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

Hidenori Fukaya; Sinya Aoki

arxiv: 2606.10462 · v1 · pith:2H2LRLTSnew · submitted 2026-06-09 · ✦ hep-lat

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

Sinya Aoki , Hidenori Fukaya This is my paper

Pith reviewed 2026-06-27 11:05 UTC · model grok-4.3

classification ✦ hep-lat

keywords chiral symmetry restorationDirac operator eigenvaluesaxial U(1) anomalyfinite temperature QCDanalyticity in quark masslattice QCDgluonic observables

0 comments

The pith

Counterexamples to arguments on chiral symmetry restoration at high temperature violate the analyticity of gluonic observables in squared quark mass.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper replies to a comment offering counterexamples meant to refute earlier claims about the eigenvalue density of the Dirac operator, chiral symmetry restoration, and the axial U(1) anomaly in QCD at finite temperature. The reply demonstrates that those counterexamples fail to satisfy the requirement that every gluonic observable remain an analytic function of the squared quark mass. It also identifies a technical error in the comment itself. A sympathetic reader would therefore conclude that the comment does not succeed in undermining the original arguments.

Core claim

The proposed counterexamples intended to refute the argument violate the crucial assumption of QCD at high temperatures that every gluonic observable is an analytic function of the squared quark mass m², and the comment contains a technical mistake; therefore its arguments are not valid.

What carries the argument

The assumption that every gluonic observable is an analytic function of the squared quark mass m².

If this is right

The original arguments linking the Dirac eigenvalue density to chiral symmetry restoration and the axial U(1) anomaly remain intact.
Counterexamples to those arguments must respect analyticity in m² to be considered valid.
The technical mistake identified in the comment further weakens its refutation attempt.
Discussions of finite-temperature QCD symmetry restoration continue to rely on the analyticity property of gluonic observables.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Future attempts to construct counterexamples in this area will need to incorporate or explicitly break the analyticity condition to be taken seriously.
The analyticity assumption may connect to broader questions about how quark mass dependence controls the phase structure of hot QCD.
Lattice simulations that test gluonic observables for analyticity in m² could provide independent checks on the assumption.

Load-bearing premise

That in QCD at high temperatures every gluonic observable is necessarily an analytic function of the squared quark mass m².

What would settle it

An explicit gluonic observable computed in high-temperature QCD that is not analytic in m² would falsify the assumption and allow the counterexamples to stand.

read the original abstract

We respond to the comment by Matteo Giordano [1] on our article [2]. We find -- and [1] itself acknowledges -- that the proposed counterexamples intended to refute our argument violate a crucial assumption of QCD at high temperatures, namely that every gluonic observable is an analytic function of the squared quark mass, $m^2$. We further point out a technical mistake found in [1]. We conclude that the arguments presented in [1] are not valid.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Short rebuttal asserts that the comment's counterexamples violate analyticity of gluonic observables in m² (which the comment allegedly accepts) plus a technical mistake, but supplies no derivation, mapping, or details on either point.

read the letter

The main takeaway is that this is a short defensive reply with no new calculations, results, or derivations. It states that the counterexamples in Giordano's comment violate the assumption that every gluonic observable is analytic in m² at high temperature, claims the comment itself acknowledges this, and flags an unspecified technical mistake, so the comment's arguments do not hold.

The paper does a clear job of isolating the analyticity condition as the load-bearing assumption that the counterexamples are said to break. If the acknowledgment claim is accurate, that undercuts the counterexamples without needing extra work.

The soft spots are straightforward. The reply invokes analyticity as a crucial QCD property but does not derive it, cite a prior derivation, or show explicitly how the counterexamples produce non-analytic gluonic observables. The technical mistake is mentioned without description. This leaves the rebuttal resting on the premise rather than demonstrating the breach.

The piece is only useful to the handful of lattice QCD people already following the specific exchange on the Dirac eigenvalue density, chiral restoration, and the axial anomaly at finite temperature. Anyone outside that loop gets nothing actionable.

I would not bring it to a reading group. It does not need or deserve peer review; replies of this length are normally published directly as comments if the original comment was.

Referee Report

2 major / 0 minor

Summary. This short reply manuscript responds to a comment by Matteo Giordano on the authors' prior work concerning chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature. The authors state that the counterexamples in the comment violate the assumption that every gluonic observable is an analytic function of the squared quark mass m² (a crucial assumption of QCD at high temperatures), note that the comment itself acknowledges this, identify an unspecified technical mistake in the comment, and conclude that the comment's arguments are therefore invalid.

Significance. If the analyticity premise is rigorously justified and the counterexamples are shown to breach it, the reply would defend the original claims by removing the proposed counterexamples from consideration. The manuscript contributes to clarifying a technical debate in lattice QCD but does not supply new derivations, data, or machine-checked results; its value is therefore conditional on addressing the explicit mapping from counterexamples to non-analyticity.

major comments (2)

The central claim that the counterexamples are invalid rests on their violation of analyticity in m² for gluonic observables, yet the manuscript asserts this violation without an explicit calculation, reference to a specific equation in [1], or demonstration that the counterexamples produce non-analytic gluonic quantities. This is load-bearing because the refutation reduces to the premise rather than a shown breach.
The technical mistake identified in [1] is invoked to support invalidity of the comment but is not described or localized to a particular statement, equation, or derivation in the comment. Without this detail the claim cannot be evaluated independently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the comments on our reply manuscript. We address the major comments point by point below, providing the strongest defense consistent with the content of our manuscript.

read point-by-point responses

Referee: The central claim that the counterexamples are invalid rests on their violation of analyticity in m² for gluonic observables, yet the manuscript asserts this violation without an explicit calculation, reference to a specific equation in [1], or demonstration that the counterexamples produce non-analytic gluonic quantities. This is load-bearing because the refutation reduces to the premise rather than a shown breach.

Authors: Our manuscript states that the comment [1] itself acknowledges the violation of the analyticity assumption in m² for gluonic observables by the proposed counterexamples. This acknowledgment by the commenter directly invalidates the counterexamples under the assumptions of QCD at high temperatures that underpin our original work. No separate explicit calculation or mapping is required in the reply, as the breach is conceded within [1]. revision: no
Referee: The technical mistake identified in [1] is invoked to support invalidity of the comment but is not described or localized to a particular statement, equation, or derivation in the comment. Without this detail the claim cannot be evaluated independently.

Authors: We agree that the specific technical mistake in [1] is not described or localized in our manuscript. We will revise the manuscript to include a description of this mistake and localize it to the relevant statement or derivation in the comment. revision: yes

Circularity Check

0 steps flagged

No circularity: refutation rests on external assumption and error identification

full rationale

The paper states that counterexamples violate the analyticity-in-m² assumption for gluonic observables (presented as a standard QCD property at high T) and identifies a technical mistake in the comment. No equation, derivation step, or self-citation reduces the conclusion to a fitted input, self-defined quantity, or prior result by construction. The argument is a logical inconsistency check against an external premise plus error pointing, remaining self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that gluonic observables are analytic in m² at high T; no free parameters or invented entities appear in the abstract.

axioms (1)

domain assumption Every gluonic observable is an analytic function of the squared quark mass m² in high-temperature QCD.
Explicitly identified in the abstract as the crucial assumption violated by the proposed counterexamples.

pith-pipeline@v0.9.1-grok · 5613 in / 1133 out tokens · 26978 ms · 2026-06-27T11:05:15.659990+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages · 1 internal anchor

[1]

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

Matteo Giordano, “Comment on “Chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature”,” preceding Comment, to 4 appear in PRD
[2]

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

S. Aoki, H. Fukaya and Y. Taniguchi, “Chiral symmetry restoration, eigenvalue density of Dirac operator and axial U(1) anomaly at finite temperature,” Phys. Rev. D86, 114512 (2012) doi:10.1103/PhysRevD.86.114512 [arXiv:1209.2061 [hep-lat]]. 5

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.86.114512 2012

[1] [1]

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

Matteo Giordano, “Comment on “Chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature”,” preceding Comment, to 4 appear in PRD

[2] [2]

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

S. Aoki, H. Fukaya and Y. Taniguchi, “Chiral symmetry restoration, eigenvalue density of Dirac operator and axial U(1) anomaly at finite temperature,” Phys. Rev. D86, 114512 (2012) doi:10.1103/PhysRevD.86.114512 [arXiv:1209.2061 [hep-lat]]. 5

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.86.114512 2012