Statistical properties of the spectrum of the QCD Dirac operator at low energy

We analyze the statistical properties of the spectrum of the QCD Dirac operator at low energy in a finite box of volume L⁴ by means of partially quenched Chiral Perturbation Theory, a low-energy effective field theory based on the symmetries of QCD. We derive the two-point spectral correlation function from the discontinuity of the chiral susceptibility. For eigenvalues much smaller than m_c=F²/ΣL², where F is the pion decay constant and Σ is the absolute value of the quark condensate, our result for the two-point correlation function coincides with the result previously obtained from chiral Random Matrix Theory (chRMT). The departure from the chRMT result above that scale is due to the contribution of the nonzero momentum modes. In terms of the variance of the number of eigenvalues in an interval containing n eigenvalues on average, it amounts to a crossover from a logn-behavior to a n²logn-behavior.