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

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 quark mass $m$ in the chirally symmetric phase of quantum chromodynamics with two light quark flavors, under certain technical assumptions on the spectrum and on the dependence of observables on $m$. I argue that a crucial step of their proof is not justified, and the validity of these conclusions should be reassessed. In an additional note I refute the objections raised by S. Aoki and H. Fukaya in their reply to my comment. I show that both the $m^2$-analyticity assumption they invoke against my arguments, and the very claim that these arguments criticize, imply that all local gluonic correlators are independent of the light-quark mass at small mass. This behavior, however, is not observed in quantum chromodynamics. I then show that their claim of having found a mistake in my arguments is baseless.

Comments: Revised version of the comment on arXiv:1209.2061, to appear in PRD, and a note on the reply by the authors, arXiv:2606.10462; 8 pages