Dirac spectrum in the chirally symmetric phase of a gauge theory. II

TL;DR

This paper explores how the Dirac spectrum's features relate to the restoration of chiral symmetry and the breaking of the U(1)_A symmetry in gauge theories with two light fermion flavors, proposing specific spectral conditions and physical mechanisms involved.

Contribution

It identifies spectral features necessary for effective U(1)_A breaking in the symmetric phase and discusses a physical mechanism involving topological objects that produce these features.

Findings

Spectral density behaves as m^2 delta(λ) in the chiral limit.

Two-point eigenvalue correlation function is singular at zero.

Delocalized near-zero modes and instanton gas-like topological charge distribution.

Abstract

I discuss the consequences of the constraints imposed on the Dirac spectrum by the restoration of chiral symmetry in the chiral limit of gauge theories with two light fermion flavors, with particular attention to the fate of the anomalous $\mathrm{U}(1)_A$ symmetry. Under general, physically motivated assumptions on the spectral density and on the two-point eigenvalue correlation function, I show that effective $\mathrm{U}(1)_A$ breaking in the symmetric phase requires specific spectral features, including a spectral density effectively behaving as $m^2\delta(\lambda)$ in the chiral limit, a two-point function singular at zero, and delocalized near-zero modes, besides an instanton gas-like behavior of the topological charge distribution. I then discuss a $\mathrm{U}(1)_A$-breaking scenario characterized by a power-law divergent spectral peak tending to $O(m^4)/|\lambda|$ in the chiral limit and by a near-zero mobility edge, and argue that the mixing of the approximate zero modes associated with a dilute gas of topological objects provides a concrete physical mechanism producing the required spectral features, and so a viable mechanism for effective $\mathrm{U}(1)_A$ breaking in the symmetric phase of a gauge theory.