{"id":"5ea08331-dc49-4ec0-8cf6-90cb24ae723c","arxiv_id":"2608.13469","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"In the two-flavor NJL model in mean field, the chiral critical endpoint in the temperature versus angular velocity plane shows standard mean-field exponents: alpha ~ 0, beta ~ 1/2, gamma ~ 1, delta ~ 3.","lead":"Using a simplified quark model, this paper finds where hot, fast-spinning quark matter has a critical point. At that point, rotation does not change the universal scaling laws: the same mean-field exponents appear as in non-rotating matter.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The extracted β, γ, and δ rely on the unproven assertion that the rotational polarization J inherits the chiral order parameter's scaling; if that link fails, the exponents do not characterize the CEP.","rationale":"The reader's weakest_assumption identifies exactly the point on which the paper's central claim depends: the rotational polarization J is used as a proxy for the chiral order parameter without derivation. The paper gives a plausible physical argument—J couples to M through the thermodynamic potential—but does not show that ∂J/∂M is nonzero at the CEP, nor that the scaling directions in the (T, ω) plane coincide with the chosen paths. Since the mean-field NJL model is expected to produce mean-field exponents for M, the genuinely new and nontrivial part of the paper is the claim that J, a rotation-related observable, scales in the same way. If that claim is wrong, the extracted β, γ, δ do not characterize the CEP, even though the underlying chiral transition may still be mean-field Ising. The proposed test—recomputing the exponents from M—would settle the issue directly and is straightforward with the same numerical machinery. I agree with the conditional verdict: the computation is internally consistent, but this load-bearing identification must be checked before the scaling-class conclusion can be accepted.","tokens_in":12933,"tokens_out":11207,"duration_ms":119121,"concrete_test":"Re-extract β, γ, and δ directly from the chiral order parameter M on the same numerical grid: (i) β from ΔM = M_broken - M_restored along the first-order line; (ii) γ from the inverse curvature (∂²Ω/∂M²)^{-1} (or from the chiral susceptibility ∂M/∂m at fixed T,ω) along the temperature direction at ω = ω_CEP; and (iii) δ from (M - M_CEP)/M_CEP versus (ω - ω_CEP)/ω_CEP at T = T_CEP. If the M-based exponents match the reported J-based values within the plateaus' scatter, the order-parameter identification is validated. If they differ, the assertion that J inherits the chiral scaling is false, and the central conclusion is not supported by the presented evidence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III (after Eq. 11) asserts that J = -∂Ω/∂ω 'is coupled to the chiral critical mode through the mixed dependence of the thermodynamic potential on M and ω, and therefore exhibits the same critical scaling behavior near the CEP.' This assertion is the only connection between the numerically extracted exponents βω ≈ 1/2, γω ≈ 1, δω ≈ 3 and the chiral critical endpoint. It is not derived. J is a smooth function of T, ω, and the equilibrium value of M; its singular part near the CEP is proportional to the order-parameter singular part only if ∂J/∂M evaluated at the CEP is nonzero and no other singular contribution enters. The paper does not verify this condition. If ∂J/∂M vanished, J would not scale as M, and the exponents would describe a different or merely regular response. The δ extraction in Eq. (20) carries an additional hidden assumption: at T = T_CEP, varying ω is treated as moving along the ordering-field direction of the mean-field Ising CEP. In general, the two relevant scaling fields are linear combinations of (T - T_CEP) and (ω - ω_CEP); if the pure-ω direction has a nonzero thermal component, the critical-isotherm path is off-axis and the effective δ can approach 3 only as an artifact of the fit. These assumptions are load-bearing because the central claim—that rotation does not change the mean-field Ising scaling class—rests entirely on the J-based exponents. The mean-field universality of the chiral order parameter M itself is not in question; the question is whether J measures it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies rotating two-flavor QCD matter in the two-flavor NJL model at mean-field level. Working in a rotating frame, the authors write down a thermodynamic potential, solve the gap equation, and locate a critical endpoint in the temperature–angular-velocity (T, ω) plane, reporting T_CEP ≈ 0.0202339062 GeV and ω_CEP ≈ 0.6440126597 GeV for fixed radial coordinate r = 0.1 GeV^-1 and angular-momentum cutoff n = 5. They define effective critical exponents α_ω, β_ω, γ_ω, and δ_ω from the specific heat density, the discontinuity of the rotational polarization J = −∂Ω/∂ω, the rotational susceptibility χ_ω = ∂J/∂ω, and the critical-isotherm response of J, respectively. From numerical logarithmic slopes they report α_ω ≈ 0, β_ω ≈ 1/2, γ_ω ≈ 1, and δ_ω ≈ 3, which satisfy the mean-field scaling relations α + 2β + γ = 2 and α + β(1 + δ) = 2. The paper concludes that rotation shifts the location of the CEP but does not change the underlying mean-field Ising scaling class.","tokens_in":127,"tokens_out":19128,"duration_ms":408733,"significance":"If the claimed results are correct, the paper would provide a systematic critical-exponent analysis for rotating QCD matter in a widely used effective model, with explicit expressions for the specific heat, rotational polarization, and rotational susceptibility. The extracted exponents are consistent with each other through the standard scaling relations, and the authors are appropriately explicit that the calculation is confined to the mean-field approximation and that fluctuations are neglected. The paper also identifies concrete extensions beyond mean field. However, the central identification of J as an order-parameter-like quantity is asserted rather than derived, and the δ_ω extraction rests on an unverified assumption about the direction of the ordering field in the (T, ω) plane. These points are load-bearing for the claim that the exponents characterize the chiral CEP, so the significance of the paper will be established only after those gaps are closed.","major_comments":[{"comment":"The statement in Eq. (9) that the stationarity condition removes the implicit M(T) dependence from the second temperature derivative is not correct. One has dΩ/dT = ∂Ω/∂T at the stationary point, but d²Ω/dT² = ∂²Ω/∂T² + 2∂²Ω/∂T∂M dM/dT + ∂²Ω/∂M² (dM/dT)², so the correct specific heat is C_ω = −T[∂²Ω/∂T² − (∂²Ω/∂T∂M)²/∂²Ω/∂M²]. The correction term shown in Eq. (10), which has a numerator involving (2n+1) ε f_+f_- and a denominator involving (2n+1)²(f_+² + f_-²), does not match the required ∂²Ω/∂T∂M and ∂²Ω/∂M², the latter being 1/(2G) − B with B defined in Eq. (18). Please derive Eq. (10) explicitly and state which expression was actually used in the α_ω extraction.","section":"Eq. (9) and Eq. (10)"},{"comment":"The assertion that the rotational polarization J is 'coupled to the chiral critical mode' and therefore inherits the chiral order-parameter scaling is not derived. For β_ω to equal the chiral order-parameter exponent, the singular part of ΔJ must be proportional to ΔM; this requires, at minimum, that ∂J/∂M evaluated at the CEP is nonzero, or that a different argument establishes the proportionality when M_c ≠ 0. The paper provides no proof and no numerical check of this condition. Please show explicitly that J = J_c + A(M − M_c) + ... with A ≠ 0 in the critical region, or demonstrate numerically that ΔJ/ΔM is finite and nonzero as the CEP is approached.","section":"Eqs. (11)–(14), Sec. III"},{"comment":"Extracting δ_ω by varying pure ω at T = T_CEP implicitly assumes that the pure-ω direction coincides with the 'magnetic' scaling field of the CEP. In general, the two relevant scaling fields near the CEP are linear combinations of δT and δω; if the chosen path has a nonzero thermal component, the apparent exponent tends to 1/β = 2 in mean-field theory rather than δ = 3. The paper should determine the mixing angle, for example from the tangent direction of the first-order transition line in the (T, ω) plane at the CEP, and extract δ along the orthogonal ordering-field direction, or show explicitly that the pure-ω direction is the magnetic direction. Without this step, the agreement δ_ω ≈ 3 cannot yet be taken as evidence for the mean-field Ising class.","section":"Eq. (20) and Fig. 9"},{"comment":"The exponents are quoted from visual plateaus without quantitative uncertainties, and the two truncation parameters — the angular-momentum cutoff n = 5 and the fixed radial coordinate r = 0.1 GeV^-1 — are not varied. The CEP coordinates are quoted to ten significant digits, which is not meaningful without a convergence study in n and a test of sensitivity to r. Please report a sensitivity analysis (for example n = 3, 5, 7, 10 and r = 0.05, 0.1, 0.2 GeV^-1) and give the resulting ranges for the CEP location and for each exponent.","section":"Sec. III, numerical setup"}],"minor_comments":[{"comment":"Equation (2) contains an unexplained γ^0 μ term in the Lagrangian even though the thermodynamic potential is evaluated at zero chemical potential; please either define μ or remove the term.","section":"Eq. (2)"},{"comment":"The sign in Eq. (12) appears inconsistent with the definition J = −∂Ω/∂ω. From Eq. (4), ∂Ω/∂ω is proportional to (f_+ − f_-), so J should be proportional to (f_- − f_+); please check the sign and the corresponding expressions in Eqs. (15)–(17).","section":"Eq. (12)"},{"comment":"The direct term in Eq. (17), −∂²Ω/∂ω², evaluated from Eq. (4) gives N_f N_c/(2π² T) Σ ∫ J_n (n+1/2)² f_+f_-, whereas Eq. (17) has N_f N_c/(4π² T); please verify the prefactors in Eqs. (10), (12), and (17) for consistency.","section":"Eq. (17)"},{"comment":"For each quoted exponent, the paper should specify the range of ln|t| or ln|ω̃| over which the plateau value was averaged, together with the resulting statistical or numerical uncertainty.","section":"Figs. 3, 4, 7, 9"},{"comment":"There are several typographical and editorial issues, including a duplicated paragraph defining α_ω in Sec. III, 'Sezionedi' in the affiliation, 'V .' in the reference list, and the capitalization in the first sentence of the Conclusions; these should be corrected.","section":"General presentation"}],"recommendation":"major_revision","confidential_remarks":"The manuscript contains several algebraic inconsistencies in the central formulas (Eqs. (9)–(10), (12), and (17)) that are not discussed or reconciled. Even if the main physical claim is correct, the presentation is not yet reliable enough for publication, and I would ask the authors to re-derive the thermodynamic response functions carefully, run the proposed convergence checks, and address the scaling-field issue for δ_ω before the paper is reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: solid mean-field NJL calculation that extracts effective critical exponents in the (T,ω) plane and gets the usual Landau values (α≈0, β≈1/2, γ≈1, δ≈3). Nothing shocking, but it is the first systematic exponent extraction for a rotating NJL system, and it holds together.\n\nWhat's good: The setup is clean. They define Cω, the rotational polarization J, and the rotational susceptibility χω, and they show that χω diverges near the CEP. The numerical exponents are internally consistent and satisfy the scaling relations. The conclusion that rotation shifts the CEP but doesn't change the mean-field scaling class is stated with appropriate modesty, and the paper acknowledges that fluctuations beyond mean-field are not captured.\n\nThe main soft spot is the identification of J as a stand-in for the chiral order parameter. The paper asserts that J couples to the critical mode and therefore inherits its scaling, but it never directly checks that ∂J/∂M is non-zero at the CEP. That is load-bearing because β, γ, and δ are all extracted from J. The indirect evidence is good: the susceptibility formula includes a term proportional to (∂²Ω/∂ω∂M)², and the fact that χω actually diverges with γ≈1 tells you the coupling is present. But a direct numerical check would kill the objection. A related subtlety applies to δ: the critical isotherm is defined by varying the ordering field at the critical temperature. Varying ω at T=T_CEP is only the ordering-field direction if the thermal scaling field happens to be zero along that path. The fact that δ≈3 suggests it is, but the paper should say so.\n\nMinor issues: no error bars or fit ranges on the exponents, and no convergence study for the n=5 angular momentum cutoff or the fixed radial coordinate. At r=0.1 GeV⁻¹ the Bessel functions kill high n, so the truncation is probably fine, but a one-line check would settle it. Also, Eq. (9) makes it look like they drop the chain-rule terms when differentiating the potential with respect to T, but the explicit expression in Eq. (10) actually contains the correction. So that particular concern is mostly a presentation problem.\n\nWho is this for? People working on rotating QCD who want a mean-field benchmark for the critical behavior in the (T,ω) plane. It doesn't change the big picture, but it's a useful reference.\n\nRecommendation: send it to peer review. A competent referee can check the ∂J/∂M coupling numerically and ask for convergence checks; both are inexpensive. This is a legitimate calculation, not a toy.","headline":"A solid, workmanlike mean-field NJL extraction of effective critical exponents in the (T,ω) plane that confirms expected Landau values; the main caveat is the asserted rather than derived identification of rotational polarization as the scaling order parameter.","tokens_in":13786,"tokens_out":8731,"would_cite":true,"duration_ms":84360,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.38.Mh","11.30.Rd","05.70.Jk"],"model":"deepseek-v4-flash","headline":"Near the chiral critical endpoint, rotating quark matter displays the same mean-field scaling exponents as non-rotating matter, with angular velocity shifting only the endpoint's location.","keywords":["rotating QCD matter","NJL model","critical endpoint","critical exponents","chiral phase transition","rotational polarization","mean-field approximation","angular velocity"],"falsifier":"Extract the same exponents from the chiral condensate itself, for instance $\\beta$ from the jump $\\Delta M$ along the first-order line, $\\gamma$ from the curvature $\\partial^2\\Omega/\\partial M^2$, and $\\delta$ from the isotherm of $M$, and compare with the values obtained from $J$; unequal exponents would show that $J$ is not a faithful order-parameter proxy. A second, purely numerical check is to rerun the extraction with a larger angular-momentum cutoff $n$ and several radial positions $r$, and test whether the plateau regions of $\\alpha$, $\\beta$, $\\gamma$, $\\delta$ and the endpoint location itself survive.","tokens_in":12741,"feed_emoji":"🌀","tokens_out":17870,"duration_ms":140022,"temperature":0.7,"pith_summary":"Off-center heavy-ion collisions produce quark matter with enormous vorticity, which makes angular velocity a natural additional control parameter alongside temperature. This paper works out the thermodynamics of the two-flavor Nambu–Jona-Lasinio model in a rotating frame at mean-field level and locates the critical endpoint of the chiral transition in the temperature–angular-velocity plane. Its central claim is that near that endpoint the specific heat, the discontinuity and the susceptibility of the rotational polarization $J=-\\partial\\Omega/\\partial\\omega$, and the critical-isotherm polarization all scale with the mean-field Ising exponents $\\alpha\\simeq 0$, $\\beta\\simeq 1/2$, $\\gamma\\simeq 1$, $\\delta\\simeq 3$, which satisfy the scaling relations $\\alpha+2\\beta+\\gamma=2$ and $\\alpha+\\beta(1+\\delta)=2$. If correct, rotation shifts the location of the critical point but leaves the universality class of the chiral transition untouched, and the rotational polarization becomes a workable probe of criticality in vortical matter.","feed_headline":"Rotation moves the QCD critical point but not its universality class","feed_subtitle":"Rotation shifts the critical endpoint but leaves the chiral transition's mean-field scaling laws intact.","key_machinery":"The central object is the rotational polarization $J=-\\partial\\Omega/\\partial\\omega$, the thermodynamic conjugate of the angular velocity. It is not the chiral condensate, but the paper argues it couples to the chiral critical mode through the mixed dependence of $\\Omega$ on the constituent mass $M$ and $\\omega$, so its discontinuity, susceptibility, and isotherm response are claimed to inherit the order-parameter scaling. The companion identity is the rotational susceptibility $\\chi_\\omega=\\partial J/\\partial\\omega$, evaluated as a total derivative along the gap-equation trajectory; it splits into a direct response $-\\partial^2\\Omega/\\partial\\omega^2$ plus a chiral-fluctuation term $(\\partial^2\\Omega/\\partial\\omega\\,\\partial M)^2\\,(\\partial^2\\Omega/\\partial M^2)^{-1}$. The singular amplification of $\\chi_\\omega$ near the endpoint is driven by the vanishing curvature $\\partial^2\\Omega/\\partial M^2\\to 0$, which softens the chiral mode. Each exponent is read off from a local logarithmic slope in $\\ln|t|$ or $\\ln|\\tilde\\omega|$ between adjacent numerical points, a procedure chosen to avoid arbitrary fitting windows.","core_discovery":"Working in a co-rotating frame, the authors add the rotation to the NJL Lagrangian through orbital and spin couplings linear in $\\omega$, so the quasiparticle energies become $\\varepsilon_n = E_k + (n+\\tfrac12)\\omega$, and derive the mean-field thermodynamic potential $\\Omega(T,\\omega,M)$. The stationary condition $\\partial\\Omega/\\partial M=0$ defines the equilibrium trajectory, and the chiral critical endpoint is found at $T_{\\mathrm{CEP}}\\simeq 0.0202339062$ GeV and $\\omega_{\\mathrm{CEP}}\\simeq 0.6440126597$ GeV, where the curvature $\\partial^2\\Omega/\\partial M^2$ vanishes. Approaching the endpoint along four distinct thermodynamic paths, the paper extracts effective exponents from local logarithmic slopes: the specific heat density $C_\\omega=-T\\,\\partial^2\\Omega/\\partial T^2$ gives $\\alpha_\\omega\\simeq 0$; the jump $\\Delta J$ of the rotational polarization $J=-\\partial\\Omega/\\partial\\omega$ across the first-order line gives $\\beta_\\omega\\simeq 1/2$; the rotational susceptibility $\\chi_\\omega=\\partial J/\\partial\\omega$ diverges with $\\gamma_\\omega\\simeq 1$; and the critical-isotherm response $\\tilde J\\sim|\\tilde\\omega|^{1/\\delta}$ at $T=T_{\\mathrm{CEP}}$ gives $\\delta_\\omega\\simeq 3$. The four exponents obey $\\alpha+2\\beta+\\gamma=2$ and $\\alpha+\\beta(1+\\delta)=2$, the relations expected for Landau mean-field Ising behavior. The paper concludes that rotation extends the control-parameter space and moves the phase boundary, but does not change the mean-field critical scaling structure.","pith_inferences":["A direct internal test suggests itself: extract $\\beta$ from the jump of the constituent quark mass $M$ itself and $\\gamma$ from $\\partial^2\\Omega/\\partial M^2$, then compare with the values obtained from $J$; agreement would confirm that the rotational polarization inherits the chiral singular part rather than contributing a singular behavior of its own.","The fixed radial coordinate $r=0.1$ GeV$^{-1}$ and angular-momentum cutoff $n=5$ are used without convergence checks; repeating the extraction at larger $n$ and several $r$ would show whether the exponent plateaus are numerical artifacts or genuine scaling.","If $J$ is a faithful critical proxy, higher-order cumulants of the rotational polarization, analogues of the kurtosis used in beam-energy scans, should diverge near the endpoint with exponents tied to $\\gamma$ and $\\delta$; that would amount to a sharper, possibly measurable rotational signature of the critical point.","A fluctuation-corrected treatment could either confirm the mean-field Ising class or expose rotation-sensitive corrections, and the present calculation fixes the baseline such a comparison needs, a step the authors themselves flag."],"forward_implications":["Angular velocity joins temperature as a genuine control parameter: the $(T,\\omega)$ plane contains its own critical endpoint, and rotation shifts the phase boundary without erasing the critical point.","The rotational susceptibility $\\chi_\\omega$ diverges at the endpoint, so the response of vortical matter to changes in angular velocity offers a new probe of criticality in rotating systems.","The four exponents are extracted independently and still satisfy both standard scaling relations, an internal consistency check that supports the mean-field identification.","The rotation-induced endpoint belongs to the same mean-field Ising scaling class as the conventional $(T,\\mu_B)$ endpoint, so methods developed for the chemical-potential plane carry over to the rotational control parameter.","The work supplies a systematic characterization of rotation-induced critical phenomena, which the paper frames as the baseline for extending rotating QCD studies beyond the mean-field approximation."],"supporting_citations":[{"why":"Supplies the rotating-frame NJL Lagrangian with rotation entering through orbital and spin couplings linear in the angular velocity; the starting point of the calculation.","marker":"[28]"},{"why":"The reference the paper points to for the derivation of the thermodynamic potential in the rotating frame.","marker":"[33]"},{"why":"Provides the parameter set used in all numerics: quark mass, four-fermion coupling, and three-momentum cutoff.","marker":"[80]"},{"why":"The renormalization-group treatment of rotating QCD matter cited as the benchmark for determining the true universality class beyond mean field.","marker":"[37]"},{"why":"The original two-flavor Nambu–Jona-Lasinio model whose mean-field thermodynamics is used here.","marker":"[75]"}],"fun_headline_variants":["Rotating QCD keeps mean-field scaling laws","Rotation shifts QCD critical endpoint, not scaling class","Rotating QCD: universality class unchanged","Rotation moves QCD critical point, exponents stay","Rotating NJL: mean-field critical exponents persist"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the rotational polarization $J=-\\partial\\Omega/\\partial\\omega$ inherits the full singular behavior of the chiral order parameter near the endpoint, so that its jump, susceptibility, and isotherm scaling genuinely measure the chiral critical exponents; the paper asserts this coupling rather than deriving it.","fun_headline_variants_meta":{"raw":{"variants":["Rotating QCD keeps mean-field scaling laws","Rotation shifts QCD critical endpoint, not scaling class","Rotating QCD: universality class unchanged","Rotation moves QCD critical point, exponents stay","Rotating NJL: mean-field critical exponents persist"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000618,"raw_usage":{"total_tokens":2919,"prompt_tokens":1046,"completion_tokens":1873,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":662,"completion_tokens_details":{"reasoning_tokens":1800}},"tokens_in":662,"tokens_out":1873,"duration_ms":15484,"temperature":1.0,"reasoning_tokens":1800,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:20:33.746988+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Extract the same exponents from the chiral condensate itself, for instance $\\beta$ from the jump $\\Delta M$ along the first-order line, $\\gamma$ from the curvature $\\partial^2\\Omega/\\partial M^2$, and $\\delta$ from the isotherm of $M$, and compare with the values obtained from $J$; unequal exponents would show that $J$ is not a faithful order-parameter proxy. A second, purely numerical check is to rerun the extraction with a larger angular-momentum cutoff $n$ and several radial positions $r$, and test whether the plateau regions of $\\alpha$, $\\beta$, $\\gamma$, $\\delta$ and the endpoint location itself survive.","supporting_citations":[{"cited_title":"Parameter fitting in three-flavor Nambu--Jona-Lasinio model with various regularizations","cited_arxiv_id":"1601.02411","evidence_quote":"Provides the parameter set used in all numerics: quark mass, four-fermion coupling, and three-momentum cutoff."}],"review_version":1}