Quark matter at finite temperature and proto-quark stars with the axion effects in SU(3) Nambu-Jona-Lasinio model

He Liu; Hong-Ming Liu; Jiao-Jiao Wang; Ke Liu; Min Ju; Peng-Cheng Chu; Peng Wu; Xiao-Hua Li; Xu-Hao Wu; Ying Zhou

REVIEW 4 major objections 5 minor 90 references

Quark matter at finite temperature and proto-quark stars with the axion effects in SU(3) Nambu-Jona-Lasinio model

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Axion fields and vector interactions stiffen hot strange quark matter, letting proto-quark stars reach 2.67 solar masses before cooling down.

desk verdict Competent NJL parameter study of proto-quark stars; headline mass sequence is read off at a hand-picked θ=π and should not be treated as an axion prediction. read the letter →

arxiv 2608.06024 v1 pith:SP5MSMKT submitted 2026-08-06 nucl-th

Peng-Cheng Chu , Jiao-Jiao Wang , Ke Liu , Peng Wu , Yu-Heng Liu , He Liu , Hong-Ming Liu , Xiao-Hua Li

show 3 more authors

Min Ju Xu-Hao Wu Ying Zhou

This is my paper · ORCID

classification nucl-th PACS 21.65.Qr97.60.Jd26.60.Kp21.30.Fe95.30.Tg

keywords proto-quarkstarstrangequarkmatterNambu-Jona-Lasiniomodelaxionfieldfinitetemperatureequationofstateneutrinotrappingisentropicevolution

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that including an axion field and flavor-dependent vector interactions in the SU(3) Nambu-Jona-Lasinio model at finite temperature changes the behavior of strange quark matter and the proto-quark stars built from it. As the axion angle $\theta$ grows from $0.35\pi$ toward $\pi$, the equation of state stiffens: the minimum free energy per baryon drops, pressure rises, and the maximum mass of a proto-quark star increases at every isentropic stage. The paper also finds a thermodynamic signature: trapped neutrinos raise the electron density while lowering the core temperature relative to a neutrino-free star at the same entropy. For $\theta = \pi$, the maximum mass reaches $2.61\,M_\odot$ during the neutrino-trapped stage, peaks at $2.67\,M_\odot$ during the hot neutrino-free stage, and settles at $2.51\,M_\odot$ when the star is cold. These numbers sit close to the most massive compact-star candidates observed, which is why the result matters for identifying the nature of such objects.

What carries the argument

The central object is the SU(3) Nambu-Jona-Lasinio Lagrangian with an axion-modified 't Hooft determinant term (Eq.~1): a low-energy effective model of QCD in which the axion enters as a chiral rotation angle $\theta$ multiplying the six-fermion interaction, and a flavor-dependent vector interaction with coupling $G_V$. In the mean-field approximation, $\theta$ tilts the balance between scalar and pseudoscalar quark condensates, changing the constituent quark masses and, through the thermodynamic potential, the equation of state. The proto-quark star is evolved through three isentropic snapshots fixing entropy per baryon and lepton fraction, and the maximum stellar mass is computed from the resulting equation of state via the standard stellar-structure equations.

What would settle it

Recompute the proto-quark star maximum mass for the same PCP parameter set but with theta determined at each density and temperature by minimizing the axion effective potential; if the climb from 2.61 solar masses to 2.67 solar masses across the isentropic stages does not survive, the claimed axion-driven stiffening is an artifact of the free parameter.

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Extended reading notes

Core claim

The central discovery claimed is that axion effects and vector interactions in the SU(3) NJL model, applied at finite temperature along the isentropic evolution of a proto-quark star, noticeably alter the thermodynamics of strange quark matter and the maximum mass the star can support. Specifically, with the vector coupling fixed at $G_V = 2G_S$ and the axion angle scanned from $0.35\pi$ to $\pi$, the equation of state becomes stiffer with increasing $\theta$ and temperature: the free-energy minimum per baryon falls steeply at $T = 50$ MeV, the constituent $u$- and $d$-quark masses drop, and the entropy density rises with both temperature and vector coupling. Along the three isentropic snapshots (trapped-neutrino stage with entropy per baryon 1 and lepton fraction 0.4; hot stage with entropy per baryon 2 and no neutrinos; cold stage), the maximum proto-quark star mass for $\theta = \pi$ evolves from $2.61\,M_\odot$ to $2.67\,M_\odot$ and then down to $2.51\,M_\odot$. The paper also reports that at fixed entropy, neutrino trapping suppresses the core temperature while increasing electron number density, and that increasing $\theta$ shifts the quark flavor fractions. The paper frames these as evidence that axion fields and particle composition should be included in models of hot compact stars.

Load-bearing premise

Every theta-dependent result rests on the axion angle being fixed by hand (scanned from 0.35pi to pi) rather than determined by minimizing the axion potential, so if theta were solved dynamically the predicted stiffening and mass shifts could substantially change or vanish.

Editorial extensions

If this is right

If the axion stiffening is real, a proto-quark star can be born with a mass near $2.6$--$2.7\,M_\odot$ even though the cold quark star it leaves behind is lighter, so the observed mass of a compact object need not match the maximum mass of cold quark matter.
Neutrino trapping lowers the core temperature at fixed entropy, so the thermal and neutrino signals from a newborn quark star depend on whether neutrinos are still trapped and on the axion angle.
Increasing $\theta$ shifts the flavor fractions of $u$, $d$, and $s$ quarks and raises the electron density, which changes the weak-equilibrium composition and therefore the cooling and neutrino emissivity of the star.
Axion effects broaden the region where strange quark matter is absolutely stable at finite temperature, so the parameter space of possible quark stars is wider than zero-temperature studies suggest.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The authors leave implicit that solving for $\theta$ dynamically from the axion potential could shrink or rescale the predicted $0.1\,M_\odot$ mass shift; a dynamic treatment is the natural next check before using $2.67\,M_\odot$ as an astrophysical constraint.
The three discrete isentropic snapshots could be replaced by a continuous time evolution with neutrino diffusion; this would test whether the maximum mass genuinely peaks at $2.67\,M_\odot$ or whether the peak is an artifact of the chosen snapshots.
The same axion-modified thermodynamics could be applied to neutron-star merger remnants or to the cooling curves of newborn stars, giving observable gravitational-wave or neutrino predictions that the paper does not compute.
The reported rise in electron density with trapped neutrinos suggests that axion angle and lepton fraction together set the neutrino opacity, which could be explored with a full transport calculation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the two-minute version. This paper extends the same group's zero-temperature NJL+axion framework to finite temperature and proto-quark stars, with flavor-dependent vector interactions and the standard three-stage isentropic evolution. That extension is real and new. The stage-to-stage mass increase (2.61 to 2.67 to 2.51 solar masses at θ=π) is internally consistent, and the result that trapped neutrinos raise electron density while lowering core temperature is a genuine model prediction. The equations are standard mean-field NJL, and the paper is honestly structured as a parameter scan.

The soft spot is the axion. θ appears only as a static multiplier of the 't Hooft determinant. It is never minimized against an axion potential or tied to a field equation, and the paper's own Fig. 1 shows the free energy dropping steeply as θ approaches π. So the choice θ=π is not a small perturbation; it is the most extreme point of the scan. The headline masses are read off at that extreme. If you added the usual m_a^2 f_a^2 (1−cos θ) potential, the equilibrium θ would likely move far from π, and the advertised axion effect could largely vanish. That is the main reason the quantitative sequence isn't a prediction.

A second issue: GV=2GS is chosen so the cold star lands at 2.51 M_sun. That makes the zero-temperature maximum mass a fit target, not an outcome. The finite-temperature difference between stages is the genuine output.

Also, numerical implementation details are sparse: gap equations with pseudoscalar condensates, TOV integration, and entropy extraction are described only schematically. The paper would be much more useful with code or a data release.

Who benefits: people doing NJL-based quark-star phenomenology, especially those interested in proto-neutron-star evolution. It's a solid, incremental contribution, not a breakthrough. I would send it to peer review, because the setup is coherent and the qualitative statements are worth publishing, but I'd ask the referee to push for a dynamical θ or at least a sensitivity analysis showing how the sequence depends on θ being fixed at its maximum, plus details on the numerics.

Referee Report

4 major / 5 minor

Summary. The paper studies strange quark matter (SQM) and proto-quark stars (PQSs) in a SU(3) Nambu-Jona-Lasinio model at finite temperature, adding a 't Hooft determinant with an axion angle θ and a flavor-dependent vector interaction. The authors compute the free energy, equation of state, constituent quark masses, entropy density, and maximum stellar masses for three isentropic evolutionary stages (neutrino-trapped, neutrino-free hot, and cold). The central quantitative result is that for θ = π the maximum PQS mass rises from 2.61 M⊙ in the neutrino-trapped stage to 2.67 M⊙ in the second hot stage, then falls to 2.51 M⊙ when cold. The paper also reports that trapped neutrinos increase the electron density while lowering the core temperature relative to the neutrino-free case.

Significance. If the central claims were established, the model would offer a way to reconcile massive compact objects (PSR J0952-0607, GW190814) with quark stars and would identify finite-temperature, lepton-composition signatures of proto-quark stars. The study has a clear organizational structure, uses a standard NJL framework, and considers physically motivated isentropic stages. The finite-temperature lepton-trapping effects appear to be genuine model outputs. However, the axion-related claims are presently not predictions from axion dynamics: θ is a scanned external parameter, and the zero-temperature maximum mass is effectively calibrated by choosing GV = 2 GS. These issues are load-bearing for the headline mass sequence, so the paper needs substantial revision before the quantitative conclusions can be accepted.

major comments (4)

[Eq. (1), Sec. III, Fig. 5] The axion angle θ is introduced as a constant c-number in the 't Hooft determinant, and in Sec. III it is scanned over [0.35π, π] with the headline masses read off at θ = π. No axion kinetic term, potential, or field equation is introduced, and the mean-field free energy is never minimized with respect to θ. Since Fig. 1 shows that the free energy per baryon changes dramatically with θ (e.g., at T = 50 MeV it drops from about 728 MeV to 259 MeV as θ goes from 0.35π to π), the choice of θ is not a small perturbation. If a standard QCD axion potential m_a^2 f_a^2 (1 - cos θ) were included, its minimum at θ = 0 would compete with the NJL contribution, and the equilibrium θ could move far from π. The authors should either include the axion potential and minimize the free energy with respect to θ, or explicitly reframe the results as an exploratory scan over an external parameter. As written, the claim that 'axion effects significantly influence' the EoS and maximum mass is not supported by a dynamical axion model.
[Sec. III, GV = 2 GS] The vector coupling is set to GV = 2 GS, and the text states that this choice allows the model to describe PSR J0952-0607 and GW190814 with a zero-temperature maximum mass of 2.51 M⊙. Because the same observational constraints are then used to argue that the model supports massive quark stars, the zero-temperature maximum mass is effectively a fit target rather than a prediction. The paper should show the sensitivity of the maximum mass sequence to GV over a physically motivated range, or justify GV from independent inputs, before using the resulting masses as evidence for the model's viability.
[Sec. III, Ref. [63]] The absolute stability requirement that the minimum energy per baryon be below 930 MeV, and the resulting constraint θ > 0.35π, are imported from Ref. [63] rather than derived here. Since the finite-temperature stability curves in Fig. 1 are central to the paper's argument that SQM is strongly stabilized by the axion effect, the authors should either reproduce the zero-temperature stability calculation explicitly or clearly state that the finite-temperature calculation is an extension of an externally imposed parameter window. This matters because all θ-dependent results inherit the θ > 0.35π restriction.
[Eqs. (5)-(8), (10), Fig. 4] The numerical implementation is not described in sufficient detail. The coupled gap equations involve both scalar and pseudoscalar condensates (σ_i and η_i), and the pseudoscalar condensates can be non-vanishing when θ ≠ 0; the paper does not state how these coupled equations are solved, how many solutions exist, or which solution is selected. Likewise, the entropy density is obtained from S = -∂Ω/∂T, but the paper does not show that the thermodynamic identities are satisfied numerically when the vector interaction and the (µ_f - ~µ_f)n_f term in Eq. (10) are included. Without convergence criteria and validation of the EoS used in the TOV integration, the quantitative mass values in Fig. 5 are not reproducible.

minor comments (5)

[Throughout] There are numerous typographical errors, including 'futher', 'Spinger-Verlag', 'Astrophyical', 'Mclerran', and inconsistent reference formatting. A careful proofreading pass is needed.
[Fig. 3 caption] The saturation density n0 is used in the text but not defined; please define n0 explicitly when it first appears.
[Abstract and Introduction] The paper refers to 'axion fields' and 'axion effects', but the model contains only the constant angle θ, not a propagating axion field. The terminology should be adjusted to avoid overstating the dynamical content, or the authors should clarify that θ parameterizes the axion-induced chiral rotation.
[Fig. 5] The maximum mass curves would benefit from indicating the observational mass ranges (PSR J0952-0607, GW190814) directly on the figure, so the reader can assess the calibration without referring back to the text.
[References] References [63] and [87] appear to refer to the same work by the same group with different years; please unify the citation and ensure bibliographic consistency.

Circularity Check

3 steps flagged · score 4.0 of 10

The cold 2.51 M⊙ anchor is a fit target rather than a prediction, and the θ window is imported from a same-group paper; the finite-temperature 2.61→2.67 rise is a genuine model output.

fitted input called prediction [Sec. III, GV choice before Fig. 1; stage-III result in Fig. 5]
"The flavor-dependent vector interaction coupling constant is set as GV = 2GS. This choice allows the model to describe both the black widow pulsar PSR J0952-0607 (M = 2.35 ± 0.17 M⊙) and the secondary component (2.50M⊙ −2.67M⊙) of GW190814 as QSs at zero temperature, with the maximum star mass reaching 2.51 M⊙."

The zero-temperature maximum mass 2.51 M⊙ is the target used to fix GV=2GS. Later, Fig. 5 reports the stage-III cold result as 'decreasing to 2.51 M⊙', so the fitted target is re-presented as a model output. The finite-temperature 2.61→2.67 rise is a genuine calculation, but the normalization of the mass sequence is set by this fit.
self citation load bearing [Sec. III, θ scan interval before Fig. 1]
"In Ref. [63], the authors discuss the parameter space satisfying the absolute stability condition of SQM within the SU (3) NJL model at zero temperature. ... The results indicate that the minimum energy per baryon at zero temperature becomes smaller than 930 MeV when θ is larger than 0 .35π using the PCP parameter set, thereby satisfying the absolute stability condition for SQM within SU(3) NJL model."

Ref. [63] is P.C. Chu et al., Phys. Rev. D 110, 123031 (2024), sharing the present first author. The allowed θ interval [0.35π, π] within which all θ-dependent results are scanned is imported from this same-group publication rather than from an independent source or from minimization over θ. The current Fig. 1 does reproduce the trend, so this is only partial, but the 'axion range' used for the headline results is set by self-citation.

1 more flagged steps

other [Eq. (1) and Sec. III, Fig. 5]
"The parameter θ corresponds to the chiral rotation angle generated by the axion field contribution in SQM [77]. ... Specially, for θ = π case, the maximum star mass of PQSs rises from 2 .61 M⊙ at the 1st stage to a peak of 2 .67 M⊙ at the 2nd stage, while decreasing to 2.51M⊙ at the 3rd stage..."

θ is introduced as a constant c-number parameter in the 't Hooft term; no axion potential, kinetic term, or field equation is included, and the free energy is not minimized with respect to θ. The headline masses are read off at the endpoint θ=π of the scanned range, so calling this an 'axion-induced' stiffening presents a hand-set input as the outcome of axion physics. The reported enhancement is a direct consequence of choosing the largest scanned value.

full rationale

The derivation from the NJL Lagrangian to the EoS and TOV masses is arithmetically self-contained: Eqs. (1)–(13) define the model, and Figs. 2–8 are numerical evaluations of those equations. The finite-temperature stage comparison (2.61 versus 2.67 M⊙) is a genuine model output. However, the zero-temperature anchor is not a prediction: the vector coupling GV=2GS is explicitly chosen so that the model reproduces PSR J0952-0607 and the GW190814 secondary with maximum star mass 2.51 M⊙, and the same number is later reported as the stage-III cold value. The θ window [0.35π,π] is justified by Ref. [63], a same-first-author publication, so the 'allowed axion range' is not independent external input. The θ=π endpoint used for the headline sequence is selected rather than derived from the axion field equation. Because the central quantitative sequence is partly constructed by these fitted and hand-chosen inputs while the thermal rise itself is computed, the score is 4 rather than higher. I do not count the θ-dependence of the EoS as circular in itself, since it follows from the stated Lagrangian, but the lack of any dynamical determination of θ makes the axion-related numbers conditional on an input choice.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The quantitative claims, especially the maximum masses, depend on the hand-chosen θ range and the GV = 2 GS fit, plus a self-cited stability constraint. The model inputs are standard NJL parameters, but the axion angle is not derived.

free parameters (2)

θ (axion chiral angle) = scanned 0.35π to π
Central parameter for axion effects; the chosen range is based on the absolute stability condition from the authors' prior paper (Ref. [63]), not derived from axion dynamics.
GV vector coupling = 2 GS
Chosen to reproduce PSR J0952-0607 and GW190814 masses with cold quark stars; every mass result inherits this tuning.

assumptions (4)

domain assumption PCP NJL parameter set (Λ=630 MeV, mu=md=5.5 MeV, ms=135.7 MeV, GSΛ^2=1.781, KΛ^5=9.29) correctly describes vacuum quark physics
Taken from Refs. [86,87]; no independent check is performed in this paper.
ad hoc to paper Absolute stability of SQM requires minimum energy per baryon below 930 MeV, and this holds only for θ > 0.35π
Uses the same authors' earlier result (Ref. [63]) as an input constraint; the finite-temperature extension inherits this threshold.
domain assumption Mean-field approximation is valid for the NJL Lagrangian with axion and vector terms
Standard in NJL studies; no beyond-mean-field corrections are considered.
domain assumption Proto-quark star evolution can be represented by three isentropic snapshots (S/nB=1, Yl=0.4; S/nB=2, Yνl=0; S/nB=0, Yνl=0)
Follows previous PNS/PQS literature [74,82-85]; no dynamical evolution is simulated.

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Cite this review

Pith. "Pith review of Quark matter at finite temperature and proto-quark stars with the axion effects in SU(3) Nambu-Jona-Lasinio model." pith.science (2026). https://pith.science/paper/SP5MSMKT

@misc{pith2026260806024,
  author       = {Pith},
  title        = {Pith review of: Quark matter at finite temperature and proto-quark stars with the axion effects in SU(3) Nambu-Jona-Lasinio model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SP5MSMKT}},
  note         = {Machine review of arXiv:2608.06024}
}

read the original abstract

We investigate the thermodynamical properties of strange quark matter (SQM) and proto-quark stars (PQSs) within the SU(3) Nambu-Jona-Lasinio (NJL) model at finite temperature, specifically incorporating the effects of axion fields and vector interactions. Our results demonstrate that these interactions significantly influence the equation of state (EoS), constituent quark masses, entropy density, and the maximum star mass of PQSs at the isentropic stages along the star evolution line. Furthermore, we reveal a distinct thermodynamic signature in the early evolution: the presence of trapped neutrinos leads to a substantial increase in electron number density while simultaneously suppressing the core temperature compared to the neutrino-free case. These findings may highlight the crucial role of the axion effects, flavor-dependent vector interactions, and particle composition in determining the observable properties of compact stars at finite temperature.

Figures

Figures reproduced from arXiv: 2608.06024 by the authors.

**Figure 2.** FIG. 2: (Color online) Pressure as a function of free energy [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** FIG. 4: (Color online) The entropy density as a function of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5: (Color online) The maximum star mass at different [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 7.** Figure 7: FIG. 7: (Color online) The number densities of neutrinos and [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

**Figure 8.** Figure 8: FIG. 8: (Color online) Fractions of [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

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This behavior arises because both temperature and the number of particle species contribute to the entropy den- sity of quark matter

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