Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD.

We introduce novel relations between the derivatives [∂^{n}ρ(λ,m_{l})/∂m_{l}^{n}] of the Dirac eigenvalue spectrum [ρ(λ,m_{l})] with respect to the light sea quark mass (m_{l}) and the (n+1)-point correlations among the eigenvalues (λ) of the massless Dirac operator. Using these relations we present lattice QCD results for ∂^{n}ρ(λ,m_{l})/∂m_{l}^{n} (n=1, 2, 3) for m_{l} corresponding to pion masses m_{π}=160-55  MeV and at a temperature of about 1.6 times the chiral phase transition temperature. Calculations were carried out using (2+1) flavors of highly improved staggered quarks with the physical value of strange quark mass, three lattice spacings a=0.12, 0.08, 0.06 fm, and lattices having aspect ratios 4-9. We find that ρ(λ→0,m_{l}) develops a peaked structure. This peaked structure arises due to non-Poisson correlations within the infrared part of the Dirac eigenvalue spectrum, becomes sharper as a→0, and its amplitude is proportional to m_{l}^{2}. We demonstrate that this ρ(λ→0,m_{l}) is responsible for the manifestations of axial anomaly in two-point correlation functions of light scalar and pseudoscalar mesons. After continuum and chiral extrapolations we find that axial anomaly remains manifested in two-point correlation functions of scalar and pseudoscalar mesons in the chiral limit.