More about the axial anomaly on the lattice

TL;DR

This paper investigates the structure of the axial anomaly on finite-size lattices using Ginsparg-Wilson Dirac operators, showing its robustness and similarity to the infinite lattice case for both abelian and non-abelian gauge groups.

Contribution

It demonstrates that the axial anomaly structure on finite lattices mirrors that on infinite lattices, supported by cohomological analysis and a conjecture for non-abelian groups.

Findings

Anomaly structure persists on finite lattices for U(1) gauge group.

Proposes a conjecture for non-abelian gauge groups' anomaly form.

Anomaly structure remains robust under finite ultraviolet and infrared cutoffs.

Abstract

We study the axial anomaly defined on a finite-size lattice by using a Dirac operator which obeys the Ginsparg-Wilson relation. When the gauge group is U(1), we show that the basic structure of axial anomaly on the infinite lattice, which can be deduced by a cohomological analysis, persists even on (sufficiently large) finite-size lattices. For non-abelian gauge groups, we propose a conjecture on a possible form of axial anomaly on the infinite lattice, which holds to all orders in perturbation theory. With this conjecture, we show that a structure of the axial anomaly on finite-size lattices is again basically identical to that on the infinite lattice. Our analysis with the Ginsparg-Wilson Dirac operator indicates that, in appropriate frameworks, the basic structure of axial anomaly is quite robust and it persists even in a system with finite ultraviolet and infrared cutoffs.