Axial U(1) symmetry in the chiral symmetric phase of 2-flavor QCD at finite temperature

Abstract: We discuss the fate of the axial U(1) symmetry in 2-flavor QCD at finite temperature, where the non-singlet chiral symmetry is restored. We first summarize the previous theoretical investigation on the relation between the eigenvalue density of the Dirac operator and the axial U(1) symmetry. We show that the eigenvalue density near the origin behaves as $\lambda^\gamma$ with $\gamma > 2$ in the chirally symmetric phase, where $\lambda$ is an eigenvalue. This implies that the axial U(1) symmetry is partially restored, so that the low energy symmetry of the theory becomes SU(2)$\otimes$ SU(2)$\otimes$ Z$_4$. Secondly, we report recent numerical investigations on this issue by lattice QCD simulations with lattice chiral fermions such as Overlap or improved domain-wall fermions. Our preliminary results indicate that the eigenvalue density seems to have a gap at the origin just above $T_c$, the temperature of the chiral symmetry restoration, which implies the axial U(1) symmetry is effectively restored at high temperature. We also point out an importance of the exact lattice chiral symmetry to obtain correct results on this issue.