Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD

Abstract

We introduce novel relations between the derivatives [${\partial }^n\rho (\lambda ,m_l)/\partial m_l^n$] of the Dirac eigenvalue spectrum [$\rho (\lambda ,m_l)$] with respect to the light sea quark mass ($m_l$) and the $(n+1)$-point correlations among the eigenvalues ($\lambda$) of the massless Dirac operator. Using these relations we present lattice QCD results for ${\partial }^n\rho (\lambda ,m_l)/\partial m_l^n$ ($n=1,2,3$) for $m_l$ corresponding to pion masses $m_{\pi }=160-55$ 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 $\rho (\lambda \to 0,m_l)$ 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 $a\to 0$, and its amplitude is proportional to $m_l^2$. We demonstrate that this $\rho (\lambda \to 0,m_l)$ 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.

Comments: 17 pages, 18 figures, minor updates to match the published version in PRL