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arxiv: 2503.17291 · v3 · submitted 2025-03-21 · ⚛️ nucl-th · hep-ph· hep-th

Linear sigma model with quarks and Polyakov loop in rotation: phase diagrams, Tolman-Ehrenfest law and mechanical properties

Pracheta Singha , Sergiu Busuioc , Victor E. Ambrus , Maxim N. Chernodub This is my paper

Pith reviewed 2026-05-22 22:32 UTC · model grok-4.3

classification ⚛️ nucl-th hep-phhep-th
keywords rotationQCD phase diagramlinear sigma modelPolyakov loopchiral restorationdeconfinementTolman-Ehrenfest lawmechanical properties
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The pith

In this effective QCD model, rotation lowers the critical temperatures for both chiral restoration and deconfinement.

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

The paper investigates rotation effects on QCD-like matter using the Polyakov-enhanced linear sigma model with quarks. It computes the phase diagram in temperature, density, and angular frequency under a homogeneous approximation and causality-enforcing boundary conditions. The central finding is that both transition temperatures decrease with rising angular velocity, in disagreement with lattice results. The work also shows thermodynamic consistency with the Tolman-Ehrenfest law only in the large-volume limit and extracts mechanical response coefficients from the thermodynamic potential.

Core claim

Within the Polyakov linear sigma model with quarks in the homogeneous approximation and with spectral boundary conditions at the cylinder surface, the critical temperatures of the chiral restoration and deconfinement transitions both decrease as angular frequency increases; thermodynamic consistency with the Tolman-Ehrenfest law is recovered only for large system size, while mechanical coefficients such as the moment of inertia and K_n shape factors are obtained from derivatives of the grand potential.

What carries the argument

Polyakov-enhanced linear sigma model coupled to quarks under rotation, in the homogeneous approximation with spectral boundary conditions enforcing causality.

If this is right

  • Both the chiral and deconfinement transition lines shift to lower temperatures with increasing angular frequency.
  • Thermodynamic quantities satisfy the Tolman-Ehrenfest law only when the cylindrical volume becomes large.
  • The moment of inertia and the K_n coefficients that measure the response of the thermodynamic potential to angular velocity are directly computable from the model.
  • The phase structure depends on the interplay between temperature, baryon chemical potential, and angular frequency within the causality constraint.

Where Pith is reading between the lines

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

  • The reported decrease in transition temperatures may require inhomogeneous field configurations or altered boundary conditions to reconcile with lattice QCD.
  • The mechanical coefficients could be tested in rotating heavy-ion collision environments where vorticity is present.
  • The large-volume limit needed for Tolman-Ehrenfest consistency suggests finite-size effects dominate the thermodynamics at accessible angular velocities.

Load-bearing premise

The system can be described by spatially uniform fields together with the specific spectral boundary conditions chosen to enforce causality at the cylindrical surface.

What would settle it

A first-principles lattice simulation that finds the critical temperatures increasing or staying constant with rotation would contradict the model's reported decrease.

Figures

Figures reproduced from arXiv: 2503.17291 by Maxim N. Chernodub, Pracheta Singha, Sergiu Busuioc, Victor E. Ambrus.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Chiral and deconfinement phase diagram for the unbounded, nonrotating LSM [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Double-peak structure of the slope [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The order parameters (a) [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Variation of [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Temperature dependence of [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Phase diagram in the [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Normalized scalar condensate [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Phase diagrams for the chiral (a) and deconfinement (b) transitions at [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Phase diagram for the chiral transition in the [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Phase diagram for the deconfinement transition, shown with colored lines, computed in the [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Phase diagram in the [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Tolman-Ehrenfest prediction in the (P)LSM [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: shows —with the horizontal dashed lines— the TE prediction for the LSMq (panel a) and the PLSMq (panel b) models, obtained by solving Eqs. (109) and (132), respectively. It can be seen that the numerical results for the pseudocritical temperature Tσ of the chiral symmetry restoration transition at µ = 0 and fixed ΩR, indeed converge, in both models, to the TE predictions as the system size R is increased … view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The critical chemical potential [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Scaled moment of inertia, [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p034_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Scaled shape coefficients 2! [PITH_FULL_IMAGE:figures/full_fig_p037_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Scaled shape coefficients, 2! [PITH_FULL_IMAGE:figures/full_fig_p038_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Scaled shape coefficients, 2! [PITH_FULL_IMAGE:figures/full_fig_p038_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. (a) Expectation value of the Polyakov loop as a function of temperature at zero chemical potential for the cases of [PITH_FULL_IMAGE:figures/full_fig_p042_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. The chiral and confinement phase diagram for two different values of [PITH_FULL_IMAGE:figures/full_fig_p042_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. Temperature dependence of (a) scaled [PITH_FULL_IMAGE:figures/full_fig_p044_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Chiral and deconfinement phase diagram with chemical potential dependent [PITH_FULL_IMAGE:figures/full_fig_p045_24.png] view at source ↗
read the original abstract

We study the effect of rotation on the confining and chiral properties of QCD using the Polyakov-enhanced linear sigma model coupled to quarks. Working in the homogeneous approximation, we obtain the phase diagram at finite temperature, baryon density and angular frequency, taking into account the causality constraint enforced by the spectral boundary conditions at a cylindrical surface. We explicitly address various limits with respect to system size $R$, angular frequency $\Omega$ and chemical potential $\mu$. We demonstrate that, in this model, the critical temperatures of both the chiral restoration and the deconfinement transitions diminish in response to the increasing rotation, being in contradiction with the first-principle lattice results. We demonstrate that consistency between the thermodynamics of the model and the Tolman-Ehrenfest law is achieved in the limit of large volume. We also compute the mechanical characteristics of the rotating plasma, such as the moment of inertia and the $K_n$ shape coefficients describing the response of the thermodynamic potential with respect to the increase of angular velocity $\Omega$.

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 / 0 minor

Summary. The manuscript studies the Polyakov-linear sigma model coupled to quarks under rotation in the homogeneous approximation. It constructs the phase diagram in T, μ, and Ω subject to causality-enforcing spectral boundary conditions on a cylinder of radius R, reports that both chiral-restoration and deconfinement critical temperatures decrease with increasing Ω (in contrast to lattice QCD), verifies thermodynamic consistency with the Tolman-Ehrenfest law only in the large-R limit, and computes mechanical response functions including the moment of inertia and Kn shape coefficients.

Significance. If the homogeneous ansatz remains valid, the calculation supplies an explicit effective-model realization of rotation effects on the QCD phase structure together with concrete mechanical coefficients; the systematic treatment of the R, Ω, μ limits and the large-volume recovery of Tolman-Ehrenfest consistency are positive technical features. The reported disagreement with lattice data is already noted by the authors and therefore does not constitute an internal inconsistency, but it does limit immediate phenomenological weight.

major comments (2)
  1. [Abstract / homogeneous approximation] Abstract and the homogeneous-approximation paragraph: the central claim that both critical temperatures decrease with Ω is obtained exclusively under the spatially constant order-parameter ansatz. The Tolman-Ehrenfest effect and centrifugal contributions are expected to induce radial dependence; the manuscript only demonstrates consistency in the large-R limit, so the finite-R phase boundaries may shift once spatially varying configurations are allowed.
  2. [Model definition / parameter section] Parameter inheritance (implicit in the model setup): the Polyakov-linear-sigma parameters are taken from fits to non-rotating phenomenology; the Ω dependence is therefore not an independent prediction but follows by construction from those fixed values, weakening the status of the reported phase diagram as a genuine rotation-induced effect.

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.

read point-by-point responses

  1. Referee: [Abstract / homogeneous approximation] Abstract and the homogeneous-approximation paragraph: the central claim that both critical temperatures decrease with Ω is obtained exclusively under the spatially constant order-parameter ansatz. The Tolman-Ehrenfest effect and centrifugal contributions are expected to induce radial dependence; the manuscript only demonstrates consistency in the large-R limit, so the finite-R phase boundaries may shift once spatially varying configurations are allowed.

    Authors: We agree that all reported results, including the decrease of both critical temperatures with Ω, are obtained strictly within the homogeneous (spatially constant) ansatz for the order parameters. This approximation is explicitly stated throughout the manuscript and is adopted to render the rotating system tractable while enforcing causality via spectral boundary conditions. We already note that thermodynamic consistency with the Tolman-Ehrenfest law holds only in the large-R limit. Spatially inhomogeneous configurations could indeed alter the finite-R phase boundaries, but their inclusion would require a substantially more involved numerical treatment that lies beyond the present scope. We will revise the abstract and the relevant introductory paragraphs to state more explicitly that the phase diagram is obtained under the homogeneous approximation. revision: yes

  2. Referee: [Model definition / parameter section] Parameter inheritance (implicit in the model setup): the Polyakov-linear-sigma parameters are taken from fits to non-rotating phenomenology; the Ω dependence is therefore not an independent prediction but follows by construction from those fixed values, weakening the status of the reported phase diagram as a genuine rotation-induced effect.

    Authors: The model parameters are fixed by reproducing vacuum phenomenology and the non-rotating transition temperatures, as is standard for effective QCD models. Once these parameters are set, the rotation dependence arises dynamically from the quark coupling, the Polyakov-loop potential, and the causality-enforcing boundary conditions. The observed decrease of critical temperatures with Ω is therefore a direct consequence of the model's structure under rotation rather than an additional free parameter. While the quantitative location of the phase boundaries inherits the non-rotating calibration, the qualitative trend constitutes a genuine prediction of the effective theory for rotating matter. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained model calculation

full rationale

The paper fixes model parameters from non-rotating QCD phenomenology (standard effective-model practice) and then solves the effective potential in the rotating frame using the homogeneous ansatz and spectral boundary conditions. The reported decrease in critical temperatures with angular velocity emerges from the modified Dirac operator and metric, not from any fit to rotating data or self-referential definition. No load-bearing self-citation chain or ansatz smuggling is present; the central claim is a direct numerical output of the model equations and remains independently falsifiable against lattice results. This is the normal, non-circular case for an effective-model study.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central results rest on the standard parameters of the Polyakov-linear-sigma model (fitted at zero rotation) plus two domain assumptions required for the rotating calculation.

free parameters (1)
  • Polyakov-linear-sigma model parameters
    Standard vacuum and lattice-calibrated couplings and masses that are held fixed while rotation is varied.
axioms (2)
  • domain assumption Homogeneous field approximation
    Fields are taken spatially constant inside the cylinder.
  • domain assumption Spectral boundary conditions at cylindrical surface
    Enforce causality by restricting the allowed modes at radius R.

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