{"id":"b9c8ba7f-98a6-454a-9612-6a0f07bb9db5","arxiv_id":"2412.08674","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":9,"one_line_summary":"In the SU(3) NJL model, the choice among magnetic field-independent, soft cut-off, and Pauli-Villars regularization strongly changes pressure, specific heat, and sound speed of hot magnetized quark matter, and regularizing the temperature part can produce unphysical behavior.","lead":"This paper computes thermodynamic properties of hot, magnetized quark matter using three different mathematical cleanup schemes in the NJL effective model. It shows that the choice of cleanup scheme changes pressures, heat capacities, and sound speeds, especially at high temperature.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The soft cut-off regulator in Eqs. (25)-(29) multiplies already-integrated potentials, so the scheme is undefined as written and all 'Soft' curves—including the high-T causality-violation claim—are unreproducible from the text.","rationale":"I agree with the reader's weakest_assumption and have made it the primary attack. The paper's central quantitative message is a comparison of three regularization schemes; that comparison is only meaningful if each scheme is a well-defined calculational procedure. For the soft cut-off, Eqs. (28)-(29) multiply integrated quantities by a function of the integration variable and Landau level, leaving no definite value for f_Λ(p); the absence of a displayed insertion of f_Λ(p) into the integrands means the 'Soft' curves cannot be independently reproduced or checked. This directly invalidates the strongest high-temperature inference (P decreasing and c_s² rising sharply, with a causality-violation caveat) because that inference is made from the regularized soft-cut-off curves. The same notational ambiguity infects the Pauli-Villars equations, where f_P.V.(M_f) is written as a scalar prefactor instead of the standard Σ_j c_j Ω(M_j) operation, although this can be repaired by reinterpretation more easily than the soft cut-off. I also note the parameter confound (MFIR and soft cut-off use Λ=631.4 MeV, m_s=135.7 MeV, while PV uses Λ=781.2 MeV, m_s=236.9 MeV), which weakens any claim that the different curves are attributable solely to the regularization choice. However, the undefined soft-cut-off operation is the more fundamental defect. Because the reader's REJECT verdict is consistent with this assessment, I recommend UNCHANGED.","tokens_in":8687,"tokens_out":6900,"duration_ms":70534,"concrete_test":"Recompute the soft-cut-off pressure and c_s² by inserting f_Λ(p) from Eq. (25) inside the momentum integrand and Landau sum of the medium, magnetic, and vacuum contributions (before carrying out ∫dp and Σ_n), rather than multiplying the already-integrated potentials as in Eqs. (28)-(29). If the resulting P(T) and c_s²(T) match the 'Soft(With Regularization)' panels of Figs. 4 and 8, the written equations were compressed shorthand; if they do not, or if no unique insertion exists, the central temperature-dependence claim is not extractable from the paper.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (25) defines f_Λ(p)=1/(1+exp((√(p²+2|q_f|Bn)-Λ)/(0.05Λ))) as a function of the integration variable p and the Landau level n. Yet Eqs. (28) and (29) use f_Λ(p) as a multiplicative prefactor multiplying Ω_vac, Ω_Tmag, ϕ_vac, and ϕ_Tmag, all of which are already fully integrated and summed over p and n (Secs. II and III.A). Once the integrals and sums are performed, p is a dummy variable and f_Λ(p) has no definite numerical value; no prescription is given for at which momentum scale, or which Landau level, the regulator should be evaluated. The paper never displays the explicitly regulated integrands in which f_Λ(p) is inserted before ∫dp and Σ_n, so neither the 'Soft(With Regularization)' nor even the 'Soft(Without Regularization)' curves of Figs. 2 and 4-8 can be reproduced from the written equations. Because the paper's headline claim about regularizing the temperature-dependent part leading to decreasing P and a sharp rise in c_s², and the associated causality-violation remark in Sec. IV, relies directly on those soft-cut-off curves, this is a load-bearing defect, not a cosmetic notation issue. A parallel ambiguity affects the Pauli-Villars notation in Eqs. (30)-(32), where f_P.V.(M_f) is written as if it were a scalar multiplying integrated potentials rather than the standard linear combination of mass-shifted potentials.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies three regularization schemes—magnetic field-independent regularization (MFIR), soft cut-off, and Pauli–Villars—in the SU(3) NJL model for quark matter at finite temperature and magnetic field. It writes the thermodynamic potential and gap equation in each scheme, solves the gap equation numerically, and compares quark masses, pressure, thermal susceptibility, energy density, specific heat, and squared sound speed as functions of T and eB. The central assertion is that the regularization choice is a first-order modeling decision: applying the soft cut-off or Pauli–Villars regulator also to the temperature-dependent part makes the pressure fall at high T and the sound speed rise sharply, with an associated remark about possible causality violations.","tokens_in":9067,"tokens_out":8483,"duration_ms":76832,"significance":"The question is legitimate and timely for effective-model studies of magnetized quark matter; different regularization prescriptions can indeed change thermodynamics, and the manuscript provides a useful catalog of parameter sets and standard MFIR formulas. The paper does not suffer from circularity in the fitted-input/output sense, since the parameters are taken from earlier fits and the conclusions are not obtained by fitting the outputs. However, the paper's quantitative value is undermined by two underdetermined regularization implementations and by comparing the schemes under different model parameters. If these issues were repaired, the comparison could be a useful reference; as it stands, the main quantitative message is not reproducible from the written equations, and no code or data tables are provided for independent checking.","major_comments":[{"comment":"The regulator f_Λ(p) is defined in Eq. (25) as a function of the integration variable p, the Landau level n, and the flavor through |q_f|. In Eqs. (28) and (29) it is used as a multiplicative prefactor multiplying Ω_vac, Ω_Tmag, φ_vac, and φ_Tmag, which have already been integrated over p and summed over n. Once the integrals and sums are performed, f_Λ(p) has no definite numerical value, so the 'Soft(With Regularization)' and 'Soft(Without Regularization)' results in Figs. 2 and 4-8 are not defined by the equations in the paper. The explicitly regulated integrands, with f_Λ inserted before ∫dp and Σ_n, are never shown; without them the soft-cut-off curves cannot be reproduced. This is load-bearing because the paper's high-temperature conclusions about P, ε, C_V, and c_s² depend directly on those curves.","section":"Section III.B, Eqs. (25)-(29)"},{"comment":"The Pauli–Villars regulator is written as f_P.V(M_f)=Σ_j c_j f(√(M_f²+jΛ²)) and then applied as a scalar f_P.V(M_f) multiplying already-integrated potentials. The standard Pauli–Villars construction is a linear combination of the full potential evaluated at shifted masses, Ω_PV(M)=Σ_j c_j Ω(√(M²+jΛ²)); the paper does not specify which operation is meant, and the notation c_j f(...) is not dimensionally transparent because the function f is not defined. Consequently the Pauli–Villars curves in Figs. 3-8 are unreproducible from the written equations, and the comparison of 'with' versus 'without' regularization in Eq. (31) versus Eq. (32) rests on an undefined prescription.","section":"Section III.C, Eqs. (30)-(32)"},{"comment":"The three schemes are compared with different model parameters: MFIR and soft cut-off use Λ=631.4 MeV, m_s=135.7 MeV, G=1.835/Λ², K=9.29/Λ², while Pauli–Villars uses Λ=781.2 MeV, m_s=236.9 MeV, G=4.90/Λ², K=129.8/Λ². The observed differences among the schemes in Figs. 1-8 are therefore not attributable solely to the regularization scheme; they could reflect the different parameter sets. Since the paper's central claim is that the choice of regularization scheme itself matters, this parameter/scheme confounding must be addressed, for example by repeating the comparison with a common parameter set or by demonstrating that the qualitative differences survive independent parameter variations.","section":"Section IV, parameter sets"},{"comment":"The statement that regularizing the temperature-dependent part 'can lead to violations of causality' is not supported by the displayed data: the vertical axis of Fig. 8 extends only to c_s²=0.5, far below the causal bound c_s²≤1. The figure may show a sharp increase in the sound speed, but not a causality violation. The claim should either be backed by results exceeding unity or removed and replaced by a more cautious statement about the scheme dependence of c_s².","section":"Section IV, Fig. 8 and surrounding text"}],"minor_comments":[{"comment":"The text refers to Γ(x_f) as 'Euler's totient function'; this should be the gamma function.","section":"Section III.A, Eq. (19)"},{"comment":"Eq. (29) is labeled the 'non-regularized form' but still multiplies the vacuum pieces by f_Λ(p); this labeling is internally inconsistent and should be clarified, since the comparison in Figs. 2 and 4-8 depends on what 'without regularization' means.","section":"Section III.B, Eq. (29)"},{"comment":"The figure captions do not state the fixed values of eB in Figs. 1-3 and Figs. 5-8, nor are the units of the right-hand panel of Fig. 4 explained beyond the label P×10^9; without these details the curves cannot be fully interpreted or reproduced.","section":"Section IV, figure captions"},{"comment":"Several references in the regularization discussion (Refs. [9]-[12], and possibly [16]) appear to concern computer vision, fluid dynamics, or general gauge-theory techniques rather than NJL regularization; the reference list should be rechecked and replaced with appropriate QCD/NJL sources.","section":"References [9]-[12]"},{"comment":"The manuscript repeatedly writes 'Pauli-Villas' instead of 'Pauli-Villars' (e.g., captions of Figs. 3 and 4 and several paragraphs); the spelling should be unified.","section":"Throughout, Sec. IV"},{"comment":"Equations (26)-(27) repeat Eqs. (3) and (6) without explicitly defining the split into Ω_vac, Ω_Tmag, φ_vac, and φ_Tmag; defining these objects would remove part of the ambiguity in Eqs. (28)-(29).","section":"Section III.B, Eqs. (26)-(27)"},{"comment":"The statement 'No Data associated in the manuscript' is at odds with the many numerical figures; providing the numerical data underlying the figures, or at least tabulated values for representative curves, would improve reproducibility.","section":"Data Availability Statement"}],"recommendation":"major_revision","confidential_remarks":"I am leaning toward major revision rather than rejection because the two implementation ambiguities in Section III are in principle repairable, and the MFIR part of the paper is standard. However, if the authors cannot provide the explicitly regulated integrands and remove the parameter/scheme confounding, the paper's quantitative conclusions should be withdrawn. I also note that the reference list contains several entries that appear unrelated to the subject, which suggests the manuscript needs a careful scholarly revision before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the warning the paper tries to deliver is useful: in the SU(3) NJL model with a magnetic field, which regulator you use is not a minor detail, and applying the soft cut-off or Pauli-Villars regulator to the temperature-dependent part of the potential as well as the vacuum part makes the pressure fall at high T and pushes the squared speed of sound above 1/3. If that is right, it is a genuine caution for people doing magnetized NJL thermodynamics. Second, the paper as written cannot support that claim, because the soft cut-off is undefined: f_Λ(p) in Eq. (25) depends on the loop momentum and Landau level, but Eqs. (28)–(29) use it as a prefactor on already-integrated potentials. The stress-test is correct on this. The Pauli-Villars notation in Eqs. (30)–(32) has the same ambiguity, written as a prefactor instead of the standard sum over mass-shifted potentials.\n\nWhat the paper does well: the MFIR formulas are standard and correctly assembled from the cited literature, and a side-by-side comparison of the three schemes in one framework is a legitimate and needed piece of work for the NJL subfield. The with-versus-without regularization comparison is made within a fixed parameter set, so the cross-scheme parameter difference (Hatsuda–Kunihiro parameters for MFIR/soft, Carignano–Buballa for PV) does not confound that particular comparison; it only muddies the cross-scheme curves in Fig. 4. The tone is honest and the conclusion does not oversell.\n\nSoft spots, in order of importance. The undefined f_Λ operation is load-bearing: every 'Soft' curve in Figs. 2 and 4–8, plus the causality-warning remark, is unreproducible from the written equations. The fix is a careful methods rewrite that shows the regulator inserted in each integrand before the momentum integral and Landau sum. Second, the paper has no code or data, which matters for a purely numerical comparison. Third, minor: refs [9] and [11] do not support the sentences they are attached to, the Fermi–Dirac factors in Eqs. (6) and (27) have typos, and the conclusion's remarks about anomalous-magnetic-moment studies are asserted rather than shown.\n\nWho this is for: NJL practitioners choosing a regularization scheme and anyone coding magnetized quark matter thermodynamics. It deserves a serious referee rather than a desk reject—the question is real and the defect is repairable. Send it to review expecting major revision, and ask the referee to check whether the numerics can actually be regenerated from the corrected equations.","headline":"A plausible and useful warning about regularization scheme choice in magnetized NJL—undermined as written because the soft cut-off regulator multiplies already-integrated potentials, leaving the headline curves unreproducible.","tokens_in":9614,"tokens_out":12035,"would_cite":false,"duration_ms":97616,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that in the SU(3) Nambu-Jona-Lasinio model, the choice and application of a regularization scheme changes the qualitative thermodynamics of hot magnetized quark matter, including a possible violation of causality when the…","keywords":["Nambu-Jona-Lasinio model","thermomagnetic quark matter","regularization schemes","magnetic field-independent regularization","soft cutoff regularization","Pauli-Villars regularization","squared speed of sound","causality violation"],"falsifier":"Recompute the soft cutoff thermodynamic potential and condensate with $f_\\Lambda(p)$ placed inside each momentum integral and Landau sum before the integration and summation are performed, using the same parameters; if $P(T,eB)$ then rises with temperature at high $T$ and $c_s^2$ stays at or below $1/3$, the paper's central causality conclusion does not survive the corrected implementation.","tokens_in":8408,"feed_emoji":"🧲","tokens_out":12754,"duration_ms":109815,"temperature":0.7,"pith_summary":"This paper asks whether the way ultraviolet divergences are tamed in the SU(3) Nambu-Jona-Lasinio model changes the predicted properties of quark matter at high temperature in a strong magnetic field. It compares three regularization schemes — magnetic field-independent cutoff, soft cutoff, and Pauli–Villars — and finds that they disagree not only in the size of corrections but in the qualitative shape of the thermodynamics: the pressure versus magnetic field rises then falls, falls then rises, or oscillates depending on the scheme, and the strange quark mass responds differently above $T\\simeq 200$ MeV. The sharpest claim is that when the regularization is also applied to the temperature-dependent part of the soft cutoff and Pauli–Villars potentials, the pressure decreases at high temperature and the squared speed of sound $c_s^2$ grows sharply, which the paper says can lead to violations of causality. The upshot is that the choice of regularization is a first-order modeling decision in NJL studies of magnetized quark matter, not a minor technical detail.","feed_headline":"Regularization choice flips quark matter pressure at high T","feed_subtitle":"Regularizing the temperature part of the NJL potential can push the squared sound speed above 1/3, breaking causality.","key_machinery":"The central object is the mean-field thermodynamic potential of the SU(3) NJL model, split into vacuum, magnetic-field, and finite-temperature contributions, together with the gap equation that fixes the dynamical masses. The three regulators are the magnetic-field-independent cutoff expressed through Hurwitz zeta functions, the soft cutoff $f_\\Lambda(p)=\\left[1+\\exp\\left(\\left(\\sqrt{p^2+2|q_f|B\\,n}-\\Lambda\\right)/(0.05\\Lambda)\\right)\\right]^{-1}$, and the Pauli–Villars combination $f_\\mathrm{PV}(M_f)=\\sum_{j=0}^3 c_j\\sqrt{M_f^2+j\\Lambda^2}$ with $c_0=1$, $c_1=-3$, $c_2=3$, $c_3=-1$. The decisive mechanism is whether the regulator multiplies the whole potential, including the thermal-magnetic part, as in Eqs. (28) and (31), or only the vacuum part, as in Eqs. (29) and (32).","core_discovery":"The paper's central claim is that the regularization scheme in the SU(3) NJL model changes the qualitative behavior of thermodynamic quantities in hot, magnetized quark matter, not just their numerical size. Starting from the mean-field thermodynamic potential $\\Omega=\\Omega_\\text{vac}+\\Omega_\\text{mag}+\\Omega_\\text{med}$ and the gap equations for the constituent masses $M_u$, $M_d$, $M_s$, the paper computes pressure, energy density, specific heat, and squared speed of sound under three schemes. For the soft cutoff and Pauli–Villars schemes it distinguishes two variants: regulating only the vacuum part, and regulating the full temperature-dependent potential. The numerical comparison shows that the vacuum-only variants stay close to the magnetic-field-independent baseline, while the fully regulated variants make the pressure fall at high temperature, make energy density and specific heat drop instead of equilibrating, and drive $c_s^2$ sharply upward — behavior the paper reads as a possible violation of causality. The conclusion is that regularizing the temperature-dependent part is not a safe default, and the scheme choice must be made consciously for the physical question at hand.","pith_inferences":["If the scheme dependence is as strong as reported, then NJL-based conclusions about magnetic catalysis or inverse catalysis should come with a scheme-sensitivity test; otherwise the qualitative outcome may be an artifact of the cutoff.","A cleaner soft cutoff implementation would place $f_\\Lambda(p)$ inside every momentum integrand and Landau sum before integrating; doing so would remove the ambiguity in Eqs. (28)–(29) and would test directly whether the high-temperature rise in $c_s^2$ survives.","The sharp rise of $c_s^2$ past $1/3$ when the temperature part is regulated may indicate that the fully regulated variant double-counts vacuum contributions at finite temperature rather than a physical instability; a flow-equation or renormalization-group regulator could separate those effects."],"forward_implications":["If the claim holds, NJL calculations of magnetized quark matter should report which scheme was used and justify it, because the schemes give qualitatively different $P(eB)$, $c_s^2$, and strange quark mass behavior.","Applying regularization to the temperature-dependent part of the soft cutoff and Pauli–Villars potentials produces a falling high-temperature pressure and a sharply rising $c_s^2$; those variants should not be used for high-$T$ thermodynamics without checking causality.","Without temperature-part regularization, the soft cutoff and Pauli–Villars results are close to the MFIR results for energy density, specific heat, and speed of sound, with deviations concentrated in the strange quark sector above $T\\simeq200$ MeV.","MFIR is the numerically simplest scheme and separates magnetic and non-magnetic contributions cleanly, so it is the natural starting point for exploratory NJL calculations.","There is no universal scheme: the appropriate choice depends on the quantity being studied, with the Pauli–Villars variant without temperature regularization favored for extensions such as the anomalous magnetic moment."],"supporting_citations":[{"why":"Provides the magnetic field-independent regularization scheme used as the MFIR baseline.","marker":"[15]"},{"why":"Supplies the MFIR treatment of magnetized NJL matter on which the comparison curves are built.","marker":"[20]"},{"why":"Gives the split of the thermodynamic potential into vacuum, magnetic, and medium parts that all three schemes share.","marker":"[21]"},{"why":"Provides the pressure expression whose scheme dependence is the paper's main numerical comparison.","marker":"[22]"},{"why":"Introduces the soft cutoff regulator and connects its magnetic-field oscillations to vacuum-term regularization.","marker":"[23]"},{"why":"Gives the soft cutoff form for magnetized NJL thermodynamics used in the second scheme.","marker":"[24]"},{"why":"Supplies the distinction between regularized and non-regularized temperature-dependent parts of the potential and condensate.","marker":"[25]"},{"why":"Defines the Pauli–Villars regulator combination used in the third scheme.","marker":"[26]"},{"why":"Supplies the cutoff, current masses, and couplings for the MFIR and soft cutoff curves.","marker":"[27]"},{"why":"Supplies the Pauli–Villars parameter set used for the third scheme.","marker":"[28]"}],"fun_headline_variants":["How you regularize NJL flips hot quark matter's fate","Regularization scheme can break causality in magnetized quark matter","Choice of cutoff decides if quark matter pressure crashes","NJL scheme choice sends sound speed past causality limit","Thermomagnetic quark matter: regularization changes everything"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's high-temperature conclusions rely on applying the soft cutoff regulator $f_\\Lambda(p)$ as a multiplicative prefactor to already-integrated potentials in Eqs. (28) and (29), even though the regulator is only defined inside the momentum integral and Landau sum, so the regulated potential is not well defined unless the regulator is inserted at an earlier stage.","fun_headline_variants_meta":{"raw":{"variants":["How you regularize NJL flips hot quark matter's fate","Regularization scheme can break causality in magnetized quark matter","Choice of cutoff decides if quark matter pressure crashes","NJL scheme choice sends sound speed past causality limit","Thermomagnetic quark matter: regularization changes everything"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000153,"raw_usage":{"total_tokens":1179,"prompt_tokens":887,"completion_tokens":292,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":503,"completion_tokens_details":{"reasoning_tokens":214}},"tokens_in":503,"tokens_out":292,"duration_ms":3248,"temperature":1.0,"reasoning_tokens":214,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T18:07:16.844161+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the soft cutoff thermodynamic potential and condensate with $f_\\Lambda(p)$ placed inside each momentum integral and Landau sum before the integration and summation are performed, using the same parameters; if $P(T,eB)$ then rises with temperature at high $T$ and $c_s^2$ stays at or below $1/3$, the paper's central causality conclusion does not survive the corrected implementation.","supporting_citations":[{"cited_title":"Gandhi, A","cited_arxiv_id":null,"evidence_quote":"Provides the magnetic field-independent regularization scheme used as the MFIR baseline."},{"cited_title":"Adel and T","cited_arxiv_id":null,"evidence_quote":"Provides the pressure expression whose scheme dependence is the paper's main numerical comparison."},{"cited_title":"Fayazbakhsh and N","cited_arxiv_id":null,"evidence_quote":"Gives the soft cutoff form for magnetized NJL thermodynamics used in the second scheme."},{"cited_title":"Do we need to use regularization on the thermal part in the NJL model?","cited_arxiv_id":"2105.14323","evidence_quote":"Supplies the distinction between regularized and non-regularized temperature-dependent parts of the potential and condensate."},{"cited_title":"Inhomogeneous chiral condensates in three-flavor quark matter","cited_arxiv_id":"1910.03604","evidence_quote":"Supplies the Pauli–Villars parameter set used for the third scheme."}],"review_version":1}