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arxiv: 2605.29701 · v1 · pith:RWNOWUG2new · submitted 2026-05-28 · ✦ hep-ph · hep-th

Thermodynamics in symmetry-improved Cornwall-Jackiw-Tomboulis formalism: application to the low-energy effective theory of QCD

Yuepeng Guan , Mamiya Kawaguchi , Shinya Matsuzaki , Akio Tomiya This is my paper

Pith reviewed 2026-06-29 06:34 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords symmetry-improved 2PICornwall-Jackiw-Tomboulis formalismQCD thermodynamicschiral transitionlinear sigma modelpressure prescriptionsWard-Takahashi identities
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The pith

Different pressure prescriptions in symmetry-improved 2PI formalism produce stable global thermodynamics for low-energy QCD effective theory.

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

The paper establishes that the symmetry-improved Cornwall-Jackiw-Tomboulis formalism restores Ward-Takahashi identities through auxiliary sources fixed self-consistently by the equilibrium state. This construction makes the thermodynamic definition of pressure nontrivial, so several prescriptions are formulated and tested in the three-flavor linear sigma model with quarks. The analysis covers the equation of state, isentropic trajectories, sound velocity, and trace anomaly across the chiral transition. A sympathetic reader would care because the results show that global thermodynamic structure remains consistent across prescriptions, with quantitative differences confined to the crossover and first-order region. This supplies a workable route to thermodynamically consistent observables in symmetry-improved 2PI calculations.

Core claim

In the symmetry-improved Cornwall-Jackiw-Tomboulis formalism, auxiliary sources that restore Ward-Takahashi identities are fixed self-consistently by the equilibrium state; several pressure definitions (vacuum-subtracted, source-matched, and pulled-back) then yield a globally stable thermodynamic structure, while quantitative differences remain localized near the chiral crossover and first-order transition when applied to the three-flavor linear sigma model with quarks.

What carries the argument

Symmetry-improved 2PI effective action with auxiliary sources fixed self-consistently by the equilibrium state, together with alternative pressure prescriptions that remove or match the explicit source-induced energy shift.

If this is right

  • The equation of state and trace anomaly remain largely insensitive to the choice of pressure prescription away from the transition region.
  • Isentropic trajectories and adiabatic sound velocity exhibit global stability across the different prescriptions.
  • Quantitative differences between prescriptions concentrate near the crossover and first-order transition region.
  • The construction supplies a practical framework for building thermodynamically consistent observables in symmetry-improved 2PI approaches.

Where Pith is reading between the lines

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

  • The same pressure-prescription strategy could be tested in other low-energy models that include finite baryon density.
  • Direct comparison of the prescription-dependent quantities near the transition with lattice QCD data would provide a concrete test of the localized differences.
  • The approach may reduce ambiguities when thermodynamic observables are extracted from other truncated 2PI calculations that employ auxiliary sources.

Load-bearing premise

The auxiliary sources can be fixed self-consistently by the equilibrium state in a manner that permits thermodynamically consistent pressure definitions without introducing uncontrolled artifacts in the equation of state or derived quantities.

What would settle it

A computation that finds large discrepancies between the different pressure prescriptions in the pressure, trace anomaly, or sound velocity at temperatures well above or below the transition region would falsify the claim of global stability.

Figures

Figures reproduced from arXiv: 2605.29701 by Akio Tomiya, Mamiya Kawaguchi, Shinya Matsuzaki, Yuepeng Guan.

Figure 1
Figure 1. Figure 1: FIG. 1. Feynman diagrams contributing to [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Two-loop diagrams of the effective quark contri [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. A schematic illustration of the SICJT formalism [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic illustration of the original effective action [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The sketch of the equilibria for different formula [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Temperature dependence of the chiral order parameters [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Temperature dependence of the pressures under three different prescriptions at [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Phase diagrams of the D [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The pressures, normalized in three different ways as in the text, are plotted as a function of temperature for several [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Isentropic trajectories (solid colored lines) in the phase diagram for [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Adiabatic sound velocities (colored-solid lines) as a function of temperature for several fixed values of the baryon [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Normalized volumetric densities of the specific [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Normalized trace anomaly [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Normalized thermodynamic quantities obtained from the CJT formalism and from the different pressure prescriptions [PITH_FULL_IMAGE:figures/full_fig_p040_14.png] view at source ↗
read the original abstract

We study the thermodynamics of the symmetry-improved Cornwall-Jackiw-Tomboulis (SICJT) formalism and apply it to a low-energy effective theory of QCD. In the symmetry-improved formulation, Ward-Takahashi identities are restored by auxiliary sources whose values are fixed self-consistently by the equilibrium state. While this construction improves the symmetry properties of the loop-wise truncated two-particle-irreducible (2PI) theory, it also makes the thermodynamic interpretation of the pressure nontrivial. We formulate several pressure prescriptions, including the conventional vacuum-subtracted pressure, a source-matched subtraction, and a pulled-back pressure in which the explicit source-induced energy shift is removed. Using the three-flavor linear sigma model with quarks, we analyze the equation of state, isentropic trajectories, adiabatic sound velocity, and trace anomaly across the chiral transition. We find that the global thermodynamic structure is stable under the different prescriptions, while quantitative differences are concentrated near the crossover and first-order transition region. These results establish a practical framework for constructing thermodynamically consistent observables in symmetry-improved 2PI approaches.

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

0 major / 2 minor

Summary. The manuscript develops thermodynamic observables within the symmetry-improved Cornwall-Jackiw-Tomboulis (SICJT) formalism, where auxiliary sources are fixed self-consistently to restore Ward-Takahashi identities in a loop-truncated 2PI theory. Applied to the three-flavor linear sigma model with quarks, it defines three pressure prescriptions (conventional vacuum subtraction, source-matched subtraction, and pulled-back pressure removing explicit source-induced shifts) and numerically compares the resulting equation of state, isentropic trajectories, adiabatic sound velocity, and trace anomaly across the chiral crossover and first-order transition region. The central result is that the global thermodynamic structure remains stable under these prescriptions, with quantitative differences localized near the transition.

Significance. If the reported stability holds, the work supplies a concrete, practical route to thermodynamically consistent observables in symmetry-improved 2PI effective theories of QCD. The explicit self-consistent implementation of auxiliary-source fixing and the direct numerical comparison of three distinct pressure definitions constitute a clear methodological advance; the finding that global features (EOS, isentropes, sound speed, trace anomaly) are insensitive to the choice of prescription while local differences remain confined to the transition region strengthens in applying such frameworks to finite-temperature QCD studies.

minor comments (2)
  1. [§3] §3 (pressure prescriptions): the explicit expressions for the three pressure definitions are given, but the relation between the pulled-back pressure and the conventional vacuum subtraction could be stated more compactly, perhaps as a single equation showing the subtracted source term.
  2. [Figure 4] Figure 4 (trace anomaly): the legend and axis labels are clear, yet the inset zooming on the first-order transition region would benefit from an explicit statement of the temperature range displayed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the clear summary of its contributions, and the recommendation for minor revision. No specific major comments are provided in the report, so we have no point-by-point responses to address. We will incorporate minor revisions as appropriate in the next version.

Circularity Check

0 steps flagged

No significant circularity; thermodynamic stability is a numerical result

full rationale

The paper defines auxiliary sources via self-consistent solution of the equilibrium equations in the symmetry-improved 2PI framework, then explicitly constructs and numerically compares three distinct pressure prescriptions (vacuum-subtracted, source-matched, pulled-back) inside the three-flavor linear sigma model. The central claim—that global thermodynamic quantities remain stable while differences localize near the transition—is obtained by direct evaluation of the equation of state, isentropes, sound velocity and trace anomaly across those prescriptions. No step equates a fitted parameter to a prediction by construction, no uniqueness theorem is imported from prior self-citation, and no ansatz is smuggled; the reported stability is an independent numerical outcome of the model dynamics and the chosen subtraction schemes.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities.

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