The Spectral Density of the Dirac Operator above T_c

Abstract

The importance of the spectral density of the Dirac operator in studying spontaneous chiral symmetry breaking and anomalous U(1) axial symmetry breaking are reviewed. It is shown that both types of symmetry breaking can be traced to effects of modes near zero virtuality. Above T_c, where chiral symmetry is restored, it is shown on general grounds that (in the massless quark limit), the density of states vanishes at zero virtuality faster than $\lambda$, where $\lambda$ is the virtuality-- $\rho(\lambda) \sim |\lambda|^\alpha$ is not possible for $\alpha \le 1$. Isospin invariance is used to show that $\rho(\lambda) \sim m_q^{1-\alpha} |\lambda|^\alpha$ is also not possible for $\alpha \le 1$. State-of-the-art lattice calculations are reviewed in light of these constraints. In particular, it is argued that violations of these constraints by lattice calculations indicate possible large systematic errors; this raises questions about $U(1)_A$ violating effects seen on the lattice.It is also shown that above $T_c$, the Dirac spectrum has a gap near zero (in the $m_q \to 0$ limit) unless contributions from quark-line-connected and disconnected contributions conspire to cancel.