Regularization and Ward-Identity Anomalies

In the context of the free-quark model, we discuss the validity of the naïve Ward identities (WI's) for arbitrary regularized $n$-point functions of scalar, pseudoscalar, vector, and axial-vector currents. In a simple version of the regularization procedure described by Pauli and Villars, we find that the naïve vector WI's are all automatically satisfied, and that there is a compact necessary condition for the existence of an axial-vector anomaly. Subsequently, this version leads to a large number of anomalous axial-vector WI's (corresponding to the cases $n=2, 3, 4, \mathrm{and} 5$). It is shown that this number cannot be reduced, for example, to Bardeen's "minimal" solution without additional counterterms beyond those possible in the general regularization framework—in spite of the framework's well-known ambiguities. We discuss other minimal sets, as well as a symmetry-breaking model in which no further anomalies are found. The explicit forms of the WI anomalies for the general minimal solution is given along with the necessary counterterms.