Chiral condensate and Dirac spectrum of one-and two-flavor QCD at nonzero θ -angle

In previous work we showed that the chiral condensate of one-flavor QCD exhibits a Silver Blaze phenomenon when the quark mass crosses m = 0: the chiral condensate remains constant while the quark mass crosses the spectrum of the Dirac operator, which is dense on the imaginary axis. This behavior can be explained in terms of exponentially large cancellations between contributions from the zero modes and from the nonzero modes when the quark mass is negative. In these proceedings we show that a similar Silver Blaze phenomenon takes places for QCD with one flavor and arbitrary θ- angle, and for QCD with two flavors with different quark masses m₁ and m₂. In the latter case the chiral condensate remains constant when m1 crosses zero at fixed m₂ > 0 until the Dashen point m₁ = -m₂ is reached, where the chiral condensate has a discontinuity. In terms of contributions from the Dirac spectrum the shift of the discontinuity from m₁ = 0 to m₁ = -m₂ also arises from exponentially large cancellations between the zero and nonzero modes when m₁m₂ < 0. All calculations are performed in the microscopic or ϵ-domain of QCD. Results for arbitrary θ-angle are discussed as well.