Comment on "Chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature"

TL;DR

This paper critiques a recent claim that the spectral density and topological susceptibility vanish in the chirally symmetric phase of QCD, arguing that a key step in the proof is unjustified and the conclusions need reevaluation.

Contribution

The paper challenges the validity of previous proofs regarding spectral density and topological susceptibility in QCD, highlighting gaps in the argument.

Findings

The proof's key step is not justified.

The conclusions about spectral density and susceptibility may not hold.

Reassessment of the original claims is necessary.

Abstract

Aoki, Fukaya, and Taniguchi claim that both the spectral density of the Dirac operator at the origin and the topological susceptibility must vanish identically for sufficiently small but nonzero mass $m$ in the chirally symmetric phase of QCD with two light quark flavors, under certain technical assumptions on the spectrum and on the dependence of observables on $m$. Independently of these assumptions, I argue that a crucial step of their proof is not justified, and the validity of these conclusions should be reassessed.