The $\theta$-Vacua and the Leutwyler--Smilga Scaling Regime

Abstract

The partition function of QCD is studied in the Leutwyler--Smilga scaling regime for an arbitrary number of quark flavors and masses including the contributions from all winding numbers. For $N_f=2$ and degenerate quark masses, the partition function becomes independent of the quark masses at $\theta=\pi$ and subsequently the scalar chiral condensate vanishes. There is a discontinuity at $\theta=\pi$ in the first derivative of the energy density with respect to $\theta$ corresponding to the first--order phase transition in which CP is spontaneously broken, known as Dashen's phenomena. These properties are found to be insensitive to both the pattern of chiral symmetry breaking and the representation of the quark fields.