From chiral Random Matrix Theory to chiral Perturbation Theory

We study the spectrum of the QCD Dirac operator by means of the valence quark mass dependence of the chiral condensate in partially quenched Chiral Perturbation Theory (pqChPT) in the supersymmetric formulation of Bernard and Golterman. We consider valence quark masses both in the ergodic domain ($m_v \ll E_c$) and the diffusive domain ($m_v \gg E_c$). These domains are separated by a mass scale $E_c \sim F^2/\Sigma_0 L^2$ (with $F$ the pion decay constant, $\Sigma_0$ the chiral condensate and $L$ the size of the box). In the ergodic domain the effective super-Lagrangian reproduces the microscopic spectral density of chiral Random Matrix Theory (chRMT). We obtain a natural explanation of Damgaard's relation between the spectral density and the finite volume partition function with two additional flavors. We argue that in the ergodic domain the natural measure for the superunitary integration in the pqChPT partition function is noncompact. We find that the tail of the two-point spectral correlation function derived from pqChPT agrees with the chRMT result in the ergodic domain. In the diffusive domain we extend the results for the slope of the Dirac spectrum first obtained by Smilga and Stern. We find that the spectral density diverges logarithmically for nonzero topological susceptibility. We study the transition between the ergodic and the diffusive domain and identify a range where chRMT and pqChPT coincide.