irreducible gauge theories with closed algebra, it was shown that the BV procedure can be formulated in a superspace with one Grassmanian variable [7,8], where BRST transformations are realised as translations.…”