Chiral Lattice Fermions, Minimal Doubling, and the Axial Anomaly

TL;DR

This paper investigates how minimally doubled lattice fermion actions can correctly reproduce the axial anomaly in two dimensions, highlighting the need for fine tuning to preserve the anomaly and avoid fermion doubling.

Contribution

It demonstrates that minimally doubled fermion actions require fine tuning to correctly reproduce the axial anomaly without fermion doubling effects.

Findings

Fine tuning of the action and axial current is necessary for anomaly reproduction.

Conservation of the flavor non-singlet vector current also requires fine tuning.

Chiral projection of minimally doubled fermions yields an undoubled fermion with the correct anomaly.

Abstract

Exact chiral symmetry at finite lattice spacing would preclude the axial anomaly. In order to describe a continuum quantum field theory of Dirac fermions, lattice actions with purported exact chiral symmetry must break the flavor-singlet axial symmetry. We demonstrate that this is indeed the case by using a minimally doubled fermion action. For simplicity we consider the Abelian axial anomaly in two dimensions. At finite lattice spacing and with gauge interactions, the axial anomaly arises from non-conservation of the flavor-singlet current. Similar non-conservation also leads to the axial anomaly in the case of the naive lattice action. For minimally doubled actions, however, fine tuning of the action and axial current is necessary to arrive at the anomaly. Conservation of the flavor non-singlet vector current additionally requires the current to be fine tuned. Finally we determine that the chiral projection of a minimally doubled fermion action can be used to arrive at a lattice theory with an undoubled Dirac fermion possessing the correct anomaly in the continuum limit.