Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD

Abstract

We investigate the Dirac eigenvalue spectrum ($\rho (\lambda ,m_l)$) to study the microscopic origin of axial anomaly in high temperature phase of QCD. We propose novel relations between the derivatives (${\partial }^n\rho (\lambda ,m_l)/\partial m_l^n$) of the Dirac eigenvalue spectrum with respect to the quark mass ($m_l$) and the $(n+1)$-point correlations among the eigenvalues ($\lambda$) of the massless Dirac operator. Based on these relations, we present lattice QCD results for ${\partial }^n\rho (\lambda ,m_l)/\partial m_l^n$ ($n=1,2,3$) with $m_l$ 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 and the tree-level Symanzik gauge action with the physical 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, which 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 $m_l^2$. After continuum and chiral extrapolations, we find that the axial anomaly remains manifested in two-point correlation functions of scalar and pseudo-scalar mesons in the chiral limit. We demonstrate that the behavior of $\rho (\lambda \to 0,m_l)$ is responsible for it.

Comments: 9 pages, 4 figures, Contribution to the 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021 Zoom/Gather@Massachusetts Institute of Technology