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Distributions of the Phase Angle of the Fermion Determinant in QCD

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.