Asymptotic behavior of the Speed of Sound in Dense Matter

Pok Man Lo; Udita Shukla

REVIEW 3 major objections 5 minor 1 cited by

Asymptotic behavior of the Speed of Sound in Dense Matter

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Local NJL quark models drive the high-density speed of sound to the wrong limit; a nonlocal quark interaction restores the expected value 1/3.

desk verdict A clean but mostly derivative proceedings illustration of why local NJL misses the conformal limit; the paper's only new algebraic piece (Eq. 19) is unverified and sign-ambiguous. read the letter →

arxiv 2507.06741 v1 pith:Q63CF6BK submitted 2025-07-09 nucl-th astro-ph.HEhep-ph

Udita Shukla , Pok Man Lo This is my paper · ORCID

classification nucl-thastro-ph.HEhep-ph

keywords speedofsounddensematterNambu-Jona-LasiniomodelconformallimitequationstateasymptoticfreedomnonlocalinteractionFermimomentum

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 argues that effective quark models of the Nambu-Jona-Lasinio (NJL) type are unsuitable for dense-matter equations of state because they give the wrong high-density limit of the speed of sound: instead of the expected conformal value $c_s^2 = 1/3$ of a free quark gas, the local interaction forces $c_s^2 \to 1$. The authors trace the problem to the momentum-independent vector interaction, which makes the density scale only linearly with the chemical potential at large $\mu$, and then show that a momentum-dependent interaction with an asymptotically free tail restores the conformal limit: $p_F \to \mu$, $n_V \propto \mu^3$, and $c_s^2 \to 1/3$. For a separable Gaussian kernel they obtain an explicit leading-order formula for the approach to $1/3$ and verify it numerically. If correct, the result tells builders of neutron-star equations of state to avoid local NJL models and to use nonlocal interactions instead.

What carries the argument

The central object is the self-consistent vector gap equation for the momentum-dependent dressed chemical potential, $\mu'(p)=\mu+F[\mu'(p);p]$, together with the Fermi-surface condition $\mu'(p_F)=p_F$ that fixes the Fermi momentum in the nonlocal case. In the local NJL limit, $F$ reduces to a constant times the vector density and produces the problematic $\mu'\propto\mu^{1/3}$ scaling; in the nonlocal case, $F$ is suppressed for large $p=p_F$ because of asymptotic freedom, which drives $p_F\to\mu$, $n_V\propto\mu^3$, and $c_s^2\to1/3$. The worked example is a separable Gaussian interaction, whose exponential falloff makes the leading correction to $1/3$ exponentially small and computable, yielding Eq. (19).

What would settle it

Solve the nonlocal gap equation (15) numerically for the Gaussian kernel at chemical potentials up to $\mu/\Lambda = 10^4$, and compare the resulting $c_s^2$ with the asymptotic formula (19); if $c_s^2$ saturates at a value above $1/3$ or the difference from (19) fails to vanish, the claimed restoration of the conformal limit is wrong. A second check would replace the Gaussian by a slowly decaying kernel, such as $\gamma(p)=\Lambda^2/(\Lambda^2+p^2)$, and test whether $c_s^2\to1/3$ still holds.

Watch

Extended reading notes

Core claim

At the level of the gap equations, the local NJL model produces the asymptotic scaling $\mu' = \mu - (2G_V N_cN_f/3\pi^2)\mu'^3$, so $\mu' \propto \mu^{1/3}$ and $n_V \to \mu/(2G_V)$; the thermodynamic relation $c_s^2 = \partial P/\partial\varepsilon \approx \partial\ln\mu/\partial\ln n_V$ then gives $c_s^2 \to 1$ instead of $1/3$. The paper's resolution is a dynamical quark model with a momentum-dependent interaction, where the dressed chemical potential satisfies $\mu'(p)=\mu+F[\mu'(p);p]$ and the Fermi momentum is determined by $\mu'(p_F)=p_F$. Because asymptotic freedom suppresses the interaction at large momentum, $p_F$ tends to $\mu$, the vector density $n_V=N_cN_fp_F^3/(3\pi^2)$ becomes proportional to $\mu^3$, and the conformal speed of sound is recovered. Using the separable interaction $V(p,q)=V_0e^{-p^2/\Lambda^2}e^{-q^2/\Lambda^2}$, the leading correction is shown to be $c_s^2\approx \frac{1}{3}\left(1-\frac{\omega_\infty}{\mu}\left(1+\frac{2\mu^2}{\Lambda^2}\right)e^{-\mu^2/\Lambda^2}\right)$ with $\omega_\infty=G_VN_cN_f\Lambda^3/(2\sqrt{\pi}^3)$, and the approximation is checked against the full numerical solution.

Load-bearing premise

The whole resolution depends on the quark interaction becoming negligible at large momentum, so that the quark's effective chemical potential approaches the actual one; the paper assumes this falloff (built into a Gaussian-shaped interaction) rather than deriving it from QCD.

Editorial extensions

If this is right

The local NJL vector-interaction scaling $\mu' \propto \mu^{1/3}$ is structural, so no tuning of $G_V$ can make $c_s^2$ approach $1/3$.
Equations of state built on local NJL models will keep their speed of sound near 1 at high density, biasing neutron-star structure predictions made with them.
In the Gaussian nonlocal model, $c_s^2$ approaches $1/3$ from below, with the correction suppressed by $e^{-\mu^2/\Lambda^2}$; this gives a concrete prediction for how fast conformality sets in.
The momentum-dependent gap equations, rather than ad hoc form factors or density-dependent couplings, are the natural way to enforce the expected high-density behavior.

Reading between the lines

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

Going beyond the paper, the mechanism should be generic: any interaction kernel that vanishes at large momentum on the Fermi surface should restore $c_s^2\to1/3$; testing this with an algebraic form factor, e.g. $\Lambda^2/(\Lambda^2+p^2)$, would separate the principle from the Gaussian example.
A further consequence the authors do not draw: the first correction to $1/3$ is exponentially small in the Gaussian case, so any observed deviation from $1/3$ at achievable densities in neutron-star mergers would constrain the effective interaction scale $\Lambda$.
The argument also suggests that hybrid equations of state matching low-density nuclear matter to high-density quark matter should use a nonlocal quark phase, because a local NJL phase would prevent the conformal limit from being reached.

Share X Bluesky LinkedIn Reddit HN

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

Hi [Colleague],

Quick take on arXiv:2507.06741. This is a short proceedings note, and you should read it as such: the main claims are already in the literature. The local-NJL result (c_s^2 → 1 at high density) is a textbook derivation, and the remedy via momentum-dependent interactions appears in Ref [5]. What the paper adds is a compact self-consistent Fermi-momentum formulation and a specific Gaussian-separable example with a claimed leading-order asymptotic correction (Eq. 19).

The first half is solid. Equations (1)-(9) are a clean, correct recapitulation of the textbook failure. The setup in Eqs. (10)-(15) is a nice way to frame the nonlocal problem: define p_F by μ'(p_F)=p_F, then solve self-consistently. The physical intuition that a form factor falling at high momentum should restore p_F → μ and c_s^2 → 1/3 is right, and the qualitative result matches what Ivanytskyi showed in a fuller model.

The soft spots are real. Eq. (19) is the quantitative core, but it is asserted with 'straightforward to derive' and no derivation is given. More importantly, the sign is ambiguous. The self-energy in Eq. (10) has a plus sign; with a positive V0 the correction to p_F is positive, which would push c_s^2 above 1/3. Eq. (19) has a minus sign, so it only works if V0 is negative. The text never specifies the sign convention. The stress-test note on this point is correct. Also, the paper overclaims by calling NJL models 'unsuitable' for dense-matter EoS; they are imperfect but remain widely used with caveats. Finally, Fig. 1 gives no parameter values, so the comparison to the asymptotic formula is not independently checkable.

Proportionately, these are fixable issues in a short note. The central qualitative argument holds up: local NJL fails, nonlocal interactions can restore conformality. The paper is honest about citing Refs [4,5] and about relying on asymptotic freedom. It is not a groundbreaking result, but it is not incoherent either.

Who is this for? People who want a quick pedagogical illustration of the NJL speed-of-sound problem and the nonlocal cure. A serious referee could check Eq. (19), so I'd send it out rather than desk-reject, but I would ask for a derivation and a clear sign convention before accepting.

Referee Report

3 major / 5 minor

Summary. This short conference paper argues that local Nambu–Jona-Lasinio (NJL) models, with a constant vector coupling, fail to reproduce the expected conformal limit of the speed of sound in dense matter: the dressed chemical potential scales as µ' ∝ µ^{1/3}, giving c_s^2 → 1 instead of 1/3. The authors then propose that a momentum-dependent interaction consistent with asymptotic freedom restores the conformal limit. Using a separable Gaussian form factor V(p,q) = V0 exp(-p^2/Λ^2) exp(-q^2/Λ^2), they state an asymptotic formula, Eq. (19), for the leading correction to c_s^2, and show a numerical illustration in Fig. 1. The paper concludes that local NJL models are unsuitable for dense-matter equations of state, while momentum-dependent interactions offer a principled resolution.

Significance. If the central asymptotic formula is correct, the paper provides a clean, explicit demonstration of a physically important point: that the high-density behavior of the speed of sound depends sensitively on the momentum dependence of the effective interaction, and that an asymptotically free interaction can restore c_s^2 → 1/3. The derivation of the local-NJL failure (Eqs. (1)-(9)) is standard and correct. The proposed Gaussian separable model yields a concrete, falsifiable prediction for the approach to the conformal limit, Eq. (19), which can be checked against full numerical solutions and against other nonlocal models. However, the paper does not provide machine-checked proofs or reproducibility details, and the central asymptotic formula is currently unverified. The significance is therefore conditional on the authors supplying the missing derivation and clarifying the sign and normalization conventions.

major comments (3)

[Section 2, Eq. (19)] The central asymptotic formula is asserted as 'straightforward to derive' but no derivation is shown. This equation is the only quantitative bridge between the self-consistent condition (15) and the claimed restoration of the conformal limit, so it must be checkable. In particular, the text does not specify the sign of V0 in Eq. (17) nor its relation to GV appearing in Eq. (19), and it is not clear whether NcNf is absorbed into the interaction in Eq. (10). Working from Eq. (15) with V(p,q) = V0 γ(p)γ(q), the first correction is pF ≈ μ + V0 γ(μ) I∞ with positive I∞, which gives a positive correction to c_s^2 for V0 > 0. Eq. (19) has the opposite sign, so it implicitly requires V0 < 0, consistent with the NJL vector-channel sign but not stated. Please provide the derivation and state the sign convention and the definition of V0 in terms of GV.
[Section 2, Fig. 1] Fig. 1 is the only numerical demonstration of the central claim, but it lacks axis labels, parameter values (Λ, GV, NcNf, numerical grid), and a description of what 'full solution' means. Without this information the comparison between the numerical solution and the asymptotic formula (19) cannot be reproduced or assessed. Please add a caption with all parameters and a description of the numerical procedure.
[Abstract and Section 2, Eqs. (1)-(9)] The conclusion that 'a class of NJL-like models' fails to reproduce the conformal limit and are 'unsuitable for analyzing the equation of state of dense matter' is broader than what is actually shown. The derivation concerns the simplest local NJL model with a constant vector coupling; models with density-dependent couplings or nonlocal interactions (refs. [4,5]) are mentioned but not analyzed. Please narrow the claim to 'the simplest local NJL model' or provide an argument that all local variants share the same failure.

minor comments (5)

[Section 2, Eq. (10)] Eq. (10) writes the vector self-energy without an explicit NcNf factor, while Eq. (19) contains NcNf. Please clarify the convention, or define an effective interaction strength that makes the two equations consistent.
[Section 2, after Eq. (16)] The sentence 'we make use of the trend dictated by asymptotic freedom' is a modeling assumption, not a derivation. Please state explicitly that the Gaussian form factor is a model input and that the result depends on this form factor falling off sufficiently fast at large momenta.
[Section 2, Eq. (12)] The reduction of nV to Eq. (14) assumes that the Fermi surface is characterized by a single pF defined by µ'(pF) = pF. For a general momentum-dependent µ'(q), the integration domain in Eq. (12) is not exactly a sphere; this is an approximation valid asymptotically. Please state this limitation.
[Section 2, heading] The heading contains a typo: 'F ailure' should be 'Failure'.
[Section 2, Eq. (19)] The notation ω∞ is introduced without explaining its physical meaning beyond the algebraic definition; a brief note that it represents the characteristic scale of the interaction would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the NJL failure is derived from standard gap equations and the nonlocal resolution follows from an explicitly stated Gaussian ansatz, not from data fitting or load-bearing self-citation.

full rationale

The paper's derivation chain is not circular. The NJL result c_s^2 -> 1 is derived from the standard gap equations (1)-(9) using only the T=0 step-function limit; this is an independent model result obtained without fitting. The nonlocal resolution is an explicit ansatz: after writing the momentum-dependent vector gap equation (10)-(15), the paper states the asymptotic-freedom assumption in Eq. (16) that the interaction drops at large p, then chooses a separable Gaussian form factor (17)-(18) as a simple example. The conformal limit c_s^2 -> 1/3 is therefore a consequence of the stated model assumption, not a quantity fitted to data and not an output smuggled in through a citation. The one self-citation, Ref. [6] by Lo and Swanson, is used only to motivate instantaneous nonlocal interactions from Coulomb-gauge QCD; the Gaussian model and its asymptotic formula (19) are not taken from that paper, so the self-citation is not load-bearing. The numerical check in Fig. 1 compares the full model solution with the asymptotic formula, so it is a consistency test rather than a fit. The main caveat is a rigor/completeness gap, not a circular one: Eq. (19) is said to be "straightforward to derive" but no derivation is shown, and the sign convention of V0 is not fixed, leaving the displayed sign of the correction ambiguous. This is an omitted-derivation and verifiability issue that would matter for correctness, but it does not make the argument circular.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper adds no new entities. Its central results rest on the standard NJL framework and on the specific nonlocal Gaussian interaction, whose high-momentum vanishing is the direct cause of the conformal restoration. The claimed resolution is an input assumption rather than a derived prediction.

free parameters (2)

V0 (vector coupling strength)
Model input, not fitted. The central claim does not depend on its value as long as it is finite, but the numerical illustration in Fig. 1 would require a specific value.
Λ (form-factor scale)
Model input, set by hand. This exponential cutoff fixes the rate of approach to conformality (Eq. 19) but not the limit itself.

assumptions (6)

domain assumption NJL gap equations with scalar and vector channels (Eqs. 1-3)
The entire failure analysis rests on assuming these gap equations define the model.
domain assumption T=0 limit and Pauli blocking with step functions (Eq. 4)
Used to derive the high-density expressions (Eqs. 5-9).
domain assumption Chiral limit m=0 with M→0
Needed for μ' ∝ μ^{1/3} scaling; the finite-mass case is not analyzed.
domain assumption Asymptotic freedom: interaction strength drops at large momentum
This is the input that drives p_F → μ and the conformal limit; it is assumed, not derived.
ad hoc to paper Separable Gaussian form factor V(p,q)=V0 γ(p)γ(q) with γ=e^{-p^2/Λ^2}
This is a choice of model, not derived from QCD; the claim of a 'principled' resolution depends on this ansatz.
domain assumption Validity of n_V = N_c N_f p_F^3/(3π^2) with p_F defined by μ'(p_F)=p_F
Generalized from the local case; assumes the nonlocal Fermi surface is still a sphere.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Asymptotic behavior of the Speed of Sound in Dense Matter." pith.science (2026). https://pith.science/paper/Q63CF6BK

@misc{pith2026250706741,
  author       = {Pith},
  title        = {Pith review of: Asymptotic behavior of the Speed of Sound in Dense Matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q63CF6BK}},
  note         = {Machine review of arXiv:2507.06741}
}

read the original abstract

We show that a class of NJL-like models fails to reproduce the expected conformal limit of the speed of sound, making them unsuitable for analyzing the equation of state of dense matter. We then demonstrate how this issue can be resolved within a simple dynamical quark model.

Figures

Figures reproduced from arXiv: 2507.06741 by the authors.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Strongly interacting matter with criticality induced by modified excluded volume in core-collapse supernova simulations
astro-ph.HE 2026-07 conditional novelty 6.0 of 10

Modified-excluded-volume EOS with continuous van der Waals phase transition yields CCSN explosions and a several-millisecond neutrino burst, while the same EOS under Gibbs construction fails to explode.

Reference graph

Works this paper leans on

8 extracted references · 3 canonical work pages · cited by 1 Pith paper

[5]

M. A. R. Kaltenborn, N.-U. F. Bastian and D. B. Blaschke, Quark-nuclear hybrid star equation of state with excluded volume effects , https://doi.org/10.1103/PhysRevD.96.056024 Phys. Rev. D 96 (2017) 056024 [ https://arxiv.org/abs/1701.04400 1701.04400 ]

work page Pith review arXiv 2017
[1]

write newline

" write newline "" before.all 'output.state := FUNCTION blank.sep after.quote 'output.state := FUNCTION fin.entry output.state after.quoted.block = 'skip 'add.period if write newline FUNCTION new.block output.state before.all = 'skip output.state after.quote = after.quoted.block 'output.state := after.block 'output.state := if if FUNCTION new.sentence out...

work page
[2]

G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song and T. Takatsuka, From hadrons to quarks in neutron stars: a review , https://doi.org/10.1088/1361-6633/aaae14 Rept. Prog. Phys. 81 (2018) 056902 [ https://arxiv.org/abs/1707.04966 1707.04966 ]

arXiv 2018
[3]

S. P. Klevansky, The Nambu—Jona-Lasinio model of quantum chromodynamics , https://doi.org/10.1103/RevModPhys.64.649 Rev. Mod. Phys. 64 (1992) 649

doi:10.1103/revmodphys.64.649 1992
[4]

Buballa, NJL model analysis of quark matter at large density , https://doi.org/10.1016/j.physrep.2004.11.004 Phys

M. Buballa, NJL model analysis of quark matter at large density , https://doi.org/10.1016/j.physrep.2004.11.004 Phys. Rept. 407 (2005) 205 [ https://arxiv.org/abs/hep-ph/0402234 hep-ph/0402234 ]

arXiv 2005
[6]

Ivanytskyi, Asymptotically conformal color-flavor-locked quark matter within a nonlocal chiral quark model , https://doi.org/10.1103/PhysRevD.111.034004 Phys

O. Ivanytskyi, Asymptotically conformal color-flavor-locked quark matter within a nonlocal chiral quark model , https://doi.org/10.1103/PhysRevD.111.034004 Phys. Rev. D 111 (2025) 034004 [ https://arxiv.org/abs/2409.05859 2409.05859 ]

arXiv 2025
[7]

P. M. Lo and E. S. Swanson, Confinement Models at Finite Temperature and Density , https://doi.org/10.1103/PhysRevD.81.034030 Phys. Rev. D 81 (2010) 034030 [ https://arxiv.org/abs/0908.4099 0908.4099 ]

work page Pith review arXiv 2010
[8]

Reinhardt, G

H. Reinhardt, G. Burgio, D. Campagnari, E. Ebadati, J. Heffner, M. Quandt et al., Hamiltonian approach to QCD in Coulomb gauge - a survey of recent results , https://doi.org/10.1155/2018/2312498 Adv. High Energy Phys. 2018 (2018) 2312498 [ https://arxiv.org/abs/1706.02702 1706.02702 ]

arXiv 2018

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [5]

M. A. R. Kaltenborn, N.-U. F. Bastian and D. B. Blaschke, Quark-nuclear hybrid star equation of state with excluded volume effects , https://doi.org/10.1103/PhysRevD.96.056024 Phys. Rev. D 96 (2017) 056024 [ https://arxiv.org/abs/1701.04400 1701.04400 ]

work page Pith review arXiv 2017

[2] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION blank.sep after.quote 'output.state := FUNCTION fin.entry output.state after.quoted.block = 'skip 'add.period if write newline FUNCTION new.block output.state before.all = 'skip output.state after.quote = after.quoted.block 'output.state := after.block 'output.state := if if FUNCTION new.sentence out...

work page

[3] [2]

G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song and T. Takatsuka, From hadrons to quarks in neutron stars: a review , https://doi.org/10.1088/1361-6633/aaae14 Rept. Prog. Phys. 81 (2018) 056902 [ https://arxiv.org/abs/1707.04966 1707.04966 ]

arXiv 2018

[4] [3]

S. P. Klevansky, The Nambu—Jona-Lasinio model of quantum chromodynamics , https://doi.org/10.1103/RevModPhys.64.649 Rev. Mod. Phys. 64 (1992) 649

doi:10.1103/revmodphys.64.649 1992

[5] [4]

Buballa, NJL model analysis of quark matter at large density , https://doi.org/10.1016/j.physrep.2004.11.004 Phys

M. Buballa, NJL model analysis of quark matter at large density , https://doi.org/10.1016/j.physrep.2004.11.004 Phys. Rept. 407 (2005) 205 [ https://arxiv.org/abs/hep-ph/0402234 hep-ph/0402234 ]

arXiv 2005

[6] [6]

Ivanytskyi, Asymptotically conformal color-flavor-locked quark matter within a nonlocal chiral quark model , https://doi.org/10.1103/PhysRevD.111.034004 Phys

O. Ivanytskyi, Asymptotically conformal color-flavor-locked quark matter within a nonlocal chiral quark model , https://doi.org/10.1103/PhysRevD.111.034004 Phys. Rev. D 111 (2025) 034004 [ https://arxiv.org/abs/2409.05859 2409.05859 ]

arXiv 2025

[7] [7]

P. M. Lo and E. S. Swanson, Confinement Models at Finite Temperature and Density , https://doi.org/10.1103/PhysRevD.81.034030 Phys. Rev. D 81 (2010) 034030 [ https://arxiv.org/abs/0908.4099 0908.4099 ]

work page Pith review arXiv 2010

[8] [8]

Reinhardt, G

H. Reinhardt, G. Burgio, D. Campagnari, E. Ebadati, J. Heffner, M. Quandt et al., Hamiltonian approach to QCD in Coulomb gauge - a survey of recent results , https://doi.org/10.1155/2018/2312498 Adv. High Energy Phys. 2018 (2018) 2312498 [ https://arxiv.org/abs/1706.02702 1706.02702 ]

arXiv 2018