From chiral random matrix theory to chiral perturbation theory

We study the spectrum of the QCD Dirac operator by means of the valence qua rk mass dependence of the chiral condensate in partially quenched Chiral Pe rturbation Theory (pqChPT) in the supersymmetric formulation of Bernard and Golterman. We consider valence quark masses both in the ergodic domain (m( upsilon) << E-c) and the diffusive domain (m(upsilon) >> E-c). These domain s are separated by a mass scale E-c similar to 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 m icroscopic spectral density of chiral Random Matrix Theory (chRMT). We obta in a natural explanation of Damgaard's relation between the spectral densit y and the finite volume partition function with two additional flavors. We argue that in the ergodic domain the natural measure for the superunitary i ntegration in the pqChPT partition function is non-compact. We find that th e tail of the two-point spectral correlation function derived from pqChPT a grees 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 logarithmical ly for non-zero topological susceptibility. We study the transition between the ergodic and the diffusive domains and identify a range where chRMT and pqChPT coincide. (C) 1999 Elsevier Science B.V.