Discretization errors in the spectrum of the Hermitian Wilson-Dirac operator

TL;DR

This paper analyzes how discretization errors affect the low-energy spectrum of the Hermitian Wilson-Dirac operator in lattice QCD, revealing conditions under which the method is valid and how the spectral gap behaves near the chiral limit.

Contribution

It extends continuum spectral analysis methods to Wilson fermions, incorporating discretization errors via chiral perturbation theory and identifying conditions for the method's applicability.

Findings

The leading discretization errors are proportional to a^2.

The method's validity depends on the sign of a low energy constant.

The spectral gap vanishes faster near the chiral limit than in the continuum.

Abstract

I study the leading effects of discretization errors on the low energy part of the spectrum of the Hermitian Wilson-Dirac operator in infinite volume. The method generalizes that used to study the spectrum of the Dirac operator in the continuum, and uses partially quenched chiral perturbation theory for Wilson fermions. The leading-order corrections are proportional to a^2 (a being the lattice spacing). At this order I find that the method works only for one choice of sign of one of the three low energy constants describing discretization errors. If these constants have the relative magnitudes expected from large N_c arguments, then the method works if the theory has an Aoki phase for m of order a^2, but fails if there is a first-order transition. In the former case, the dependence of the gap and the spectral density on m and a^2 are determined. In particular, the gap is found to vanish more quickly as m_pi^2-> 0 than in the continuum. This reduces the region where simulations are safe from fluctuations in the gap.