Distributions of the Phase Angle of the Fermion Determinant in QCD

TL;DR

This paper investigates the distribution of the phase angle of the fermion determinant in QCD at non-zero chemical potential, revealing Gaussian and Lorentzian behaviors and analyzing their impact on baryon number and chiral condensate.

Contribution

It provides a detailed analysis of the phase angle distribution in QCD using chiral perturbation theory and exact results in one dimension, highlighting the transition between Gaussian and Lorentzian distributions.

Findings

Phase angle distribution is Gaussian at small chemical potential.

Distribution becomes Lorentzian when the quark mass is inside the Dirac spectrum.

Severe cancellations occur in the integration over the phase angle.

Abstract

The distribution of the phase angle and the magnitude of the fermion determinant as well as its correlations with the baryon number and the chiral condensate are studied for QCD at non zero quark chemical potential. Results are derived to one-loop order in chiral perturbation theory. We find that the distribution of the phase angle is Gaussian for small chemical potential and a periodic Lorentzian when the quark mass is inside the support of the Dirac spectrum. The baryon number and chiral condensate are computed as a function of the phase of the fermion determinant and we discuss the severe cancellations which occur upon integration over the angle. We compute the distribution of the magnitude of the fermion determinant as well as the baryon number and chiral condensate at fixed magnitude. Finally, we consider QCD in one Euclidean dimension where it is shown analytically, starting from the fundamental QCD partition function, that the distribution of the phase of the fermion determinant is a periodic Lorentzian when the quark mass is inside the spectral density of the Dirac operator.