Dirac spectrum and chiral condensate for QCD at fixed $\theta$-angle

TL;DR

This paper investigates how the chiral condensate in QCD behaves at nonzero theta angles, revealing complex oscillations in the Dirac spectrum that influence the condensate's discontinuity and providing new analytical formulas for spectral densities.

Contribution

It introduces a decomposition of the spectral density into zero modes, quenched, and dynamical parts, with new formulas for each, especially at nonzero theta angles.

Findings

The quenched and dynamical spectral densities oscillate exponentially with volume.

The divergence in the quenched condensate at nonzero theta is canceled by zero mode contributions.

New analytical expressions for microscopic spectral densities at nonzero theta for one- and two-flavor QCD.

Abstract

We analyze the mass dependence of the chiral condensate for QCD at nonzero $\theta$-angle and find that in general the discontinuity of the chiral condensate is not on the support of the Dirac spectrum. To understand this behavior we decompose the spectral density and the chiral condensate into contributions from the zero modes, the quenched part, and a remainder which is sensitive to the fermion determinant and is referred to as the dynamical part. We obtain general formulas for the contributions of the zero modes. Expressions for the quenched part, valid for an arbitrary number of flavors, and for the dynamical part, valid for one and two flavors, are derived in the microscopic domain of QCD. We find that at nonzero $\theta$-angle the quenched and dynamical part of the Dirac spectral density are strongly oscillating with an amplitude that increases exponentially with the volume $V$ and a period of order of $1/V$. The quenched part of the chiral condensate becomes exponentially large at $\theta\ne0$, but this divergence is canceled by the contribution from the zero modes. The oscillatory behavior of the dynamical part of the density is essential for moving the discontinuity of the chiral condensate away from the support of the Dirac spectrum. As important by-products of this work we obtain analytical expressions for the microscopic spectral density of the Dirac operator at nonzero $\theta$-angle for both one- and two-flavor QCD with nonzero quark masses.