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 θdependence of the QCD vacuum is studied in detail. For N_f= 2 and 3, the chiral condensate decreases monotonically as θincreases from zero to πand the chiral condensate develops a cusp at θ=πfor degenerate quark masses in the macroscopic limit. We find a discontinuity at θ=πin the first derivative of the energy density with respect to θfor degenerate quark masses. This corresponds to the first--order phase transition in which CP is spontaneously broken, known as Dashen's phenomena.
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