Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD
May 16, 2022
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 $\rho(\lambda\to0,m_l)$ 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 $a\to0$, 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 $\rho(\lambda\to0,m_l)$ is responsible for it.
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