Anomaly Cancellation Condition in Lattice Gauge Theory

TL;DR

This paper demonstrates that gauge anomalies on a finite lattice can be canceled by local counterterms, ensuring consistency with continuum theory and applicable to lattice chiral gauge theories with weak gauge fields.

Contribution

It establishes conditions under which lattice gauge anomalies can be removed by local counterterms, extending continuum anomaly cancellation to lattice formulations.

Findings

Gauge anomalies can be canceled by local counterterms on the lattice.

The result applies to Ginsparg-Wilson fermions with weak gauge fields.

An analysis of Wess-Zumino consistency conditions on the lattice was performed.

Abstract

We study the gauge anomaly ${\cal A}$ defined on a 4-dimensional infinite lattice while keeping the lattice spacing finite. We assume that (I) ${\cal A}$ depends smoothly and locally on the gauge potential, (II) ${\cal A}$ reproduces the gauge anomaly in the continuum theory in the classical continuum limit, and (III) U(1) gauge anomalies have a topological property. It is then shown that the gauge anomaly ${\cal A}$ can always be removed by local counterterms to all orders in powers of the gauge potential, leaving possible breakings proportional to the anomaly in the continuum theory. This follows from an analysis of nontrivial local solutions to the Wess-Zumino consistency condition in lattice gauge theory. Our result is applicable to the lattice chiral gauge theory based on the Ginsparg-Wilson Dirac operator, when the gauge field is sufficiently weak $\|U(n,\mu)-1\|<\epsilon'$, where $U(n,\mu)$ is the link variable and $\epsilon'$ a certain small positive constant.