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arxiv: 2509.24611 · v3 · pith:F45V3KFOnew · submitted 2025-09-29 · ✦ hep-ph

Non-extensive NJL model study of QCD phase structure with chiral imbalance and strong magnetic fields

Xiang-Qiong Liu , Sheng-Qin Feng This is my paper

Pith reviewed 2026-05-25 07:48 UTC · model grok-4.3

classification ✦ hep-ph
keywords NJL modelTsallis statisticsQCD phase diagramchiral imbalancemagnetic catalysisnon-extensive statisticschiral symmetry restorationstrong magnetic fields
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The pith

Non-extensive NJL model shows higher Tsallis q lowers critical temperature for chiral symmetry restoration under magnetic fields and chiral imbalance.

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

The paper applies Tsallis non-extensive statistics within the two-flavor NJL model to examine QCD phase structure in the presence of strong magnetic fields and chiral chemical potential. It establishes that the critical temperature decreases as the non-extensive parameter q rises, indicating that non-equilibrium conditions favor chiral symmetry restoration at lower temperatures. The chiral chemical potential modifies the system's response to magnetic fields, producing a transition from magnetic catalysis to inverse magnetic catalysis under specific conditions. For q greater than 1 the critical temperature displays non-monotonic dependence on magnetic field strength. Thermodynamic quantities such as pressure become anisotropic and the speed of sound develops a dip near the critical temperature that moves to lower values with increasing q.

Core claim

Within the two-flavor NJL model extended by Tsallis statistics, the critical temperature Tc decreases with rising q, non-equilibrium effects promote chiral symmetry restoration at lower temperatures, the chiral chemical potential μ5 induces a change from magnetic catalysis to inverse magnetic catalysis, and for q > 1 the dependence of Tc on magnetic field eB becomes non-monotonic while pressure turns anisotropic and the speed-of-sound dip shifts to lower temperatures.

What carries the argument

Tsallis non-extensive statistics parameter q inserted into the two-flavor NJL model, which alters the quark distribution functions and thereby encodes non-equilibrium effects alongside magnetic field and chiral chemical potential μ5.

If this is right

  • Pressure becomes direction-dependent once magnetic fields are strong.
  • The speed of sound exhibits a dip near Tc whose location moves to lower temperatures as q increases.
  • Magnetic catalysis gives way to inverse magnetic catalysis once μ5 is sufficiently large.
  • Tc versus eB changes from monotonic to non-monotonic when q exceeds 1.

Where Pith is reading between the lines

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

  • Heavy-ion collision experiments could test the predicted lowering of Tc by comparing effective temperatures extracted from different centrality or rapidity bins.
  • The same non-extensive modification might be applied to other effective models such as the Polyakov-loop extended NJL to check consistency of the phase boundary shift.
  • If confirmed, the anisotropic pressure result would affect hydrodynamic modeling of the quark-gluon plasma in the presence of strong early-time magnetic fields.

Load-bearing premise

The Tsallis parameter q combined with the standard two-flavor NJL regularization and coupling constants sufficiently represents non-equilibrium effects throughout the QCD medium under strong magnetic fields and chiral imbalance.

What would settle it

Observation in heavy-ion collision data or lattice QCD that the critical temperature either fails to decrease with a measured non-equilibrium indicator or shows no non-monotonic variation with magnetic field strength when q exceeds 1.

Figures

Figures reproduced from arXiv: 2509.24611 by Sheng-Qin Feng, Xiang-Qiong Liu.

Figure 1
Figure 1. Figure 1: FIG. 1. The quark dynamical mass as a function of temperature [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dependence of the pseudocritical temperature [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Dependence of the pseudocritical temperature [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Chiral number density [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Chiral number density [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Pressure anisotropy under strong magnetic fields at fi [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The squared speed of sound [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
read the original abstract

Based on the two-flavor NJL model with Tsallis non-extensive statistics, this work explores the QCD phase structure and thermodynamic properties under strong magnetic fields and chiral imbalance. The Tsallis parameter $q$ captures non-equilibrium effects relevant to heavy-ion collisions. Key findings reveal that the critical temperature $T_c$ decreases with increasing $q$, indicating that non-equilibrium conditions promote chiral symmetry restoration at lower temperatures. The chiral chemical potential $\mu_5 $ significantly alters the magnetic response, with a transition from magnetic catalysis to inverse magnetic catalysis under certain conditions. For $q > 1$, non-monotonic behavior of $T_c$ with magnetic field $eB$ emerges. Pressure becomes anisotropic under strong $eB$, and the speed of sound exhibits a dip near $T_c$, shifting to lower temperatures with larger $q$. These results highlight how non-extensive statistics, chiral imbalance, and magnetic fields collectively influence the QCD phase diagram and thermodynamic observables, offering insights for interpreting heavy-ion collision data.

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.

Referee Report

2 major / 2 minor

Summary. The paper studies the two-flavor NJL model augmented by Tsallis non-extensive statistics (parameter q) to model non-equilibrium effects in the QCD phase diagram. It reports that the chiral critical temperature Tc decreases with increasing q, that the chiral chemical potential μ5 can drive a transition from magnetic catalysis to inverse magnetic catalysis, and that Tc(eB) becomes non-monotonic for q>1; additional results concern anisotropic pressure and the speed of sound near Tc.

Significance. If the numerical findings prove robust, the work would supply concrete, falsifiable predictions for how non-equilibrium statistics modify the QCD phase structure under strong magnetic fields and chiral imbalance, directly relevant to heavy-ion collision phenomenology. No machine-checked derivations or parameter-free results are presented.

major comments (2)
  1. [Model section / gap equation] Model formulation (gap equation and thermodynamic potential): the standard NJL parameters G and Λ are retained from vacuum fits performed at q=1, yet the Tsallis occupation numbers modify both the gap equation and the relation between the potential and pressure; no re-fitting procedure or demonstration that results are insensitive to the 3D cutoff for q≠1 is supplied, rendering the reported Tc(q) and catalysis-to-inverse-catalysis transition dependent on an unexamined model choice.
  2. [Results section (Tc vs eB and μ5)] Numerical results on magnetic response: the claimed non-monotonic Tc(eB) for q>1 and the μ5-induced change in magnetic catalysis rest on the Landau-level summation performed with the unmodified cutoff; without convergence checks or alternative regularizations shown for q>1, these central behaviors cannot be distinguished from cutoff artifacts.
minor comments (2)
  1. [Abstract] Abstract does not specify the regularization scheme, the precise fitting procedure for G and Λ, or any convergence tests, making it difficult for readers to assess the technical reliability of the quoted key findings.
  2. [Model section] Notation for the Tsallis distribution and its insertion into the NJL integrals should be written explicitly (e.g., the form of the q-generalized Fermi-Dirac factor) to allow direct reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and commit to revisions that strengthen the presentation of the model choices and numerical robustness.

read point-by-point responses

  1. Referee: Model formulation (gap equation and thermodynamic potential): the standard NJL parameters G and Λ are retained from vacuum fits performed at q=1, yet the Tsallis occupation numbers modify both the gap equation and the relation between the potential and pressure; no re-fitting procedure or demonstration that results are insensitive to the 3D cutoff for q≠1 is supplied, rendering the reported Tc(q) and catalysis-to-inverse-catalysis transition dependent on an unexamined model choice.

    Authors: We agree that further justification is warranted. The parameters G and Λ are fixed by vacuum phenomenology at equilibrium (q=1), which is the conventional approach when extending the NJL model to non-extensive statistics; a q-dependent re-fit would introduce uncontrolled additional parameters without a clear physical prescription. Nevertheless, to address the referee's concern we will add an explicit sensitivity study in the revised manuscript: we vary the cutoff Λ by ±10% around the vacuum value and recompute the key quantities (Tc(q), the μ5-driven transition, and Tc(eB) for q>1). The qualitative trends remain unchanged, demonstrating that the reported behaviors are not artifacts of the specific parameter choice. A brief discussion of this test will be inserted in the model section. revision: yes

  2. Referee: Numerical results on magnetic response: the claimed non-monotonic Tc(eB) for q>1 and the μ5-induced change in magnetic catalysis rest on the Landau-level summation performed with the unmodified cutoff; without convergence checks or alternative regularizations shown for q>1, these central behaviors cannot be distinguished from cutoff artifacts.

    Authors: We acknowledge that explicit convergence tests for q>1 are needed. The 3D cutoff is applied in the standard manner for magnetized NJL calculations, with the Landau-level sum truncated once higher levels contribute negligibly. In the revised version we will include supplementary figures that (i) increase the maximum Landau-level index until the gap equation stabilizes to better than 1% and (ii) compare the non-monotonic Tc(eB) and the μ5-induced catalysis reversal against a modest variation of the cutoff scheme. These checks will confirm that the reported non-monotonicity and the μ5-driven transition persist and are not regularization artifacts. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard model study with independent content.

full rationale

The paper applies the two-flavor NJL model extended by Tsallis statistics to compute phase structure under magnetic fields and μ5. Reported behaviors (Tc(q), catalysis transition, non-monotonic Tc(eB) for q>1) are direct numerical outputs from the model's gap equation and thermodynamic potential after inserting the Tsallis distribution. This constitutes a standard phenomenological exploration rather than any reduction of a claimed prediction to its inputs by construction. No self-citations are load-bearing, no uniqueness theorems are invoked, no ansatz is smuggled, and parameters are used in their conventional role for the model. The derivation chain is self-contained against the model's own equations.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard NJL effective-model framework and the applicability of Tsallis statistics; both are domain assumptions rather than derived results. No new entities are postulated.

free parameters (2)
  • Tsallis non-extensivity parameter q
    Introduced to capture non-equilibrium effects; its values are scanned to observe changes in Tc.
  • NJL coupling strength and cutoff scale
    Standard NJL parameters, normally fixed to vacuum observables, that control the gap equation solutions.
axioms (2)
  • domain assumption The two-flavor NJL model supplies an adequate effective description of chiral dynamics under external magnetic fields and chemical potentials.
    Invoked as the base framework for all calculations.
  • domain assumption Tsallis statistics with a single parameter q is applicable to the thermodynamic description of the QCD medium created in heavy-ion collisions.
    Used to incorporate non-equilibrium effects into the partition function.

pith-pipeline@v0.9.0 · 5707 in / 1555 out tokens · 49892 ms · 2026-05-25T07:48:24.748353+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    The Tsallis parameter q captures non-equilibrium effects... Tc decreases with increasing q... transition from magnetic catalysis to inverse magnetic catalysis... non-monotonic behavior of Tpc with magnetic field eB emerges (abstract, Sec. III).

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    By fitting the meson decay constant fπ=92.4 MeV and vacuum chiral condensate... adopt the parameters Λ=620 MeV, GΛ²=2.2 (Sec. III).

What do these tags mean?
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The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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