Remark on Pauli-Villars Lagrangian on the Lattice

It is interesting to superimpose the Pauli-Villars regularization on the lattice regularization. We illustrate how this scheme works by evaluating the axial anomaly in a simple lattice fermion model, the Pauli-Villars Lagrangian with a gauge non-invariant Wilson term. The gauge non-invariance of the axial anomaly, caused by the Wilson term, is remedied by a compensation among Pauli-Villars regulators in the continuum limit. A subtlety in Frolov-Slavnov's scheme for an odd number of chiral fermions in an anomaly free complex gauge representation, which requires an infinite number of regulators, is briefly mentioned.