Classical Limits of Scalar and Tensor Gauge Operators Based on the Overlap Dirac Matrix

TL;DR

This paper demonstrates that the classical limit of the trace of the overlap Dirac operator and its tensor component on the lattice accurately reproduces the continuum gauge field strength tensor, confirming a key theoretical proposal.

Contribution

It provides analytical and numerical evidence that the trace of the overlap Dirac operator relates correctly to the gauge field strength in the classical limit, validating a lattice gauge formulation based on the Dirac operator.

Findings

The trace of the overlap Dirac operator is proportional to the gauge field strength tensor.

The proportionality constants are computed with high precision across various parameters.

The results hold consistently in both finite and infinite volume settings.

Abstract

It was recently proposed by the second author to consider lattice formulations of QCD in which complete actions, including the gauge part, are built explicitly from a given Dirac operator D. In a simple example of such theory, the gauge action is proportional to the trace of Ginsparg-Wilson operator D chosen to define the quark dynamics. This construction relies on the proposition that the classical limit of lattice gauge operator tr D(x,x) is proportional to tr F.F(x) (up to an additive constant). Here we show this for the case of the overlap Dirac operator using both analytical and numerical methods. We carry out the same analysis also for the tensor component of D, which is similarly related to the field-strength tensor F, and obtain results identical to our previous derivation that used different approach. The corresponding proportionality constants are computed to high precision for wide range of the negative mass parameter values, and it is verified that they are the same in finite and infinite volumes.