Constraints on the Dirac spectrum from chiral symmetry restoration and the fate of $\mathrm{U}(1)_A$ symmetry

TL;DR

This paper investigates how chiral symmetry restoration in QCD with two light quarks constrains the Dirac spectrum and examines the persistence of $ ext{U}(1)_A$ symmetry breaking, revealing a topological origin for near-zero spectral peaks.

Contribution

It establishes constraints on the Dirac spectral density during chiral symmetry restoration and links the near-zero mode peak to topological effects, advancing understanding of symmetry breaking.

Findings

$ ext{U}(1)_A$ remains broken if a near-zero spectral peak develops.

Restoration of $ ext{SU}(2)_A$ requires the spectral peak to be topological in origin.

The spectral density near zero is constrained to behave as $O(m^4)/ ext{}\lambda$ in the chiral limit.

Abstract

I discuss chiral symmetry restoration in the chiral limit $m\to 0$ of QCD with two light quark flavours of mass $m$, focussing on its consequences for scalar and pseudoscalar susceptibilities, and on the resulting constraints on the Dirac spectrum. I show that $\mathrm{U}(1)_A$ symmetry remains broken in the $\mathrm{SU}(2)_A$ symmetric phase if the spectral density $\rho(\lambda;m)$ develops a singular near-zero peak, tending to $O(m^4)/\lambda$ in the chiral limit. Moreover, $\mathrm{SU}(2)_A$ restoration requires that the number of modes in the peak be proportional to the topological susceptibility, indicating that such a peak must be of topological origin.