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

Distributions of the Phase Angle of the Fermion Determinant in QCD