Chiral Condensate and Dirac Spectrum of One-Flavor and Two-Flavor QCD at Nonzero $\theta$-Angle
In previous work  we showed that the chiral condensate of one flavor QCD shows 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 is explained in terms of exponentially large cancellations between contributions from the zero modes and the nonzero modes for negative quark mass. In this talk we explain that a similar Silver Blaze phenomenon takes places for QCD with one flavor and arbitrary $\theta$-angle , and for QCD with two flavors with different quark masses $m_1$ and $m_2$ . In the latter case the Silver Blaze phenomenon is that the chiral condensate remains constant when $m_2$ crosses zero at fixed $m_1>0$ until the Dashen point $m_2 = -m_1$, where the chiral condensate has a discontinuity. In terms of contributions from the Dirac spectrum the shift of the discontinuity from $m_2=0$ to $m_2=-m_1$ also arises from exponentially large calculations between the zero modes and nonzero modes when $m_2 <0$. All calculations are performed in the microscopic or $\epsilon$ domain of QCD and results for arbitrary $\theta$-angle are discussed as well.
 J.J.M. Verbaarschot and T. Wettig, Phys. Rev. D90 (2014) 113.  M. Kieburg, J.J.M. Verbaarschot and T. Wettig, (2017) in preparation.
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