Anomalies in the Fujikawa method using parameter-dependent regulators.

Abstract

We propose an extended definition of the regularized Jacobian which allows the calculation of anomalies using parameter-dependent regulators in the Fujikawa approach. This extension incorporates the basic Green's function of the problem in the regularized Jacobian, allowing us to interpret a specific regularization procedure as a way of selecting the finite part of the Green's function, in complete analogy with what is done at the level of the effective action. In this way we are able to consider the effect of counterterms in the regularized Jacobian in order to relate different regularization procedures. We also discuss the ambiguities that arise in our prescription due to some freedom in the place where we can insert the regulator, using charge-conjugation invariance as a guiding principle. The method is applied to the case of vector and axial-vector anomalies in two- and four-dimensional quantum electrodynamics. In the first situation we recover the standard family of anomalies calculated by the point-splitting regularization prescription. We also study in detail an alternative choice in the position of the regulator and we calculate explicitly all the currents that generate the families of anomalies that we are considering. Next we extend the calculation to four dimensions, using the same prescriptions as before, and we compare the results with those obtained from the point-splitting calculation, which we also perform in the case of the vector anomaly. A discussion of the relation among the results obtained by different regularization prescriptions is given in terms of the allowed counterterms in the regularized Jacobian, which are highly constrained by the requirement of charge-conjugation invariance.