Comment 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 between 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. ⋆ e-mail: [email protected] It seems interesting to put the Pauli–Villars type Lagrangian level regularization on the lattice. The interest is twofold: Firstly, the Pauli–Villars regularization [1] for fermion one-loop diagrams can be expressed as a Lagrangian of regulators (bosonic and fermionic spinors). In the actual perturbative calculation, however, the Lagrangian has to be supplemented with additional prescriptions, such that the momentum of propagators has to be assigned in the same way for all the fields, and the integrand in the momentum integral has to be summed before the integration. This point may conceptually be somewhat unsatisfactory, but once the Lagrangian is put on the lattice, those prescriptions are automatically implemented, providing the non-perturbative meaning. More interestingly and more importantly, ”superimposing” a different kind of regularization on the lattice regularization may give some clue to the lattice regularization of the chiral gauge theory. No manifestly gauge invariant lattice formulation of the chiral gauge theory, being consistent with the unitarity and the locality, is yet known [2]. In particular, for a chiral fermion in a complex gauge representation, it is impossible to introduce in a gauge invariant way the Wilson term [3] to eliminate the unwanted species doublers. † The difficulty of a manifestly gauge invariant lattice formulation of the chiral gauge theory is highlighted by the No-Go theorem [4]. The basic idea of “superimposing” is quite simple. Let us consider, for example, the naive momentum cutoff regularization applied to fermion one-loop diagrams in QED. This regularization breaks the gauge invariance, generating gauge noninvariant contributions. However we may use in addition say, the gauge invariant dimensional regularization. With this superimposed regularization, the infinite momentum cutoff limit can be taken and we are left with the gauge invariant expression in the dimensional regularization. Of course in this example there is no real need to break the gauge invariance by introducing the momentum cutoff, † For chiral fermions in real-positive gauge representation, and for even number of chiral fermions in pseudoreal representation, it is possible to introduce a gauge invariant (Majorana-type) Wilson term.