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The paper argues that the lattice-observed peak in the squared speed of sound of dense two-color QCD appears once the Nambu-Jona-Lasinio model is regularized with the Medium Separation Scheme instead of a sharp cutoff, and that the traditio

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2026-08-04 10:19 UTC pith:6YFVDMZ2

load-bearing objection Interesting but not trustworthy as printed: the MSS-vs-TRS sound-speed comparison is plausible and not fitted, but the baryon density in Eq. (3.10) is a factor of two larger than Eq. (2.4) allows, and the appendix hides several nontrivial steps. the 2 major comments →

arxiv 2510.10198 v2 pith:6YFVDMZ2 submitted 2025-10-11 hep-ph hep-lathep-thnucl-th

Speed of sound peak in two-color dense QCD: confronting effective models with lattice data

Arthur E.B. Pasqualotto , Dyana C. Duarte , Ricardo L. S. Farias , Rudnei O. Ramos This is my paper
classification hep-ph hep-lathep-thnucl-th
keywords two-color QCDNambu-Jona-Lasinio modelspeed of soundmedium separation schemediquark condensateequation of statechemical potentiallattice QCD
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper argues that the peak in the squared speed of sound seen in lattice simulations of dense two-color QCD, which was missed by earlier effective-model treatments, appears when the Nambu-Jona-Lasinio model is regularized with the Medium Separation Scheme (MSS) rather than a sharp momentum cutoff. The MSS separates all chemical-potential dependence out of the divergent loop integrals before cutting them off, leaving only vacuum quantities regularized. With this change, the equation of state softens at high density, the diquark condensate grows with chemical potential, and the computed c_s^2 traces the lattice peak. If this is right, the main obstacle to describing the observable was not the effective model itself but how its ultraviolet divergences were handled.

Core claim

In a two-color, two-flavor NJL model with scalar and diquark interactions at zero temperature and finite baryon chemical potential, regularizing the gap equations with the Medium Separation Scheme yields a diquark gap Δ that increases monotonically with μ_B and a squared speed of sound c_s^2 that develops the peak observed in lattice simulations of two-color QCD. The same model with the traditional sharp cutoff produces a diquark gap that eventually drops to zero and no sound-speed peak. The paper also shows that MSS results for pressure, energy density, and baryon density track the lattice data and approach the free-fermion limit at high density, and that c_s^2 is well approximated by a com

What carries the argument

The Medium Separation Scheme (MSS): a regularization procedure that isolates the finite-density part of loop integrals from their ultraviolet-divergent part by iterating a subtraction identity (Eq. (A3)) until all dependence on chemical potential and in-medium masses sits in finite integrals. Only the remaining vacuum integrals, which depend solely on the vacuum quark mass M_0, are regularized with the cutoff Λ. The scheme enters the gap equations for M and Δ and the baryon-density integral, and it is what converts the NJL equation of state from a stiff curve without a peak into a softer curve whose c_s^2 matches the lattice peak.

Load-bearing premise

The MSS subtraction expansion must converge and remove essentially all chemical-potential dependence from the divergent integrals in the regime μ_B near the cutoff Λ; if medium effects leak back into the regularized pieces, the predicted diquark gap and sound-speed peak could be artifacts of the regularization rather than physics.

What would settle it

Compute the MSS gap equations including the next-order residual integrals or extend the calculation to μ_B beyond Λ: if the peak in c_s^2 shifts markedly, disappears, or changes sign, the subtraction series is not converged and the claimed reproduction of the lattice peak is not robust. A direct lattice determination of the diquark gap Δ(μ_B) in two-color QCD that disagrees with the MSS curve would also undercut the mechanism.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • MSS-regularized NJL reproduces the lattice-observed peak in the squared speed of sound in two-color QCD, while the traditional cutoff scheme does not.
  • The diquark gap Δ increases with chemical potential and remains positive within the model's validity range, matching the trend expected from perturbative QCD at high density.
  • Thermodynamic observables computed with MSS—pressure, energy density, baryon density—agree better with lattice data and approach the Stefan-Boltzmann limit at high μ_B.
  • Because two-color QCD has no sign problem, this gives a direct quantitative test bed for regularization schemes in effective models at finite density.
  • The analytic approximation for c_s^2 captures the MSS curve and may guide analogous estimates for three-color dense matter where lattice comparison is harder.

Where Pith is reading between the lines

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

  • Because the peak appears only after medium effects are removed from divergent integrals, the same prescription may shift predictions for other density-sensitive observables, such as susceptibilities or transport coefficients, in three-color NJL applications.
  • A sharper test would be to apply MSS to the same model at finite temperature and compare with two-color lattice data: the zero-temperature peak should persist or move systematically with T in a calculable way.
  • The analytic formula suggests the peak height and location are controlled by the ratio Δ/μ_B; if future data independently constrain the diquark gap, one could check whether that ratio, rather than the regularization detail, is the proximate cause of the peak.
  • Keeping the next subtraction term in the MSS expansion would show whether the scheme's series is converged in the regime μ_B near Λ; a large shift in the peak would signal that the result is regularization-sensitive.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. This paper studies the squared speed of sound c_s^2 in two-color, two-flavor QCD at zero temperature and finite baryon chemical potential, using an NJL model with scalar and diquark interactions. The authors compare the traditional sharp-cutoff regularization (TRS) with the Medium Separation Scheme (MSS), in which medium-dependent parts are subtracted from the divergent integrals and only vacuum integrals are regulated. They find that in the MSS the diquark gap and thermodynamic quantities stay closer to chiral perturbation theory and to recent lattice data, and the MSS c_s^2 curve develops a peak qualitatively resembling the lattice result, while the TRS curve does not. An analytic approximation for c_s^2 is also given. The central claim is that the regularization scheme, not the model itself, was the main obstacle to describing this observable.

Significance. If correct, the result would be a useful resolution of a puzzling discrepancy and would strengthen the case for MSS as a physically sensible regulator in dense effective models. The paper fixes all parameters from vacuum inputs (Table I), compares with lattice data from Ref. [38] without using those data as input, and provides an analytic approximation. However, the derivation contains a factor-of-2 error in the baryon density that compromises the thermodynamic quantities, and the central MSS subtraction is not sufficiently documented. The comparison with lattice data is also qualitative rather than quantitative. For these reasons the central claim is not yet established.

major comments (2)
  1. [Eq. (3.10)] The baryon density is off by a factor of 2. Starting from Eq. (2.4), the exact thermodynamic derivative is n_B = -∂Ω/∂μ_B = N_c N_f Σ_s ∫ d^3p/(2π)^3 s(E + s μ_B/2)/(2E_s^Δ) = N_c N_f [ (1/2)I_n + (μ_B/4)I_Δ ], with I_n,I_Δ defined in Eqs. (3.4) and (A18). The printed expression has N_c N_f [I_n + (μ_B/2)I_Δ], i.e., twice the correct result. In the massless Δ=0 limit it gives n_B/μ_B^3 = 1/(12π^2) instead of 1/(24π^2); the resulting ε = -P + μ_B n_B therefore changes and c_s^2 changes quantitatively (a free gas gives 1/7 instead of 1/3). If the numerical code differentiates Ω rather than using Eq. (3.10), the text still needs correction; if it uses Eq. (3.10) as written, Figs. 3–5 and the central peak claim are not reliable. Please re-derive Eqs. (3.10)/(3.12) and repeat the numerics.
  2. [Appendix A, Eqs. (A3)–(A20)] The MSS subtraction is the technical heart of the paper, but the steps labelled 'straightforward algebraic manipulations' are not shown. In particular, one must verify that after two (or three) iterations of Eq. (A3) the residual integrals I2, I3, I6 are finite, that the truncated expansion is accurate for μ_B≲Λ, and that Eq. (A4) (which appears garbled in the typesetting) and Eq. (A19) are complete. Given the factor error in Eq. (3.10), these algebraic manipulations cannot be taken on faith. Please provide the full derivation or a supplementary notebook and state precisely how each I_n is evaluated. Without this, the central result is not independently checkable.
minor comments (6)
  1. [Fig. 4 caption] The caption spells 'barion' instead of 'baryon'. Also, the axis values of n_B/μ_B^3 (2–16) seem inconsistent with the free-gas limit in natural units (≈0.004) unless a normalization is being used; please specify the units or normalization.
  2. [Eq. (A4)] The displayed expansion appears garbled: after the first two terms the following terms are not joined by plus signs. Please fix the typesetting so the identity is unambiguous.
  3. [Figs. 3 and 5] The agreement with lattice data is assessed only visually. Provide a quantitative measure (e.g., a chi-square over a stated μ_B range) or at least explicitly identify the range over which the MSS curve is claimed to reproduce the lattice points.
  4. [Sec. IV, Figs. 4–5] Curves are plotted up to μ_B/mπ=8–12, well beyond the stated validity limit Λ≈657 MeV≈4.7 mπ. Either mark the validity region on the figures or reduce the plotted range so the claimed comparison is not extrapolated.
  5. [References] Refs. [71] and [78] are the same arXiv:2507.14343 preprint; [71] lacks journal metadata and [78] is incompletely formatted. Ref. [79] also lacks full metadata. Please clean up the reference list.
  6. [Eq. (4.1)] The analytic approximation for c_s^2 is asserted to be a good approximation to the MSS result, but no derivation is given in this paper. Please either derive it or spell out its precise relation to the result in Ref. [89], including its regime of validity.

Circularity Check

0 steps flagged

No significant circularity: the MSS sound-speed result is a vacuum-calibrated model prediction compared with lattice data, with the MSS decomposition derived in the appendix rather than imported solely by self-citation.

full rationale

The central claim is not a fitted reproduction of the target data. The three parameters (G, m_q, Lambda) in Table I are fixed from vacuum quantities (rescaled f_pi, m_pi, and quark condensate), not from the lattice pressure/energy density or sound speed, and the lattice data appear only for comparison in Figs. 3 and 5. The MSS regularization is not taken on faith from the authors' prior papers: Appendix A re-derives the subtraction identities (A3)-(A4) and (A19), defines the residual integrals I1-I6, and obtains the gap equations and baryon density used in the numerics. The analytical c_s^2 expression, Eq. (4.1), is quoted as 'similar to the expression found in Ref. [89]' and is only said to approximate the MSS curve; it is not used as the source of the peak. Self-citations to [70,72-79] are present, but they support the method and are accompanied by an explicit derivation in this paper, so they are not load-bearing in a circular sense. The possible missing factor of 1/2 in the baryon density, Eq. (3.10), flagged by the skeptic, is a numerical/accuracy concern that would affect the equation of state and c_s^2, but it is not a case of the prediction being equivalent by construction to its inputs. Overall, no circular step of the kind defined in the requested taxonomy is identifiable.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

No new particles or forces are introduced. The central new input is the MSS regularization prescription; its validity is the main load-bearing assumption. Model parameters are inherited from prior vacuum fits, so they are listed as free parameters but were not tuned to the target lattice data.

free parameters (3)
  • Λ (three-momentum cutoff) = 657 MeV
    Fixed in prior work (Refs [83,84]) by N_c-scaling vacuum inputs; not fitted to the lattice sound-speed data, but the MSS split depends on it.
  • G (scalar/diquark coupling) = 7.20 GeV^-2
    Fixed from vacuum f_π and quark condensate via N_c scaling; not fitted to the target observable.
  • m_q (current quark mass) = 5.3 MeV
    Set to reproduce m_π=140 MeV; standard input, not fitted to the sound-speed peak.
axioms (5)
  • domain assumption NJL Lagrangian (2.1) with G_s=G_d from Fierz describes two-color QCD in the mean-field regime.
    Sec. II; standard two-color Fierz relation [81], but an effective-model assumption.
  • domain assumption Mean-field approximation for the thermodynamic potential (2.2)-(2.4).
    Sec. II; ignores meson/diquark fluctuations and finite-T corrections.
  • domain assumption The Medium Separation Scheme expansion (A3) removes all medium dependence from divergent integrals, leaving finite residuals I2, I3, I6.
    App. A, Eqs. (A3)-(A19); if this split is inexact, the gap equations (3.8)-(3.9) and the sound-speed peak would not be reliable.
  • domain assumption N_c scaling of f_π and ⟨q̄q⟩ fixes the two-color NJL parameters in Table I.
    Sec. II footnote; follows Refs [83,84]; the quantitative comparison with lattice depends on this parameter choice.
  • domain assumption The lattice QC2D data of Ref [38] are a valid benchmark in the same regime.
    Used in Figs. 3 and 5 as ground truth; no continuum or systematic-error discussion is included.

pith-pipeline@v1.3.0-alltime-deepseek · 18710 in / 16324 out tokens · 139067 ms · 2026-08-04T10:19:30.054271+00:00 · methodology

0 comments
read the original abstract

Lattice simulations of two-color, two-flavor Quantum Chromodynamics (QCD) at finite quark chemical potential have revealed a distinctive peak structure in the sound velocity. Although chiral perturbation theory (ChPT) and the Nambu-Jona-Lasinio (NJL) model have been employed to explain this phenomenon, neither approach has fully captured the observed behavior. To address this discrepancy, we have extended the NJL framework by incorporating the Medium Separation Scheme (MSS). This approach isolates medium contributions from divergent integrals, allowing for a more accurate treatment of finite-density effects. Our results indicate a clear increase in the diquark gap ($\Delta$) with increasing chemical potential, consistent with what is also seen in perturbative QCD predictions at high densities. {}Furthermore, the MSS-modified NJL model successfully reproduces the observed peak in the sound velocity.

Figures

Figures reproduced from arXiv: 2510.10198 by Arthur E.B. Pasqualotto, Dyana C. Duarte, Ricardo L. S. Farias, Rudnei O. Ramos.

Figure 2
Figure 2. Figure 2: The pressure density 𝑃 as a function of energy density 𝜀 (all in units of MeV/fm3 ), for the MSS (continuous green line), TRS (dashed black line) and ChPT (dotted red line). The solid and dashed gray lines correspond to the values of constant speed of sound EoS, 𝑐 2 𝑠 = 1 and 𝑐 2 𝑠 = 1/3, respectively. The numerical results shown in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The energy density (panel a) and pressure (panel b) scaled [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The barion number density divided by baryonic chemical [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Speed of sound squared, 𝑐 2 𝑠 , as a function of baryonic chemical potential normalized by the pion mass, 𝜇𝐵/𝑚𝜋, for ChPT (red dotted line), TRS (black dashed line), the analytical expression Eq. (4.1) (blue dot-dashed line), and MSS (green solid line). Blue dots with error bars indicate the lattice datapoints (data taken from Ref. [38]). V. FINAL REMARKS In this paper, we have presented a comprehensive an… view at source ↗

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Forward citations

Cited by 2 Pith papers

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  2. Dense and Cold Magnetized Quark Matter: A Review of Magnetic-Field-Independent Regularization and the Medium Separation Scheme

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Reference graph

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    Gap equation forΔin the MSS The gap equation forΔin the TRS implementation consists of introducing a three-dimensional cutoffΛto regularize all UV divergent momentum integrals. Conversely, for MSS we begin by rewriting (3.4) as follows 𝐼Δ = ∑︁ 𝑠=±1 ∫ 𝑑3𝑝 (2𝜋)3 1 𝐸𝑠 Δ = 1 𝜋 ∫ +∞ −∞ 𝑑𝑝 4 ∫ 𝑑3𝑝 (2𝜋)3 1 𝑝2 4+(𝐸 𝑠 Δ)2 ,(A1) such that 1 2 ∑︁ 𝑠=±1 ∫ 𝑑3𝑝 (2𝜋)3 1 ...

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