Mesoscopic QCD and the θ-vacua

Abstract

The partition function of QCD is analyzed for an arbitrary number of flavors, N f , and arbitrary quark masses including the contributions from all topological sectors in the Leutwyler–Smilga regime. For given N f and arbitrary vacuum angle, θ, the partition function can be reduced to N f −2 angular integrations of single Bessel functions. For two and three flavors, the θ dependence of the QCD vacuum is studied in detail. For N f =2 and 3, the chiral condensate decreases monotonically as θ increases from zero to π and the chiral condensate develops a cusp at θ= π for degenerate quark masses in the macroscopic limit. We find a discontinuity at θ= π in the first derivative of the energy density with respect to θ for degenerate quark masses. This corresponds to the first-order phase transition in which CP is spontaneously broken, known as Dashen's phenomena.