Mesoscopic QCD and the θ-vacua Original Research Article

The partition function of QCD is analyzed for an arbitrary number of flavors, Nf, and arbitrary quark masses including the contributions from all topological sectors in the Leutwyler–Smilga regime. For given Nf and arbitrary vacuum angle, θ, the partition function can be reduced to Nf−2 angular integrations of single Bessel functions. For two and three flavors, the θ dependence of the QCD vacuum is studied in detail. For Nf=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 Dashenʹs phenomena.