Statistical properties of the spectrum of the QCD Dirac operator at low energy

TL;DR

This paper investigates the statistical properties of the low-energy spectrum of the QCD Dirac operator using chiral perturbation theory, revealing a crossover from random matrix theory predictions to nonzero momentum mode effects.

Contribution

It derives the two-point spectral correlation function for the QCD Dirac operator at low energy and describes the transition from chRMT behavior to nonzero momentum mode contributions.

Findings

Two-point correlation function matches chRMT at small eigenvalues.

Identifies a scale where deviations from chRMT occur.

Describes a crossover in eigenvalue number variance behavior.

Abstract

We analyze the statistical properties of the spectrum of the QCD Dirac operator at low energy in a finite box of volume $L^4$ by means of partially quenched Chiral Perturbation Theory (pqChPT), a low-energy effective field theory based on the symmetries of QCD. We derive the two-point spectral correlation function from the discontinuity of the chiral susceptibility. For eigenvalues much smaller than $E_c=F^2/\Sigma L^2$, where $F$ is the pion decay constant and $\Sigma$ is the absolute value of the quark condensate, our result for the two-point correlation function coincides with the result previously obtained from chiral Random Matrix Theory (chRMT). The departure from the chRMT result above that scale is described by the contribution of the nonzero momentum modes. In terms of the variance of the number of eigenvalues in an interval containing $n$ eigenvalues on average, it results in a crossover from a $\log n$-behavior to a $n^2 \log n$-behavior.