Individual eigenvalue distributions of crossover chiral random matrices and low-energy constants of SU(2)$\times$U(1) lattice gauge theory

TL;DR

This paper calculates the distributions of low-lying eigenvalues in a crossover random matrix ensemble and uses these to determine low-energy constants in SU(2) lattice gauge theory with U(1) perturbations.

Contribution

It introduces a method to compute eigenvalue distributions for a crossover ensemble and applies it to extract physical constants from lattice gauge theory spectra.

Findings

Eigenvalue distributions fit lattice data well

Linear relation between crossover parameter and U(1) strength

Precise determination of pseudo-scalar decay constant and chiral condensate

Abstract

We compute individual distributions of low-lying eigenvalues of a chiral random matrix ensemble interpolating symplectic and unitary symmetry classes by the Nystr\"om-type method of evaluating the Fredholm Pfaffian and resolvents of the quaternion kernel. The one-parameter family of these distributions are shown to fit excellently the Dirac spectra of SU(2) lattice gauge theory with a constant U(1) background or dynamically fluctuating U(1) gauge field, which weakly breaks the pseudo-reality of the unperturbed SU(2) Dirac operator. Observed linear dependence of the crossover parameter with the strength of U(1) perturbations leads to precise determination of the pseudo-scalar decay constant, as well as the chiral condensate in the effective chiral Lagrangian of AI class.