A Perturbative Study of a General Class of Lattice Dirac Operators

TL;DR

This paper investigates a broad class of lattice Dirac operators satisfying a generalized Ginsparg-Wilson relation, analyzing their quantum corrections and anomalies to ensure they correctly reproduce chiral and Weyl anomalies.

Contribution

It introduces a generalized algebraic framework for lattice Dirac operators and demonstrates their consistency with gauge invariance and anomaly calculations at one-loop level.

Findings

Ward identity is satisfied by the self-energy tensor.

Correct chiral and Weyl anomalies are obtained.

Infra-red divergences are properly managed in the analysis.

Abstract

A perturbative study of a general class of lattice Dirac operators is reported, which is based on an algebraic realization of the Ginsparg-Wilson relation in the form $\gamma_{5}(\gamma_{5}D)+(\gamma_{5}D)\gamma_{5} = 2a^{2k+1}(\gamma_{5}D)^{2k+2}$ where $k$ stands for a non-negative integer. The choice $k=0$ corresponds to the commonly discussed Ginsparg-Wilson relation and thus to the overlap operator. We study one-loop fermion contributions to the self-energy of the gauge field, which are related to the fermion contributions to the one-loop $\beta$ function and to the Weyl anomaly. We first explicitly demonstrate that the Ward identity is satisfied by the self-energy tensor. By performing careful analyses, we then obtain the correct self-energy tensor free of infra-red divergences, as a general consideration of the Weyl anomaly indicates. This demonstrates that our general operators give correct chiral and Weyl anomalies. In general, however, the Wilsonian effective action, which is supposed to be free of infra-red complications, is expected to be essential in the analyses of our general class of Dirac operators for dynamical gauge field.