Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD
Heng-Tong Ding, S.-T. Li, Swagato Mukherjee, Akio Tomiya, X.-D. Wang, Y. Zhang
Abstract
We introduce novel relations between the derivatives [${\ensuremath{\partial}}^{n}\ensuremath{\rho}(\ensuremath{\lambda},{m}_{l})/\ensuremath{\partial}{m}_{l}^{n}$] of the Dirac eigenvalue spectrum [$\ensuremath{\rho}(\ensuremath{\lambda},{m}_{l})$] with respect to the light sea quark mass (${m}_{l}$) and the ($n+1$)-point correlations among the eigenvalues ($\ensuremath{\lambda}$) of the massless Dirac operator. Using these relations we present lattice QCD results for ${\ensuremath{\partial}}^{n}\ensuremath{\rho}(\ensuremath{\lambda},{m}_{l})/\ensuremath{\partial}{m}_{l}^{n}$ ($n=1$, 2, 3) for ${m}_{l}$ corresponding to pion masses ${m}_{\ensuremath{\pi}}=160--55\text{ }\text{ }\mathrm{MeV}$ 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 $\ensuremath{\rho}(\ensuremath{\lambda}\ensuremath{\rightarrow}0,{m}_{l})$ 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 $a\ensuremath{\rightarrow}0$, and its amplitude is proportional to ${m}_{l}^{2}$. We demonstrate that this $\ensuremath{\rho}(\ensuremath{\lambda}\ensuremath{\rightarrow}0,{m}_{l})$ 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.