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

Kieburg M, Verbaarschot JJM, Wettig T. Dirac spectrum and chiral condensate for QCD at fixed $\Theta$ angle. PHYSICAL REVIEW D. 2019;99(7): 074515.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 parts 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 not equal 0, 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