ABSTRACT:
The partition functions of gauge theories with spontaneously broken
chiral symmetries are 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. In the
Leutwyler--Smilga regime, the theories only depend on simple
combinations of quark masses, volume, chiral condensate and vacuum
angle. We consider the cases of quarks in the adjoint and fundamental
representation separately. For two and three flavors, the heta
dependence of the QCD vacuum is studied in detail. We find a
discontinuity at heta=pi in the first derivative of the energy
density with respect to heta for degenerate quark masses. This
corresponds to the first--order phase transition in which CP is
spontaneously broken, known as Dashen's phenomena. We derive simple
expressions for the chiral condensate and the topological density and
show that they are in fact related. By examining the zeros of the
various partition functions, we elucidate the mechanism leading to
Dashen's phenomena in QCD. |