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Temperature-fluctuation cumulants in interacting hadron gases match lattice QCD below the chiral crossover, split along freeze-out at low energies, and trace the nuclear liquid-gas Widom line through a c2 minimum and a c3 sign change.

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2026-08-04 01:27 UTC pith:EUMIYBRO

load-bearing objection First HRG calculation of temperature-fluctuation cumulants; plausible and honest, but the nuclear liquid-gas fingerprint rests on an unvalidated high-density VDW extrapolation. the 3 major comments →

arxiv 2607.24616 v2 pith:EUMIYBRO submitted 2026-07-27 nucl-th nucl-ex

Interaction fingerprints in temperature-fluctuation cumulants from the QCD crossover to nuclear liquid-gas criticality

Debasish Mallick This is my paper
classification nucl-th nucl-ex
keywords temperature fluctuationscumulantshadron resonance gasvan der Waals interactionnuclear liquid-gas transitionWidom lineQCD phase diagrammean transverse momentum fluctuations
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.

This paper tries to establish that temperature-fluctuation cumulants—quantities that encode the event-by-event variation of the mean transverse momentum through the specific heat—can act as a thermometer for thermodynamic structures in the QCD phase diagram. The paper computes the second, third, and fourth cumulants in three hadron resonance gas models: an ideal gas, an excluded-volume gas with baryon repulsion constrained by lattice QCD, and a van der Waals gas that reproduces the nuclear ground state. At zero baryon density, all three models agree with lattice thermodynamics up to the chiral crossover and separate above it. Along the chemical freeze-out curve they remain close for collision energies above about 20 GeV and diverge strongly at lower energies, where baryonic interactions dominate the thermal response. In the van der Waals model, the variance c2 develops a minimum along the Widom line of the nuclear liquid-gas transition and c3 changes sign across it, giving a concrete hadronic signature that low-energy heavy-ion experiments could search for.

Core claim

The central claim is that temperature cumulants of a hadron resonance gas are nearly insensitive to the interaction scheme at high collision energies and at zero baryon density, but become sharply model-dependent in the low-energy freeze-out region. The van der Waals model, which reproduces the nuclear ground state, predicts that c2 has a minimum along the nuclear liquid-gas Widom line and that c3 changes sign across it, with the minimum deepening and moving to lower temperatures as the chemical potential approaches the liquid-gas critical point near (Tc, mu_c) ≈ (19.7, 908) MeV. The paper identifies the baryon sector as the carrier of this interaction dependence through a meson-baryon secto

What carries the argument

The machinery is the set of temperature-fluctuation cumulants c_n defined through dimensionless pressure derivatives χ_n = T^(n−4) ∂^n p/∂T^n at fixed chemical potentials, with c2 = 1/χ2, c3 = −χ3/χ2^3, and c4 = 3χ3^2/χ2^5 − χ4/χ2^4. These formulas convert the equation of state into observables whose sensitivity grows with each derivative, so interaction effects that are small in the pressure show up clearly in the cumulants. The van der Waals HRG with parameters a = 329 MeV fm^3 and b = 3.42 fm^3, fixed by the nuclear ground state, supplies a nuclear liquid-gas critical point and a Widom line along which the variance minimum and the c3 sign change develop. A sector decomposition into meson

Load-bearing premise

The central liquid-gas fingerprint rests on the assumption that the van der Waals hadron resonance gas with parameters fixed by the nuclear ground state (a = 329 MeV fm^3, b = 3.42 fm^3) remains quantitatively valid for baryonic matter at the densities reached along low-energy freeze-out (mu_B up to about 910 MeV); the paper does not validate this equation of state against finite-density lattice QCD or other independent high-density data, and the freeze-out parametrization is

What would settle it

A finite-density lattice QCD calculation of the baryon pressure at chemical potentials mu_B = 850–910 MeV and temperatures T = 20–50 MeV would settle whether the van der Waals equation of state is approximately correct; if the lattice pressure deviates markedly from the VDW prediction, the c2 minimum and c3 sign change are model artifacts. Alternatively, measurements of mean-pT fluctuation cumulants in fixed-target Au+Au collisions at sqrt(s_NN) = 2–3 GeV that fail to show the predicted c2 minimum and positive-c3 band as the trajectory approaches the Widom line would falsify the fingerprint.

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

If this is right

  • At zero baryon density, c2 is fixed directly by the lattice specific heat, and the interacting HRG models match lattice QCD on the hadronic side of the chiral crossover (up to T ≈ 155–165 MeV); above that temperature the hadronic description ceases to apply.
  • Along the chemical freeze-out curve the three models are close and nearly energy independent for sqrt(s_NN) ≥ 20 GeV, but separate strongly below about 15 GeV; the separation is carried almost entirely by the baryon sector and reflects the competition between repulsive and attractive interactions.
  • In the van der Waals model, c2 develops a minimum that deepens as the chemical potential approaches the nuclear liquid-gas critical point, and c3 changes sign in a narrow band that follows the high-temperature side of the Widom line; this pattern is a hadronic-baseline signature, distinct from expected chiral-critical non-monotonic structures.
  • The low-energy continuation of the chemical freeze-out curve runs parallel to and only 5–9 MeV below the Widom line for sqrt(s_NN) between 2 and 3 GeV, so low-energy fixed-target collisions probe a region where temperature cumulants may be sensitive to nuclear liquid-gas thermodynamics.
  • Even the ideal HRG produces a mild non-monotonicity along the freeze-out curve (a shallow c2 minimum near sqrt(s_NN) ≈ 11 GeV) purely from the trajectory, and the extrema of c2, c3, and c4 fall inside the fixed-target window of 3.0–7.7 GeV where non-monotonic pT correlations have been reported; the paper positions these cumulants as a quantitative hadronic baseline for such measurements.

Where Pith is reading between the lines

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

  • Editorial inference: If the predicted c2 minimum and c3 sign change survive embedding in a realistic dynamical evolution, event-by-event mean-pT cumulants could become a direct experimental probe of the nuclear liquid-gas Widom line at low collision energies, complementing conserved-charge cumulants.
  • Editorial inference: The sector decomposition suggests a testable separation: because meson cumulants are model-independent and vary slowly with energy while baryon cumulants grow and interact, the energy dependence of the third or fourth cumulant of mean-pT fluctuations could in principle constrain the strength of baryonic repulsion and attraction separately.
  • Editorial inference: The paper does not validate the van der Waals equation of state against finite-density lattice QCD at mu_B ≈ 850–910 MeV; a lattice computation of the baryon pressure in that region would either substantiate the Widom-line fingerprint or show it to be a model artifact.
  • Editorial inference: The same cumulant construction could be applied to equations of state that include strangeness or that target the chiral critical region; the paper's finding that temperature derivatives amplify interaction effects suggests that higher-order cumulants of mean-pT distributions may be more discriminating than the variance alone in beam-energy scans.

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

3 major / 5 minor

Summary. The manuscript defines scaled temperature-fluctuation cumulants c2 through c6 as nonlinear functions of temperature derivatives of the pressure (Eqs. (2)-(6)) and evaluates them in three hadronic models: the ideal hadron resonance gas, a lattice-constrained baryon-only excluded-volume HRG, and a baryonic van der Waals HRG. It compares these models with lattice QCD at zero baryon chemical potential, follows the cumulants along the Andronic chemical freeze-out curve, and analyzes the nuclear liquid-gas region. The central results are that the models agree with lattice thermodynamics up to the chiral crossover, separate below about 15 GeV along freeze-out, and, in the VDW-HRG, c2 develops a minimum and c3 changes sign near the nuclear liquid-gas Widom line. The paper frames these as interaction fingerprints that may be relevant to STAR fixed-target measurements of mean transverse-momentum fluctuations.

Significance. This is the first HRG-based calculation of temperature-fluctuation cumulants, and the derivative observables are shown to be considerably more sensitive to interaction physics than the pressure itself. The sector decomposition in Appendix A.1 and the direct c2 comparison to published lattice specific-heat data are genuine strengths. The central liquid-gas prediction is concrete and falsifiable with fixed-target data. However, the significance is conditional: the zero-density comparison is solid, while the headline liquid-gas fingerprint rests on a high-density VDW extrapolation that is not independently validated, and the low-energy freeze-out trajectory is a continuation without propagated uncertainty.

major comments (3)
  1. [Sec. 4.4, Eq. (9), Fig. 4] The c2 minimum and c3 sign change are computed with VDW-HRG parameters a=329 MeV fm^3 and b=3.42 fm^3 fixed solely by the nuclear ground state. The region mu_B ~ 700-910 MeV and T ~ 20-100 MeV is not confronted with independent finite-density constraints, and Sec. 5 itself calls for trajectories constrained by finite-density lattice QCD. Because these features are the paper's headline result, the revision should include a sensitivity analysis of a and b over the range allowed by nuclear ground-state properties and, where possible, a comparison with an alternative high-density equation of state. Without such tests, the abstract's claim that temperature cumulants connect to thermodynamic structures 'in the QCD phase diagram' is stronger than the evidence presented.
  2. [Sec. 4.3/4.4, Eq. (10), Fig. 4(c)] The chemical freeze-out parametrization of Eq. (10) is continued below sqrt(s_NN)=3 GeV without propagating any uncertainty. The continuation lies only 5-9 MeV below the Widom line, so the statement that fixed-target energies probe nuclear liquid-gas thermodynamics depends on this extrapolation. Please specify the fitted range of Eq. (10) and either propagate the thermal-fit uncertainties or compare with an alternative low-energy freeze-out curve obtained from HADES/STAR thermal fits.
  3. [Sec. 4.2, Fig. 2, Appendix A.2] The lattice bands for c3 and c4 are obtained by differentiating the HotQCD and WB analytic parametrizations; they do not include a full propagation of the published lattice uncertainties and correlations. The paper acknowledges this, but the abstract's first comparison claims agreement with lattice QCD for all three models up to the crossover. To support that claim quantitatively, the c3/c4 comparison should either use direct lattice determinations of the higher temperature derivatives or include a proper covariance propagation; otherwise the agreement claim should be explicitly restricted to c2.
minor comments (5)
  1. [Eqs. (2)-(6)] The power notation in the rendered equations is difficult to parse. Use explicit parentheses, e.g., (chi_2)^5, so that expressions like chi_2^5 are unambiguous.
  2. [Sec. 2] Please clarify whether chi_2 is per unit volume and what volume is implicit in c2=1/chi_2, so that the relation <(Delta T)^2>=T^2/C_V is unambiguous.
  3. [Table 1] State the fitted domain of the Andronic parametrization. The 2.4 GeV row is described as a continuation, but if the original fit includes this energy, the terminology should be adjusted.
  4. [Appendix A.2] The grey curves labeled 'mu_B=0 LQCD-EoS fits mapped by T_fo' need a clearer explanation: are they the mu_B=0 lattice curves evaluated at the freeze-out temperature, and if so, why is this a meaningful high-energy reference?
  5. [Fig. 4(c)] The dashed c3=0 contour is visually difficult to distinguish from the color-map features. Consider increasing the line width or using a contrasting color.

Circularity Check

0 steps flagged

No significant circularity: the cumulant fingerprints are emergent outputs of independently calibrated EoS models.

full rationale

The derivation chain is self-contained and does not recycle its inputs as predictions. The cumulant definitions in Eqs. (2)-(4) are taken from Ref. [25], but the paper evaluates them from the pressures of three HRG models; no temperature-cumulant data are used to fix any parameter. The EV repulsion b is constrained by lattice-QCD conserved-charge fluctuations (Refs. [28,29]) and the VDW parameters a,b by the nuclear ground state (Refs. [28,30], Eq. (9)); these are independent of c2,c3,c4. The lattice comparisons in Fig. 2 use published HotQCD specific-heat points and analytic EoS parametrizations as external benchmarks, not as fit targets. The c2 minimum, c3 sign change, and low-energy freeze-out separation in Figs. 3-4 are emergent consequences of the calibrated VDW/EV equations of state: the c3=0 contour is a separate derivative calculation and is displaced from the c2 minimum by the explicit T powers in Eq. (2), so it is not a trivial derivative of the minimum. The only self-citation (Ref. [4], a review with author overlap) is contextual and non-load-bearing. The identification of the c2 valley as the Widom line in Fig. 4(c) is a labeling choice for a computed minimum locus, not an input used to derive that minimum. The acknowledged extrapolation of Eq. (10) below sqrt(s_NN)=3 GeV is a validity limitation, not a circular step.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The central calculation rests on the Chen et al. temperature-cumulant formalism, the ideal/EV/VDW HRG equations imported from Refs. [28,29], and lattice/freeze-out parametrizations. All free parameters are external fit values from prior literature; the low-energy freeze-out continuation is an extrapolation made in this paper. No new particles, forces, or conserved quantities are introduced.

free parameters (4)
  • EVHRG excluded-volume parameter b = b = 1.0 fm^3 (central; interval 0.4-1.0 fm^3 considered)
    Baryon repulsion strength taken from lattice QCD conserved-charge fluctuation constraints [28,29]; central c2/c3/c4 values use b=1.0 fm^3.
  • VDW-HRG attraction a = a = 329 MeV fm^3
    Fixed by nuclear ground-state saturation density and binding energy in Refs. [28,30]; not fitted to cumulant observables.
  • VDW-HRG repulsion b = b = 3.42 fm^3
    Same nuclear-matter fit [28,30]; controls the liquid-gas critical point location (Tc, mu_c) approx (19.7, 908) MeV.
  • Chemical freeze-out curve coefficients = T_fo: 158.4, 2.60, 0.45; mu_B: 1307.5, 0.288 (Eq. 10)
    Phenomenological parametrization fitted to particle yields [35]; the paper extends it below sqrt(s_NN)=3 GeV as a low-energy continuation.
axioms (6)
  • domain assumption Temperature-fluctuation cumulants follow from entropy derivatives of w=T s-p, giving Eqs. (4)-(6).
    Central formalism imported from Ref. [25]; standard subsystem-in-heat-bath statistical mechanics at second order, with the higher-order construction taken from Chen et al.
  • domain assumption The hadron resonance gas with PDG2020 states up to 2.6 GeV (433 states, charm excluded) approximates the hadronic partition function.
    Standard HRG modeling choice stated in Sec. 3; mass cutoff and charm exclusion affect the numerical cumulants.
  • domain assumption Baryon-only EV/VDW sector structure: mesons stay ideal, baryon and antibaryon sectors are separate, and no meson-baryon or baryon-antibaryon interactions are included.
    Taken from Refs. [28,29]; this structure is necessary for the EV and VDW results in Figs. 2-4.
  • domain assumption The van der Waals EoS with a=329 MeV fm^3 and b=3.42 fm^3 reproduces the nuclear ground state and yields a liquid-gas critical point near (19.7, 908) MeV.
    Imported from Refs. [28,30]; the Widom-line fingerprint is computed from this EoS.
  • domain assumption Analytic HotQCD/WB parametrizations of the lattice EoS can be differentiated to give c3, c4, c5, c6 at mu_B=0.
    The paper acknowledges the spread reflects parametrization differences, not a full lattice error budget (Sec. 4.2, Appendix A.2).
  • ad hoc to paper Chemical freeze-out follows Eq. (10) with strangeness neutrality and n_Q=0.4 n_B, including below sqrt(s_NN)=3 GeV.
    The low-energy rows in Table 1 and the proximity to the Widom line in Fig. 4(c) rely on extrapolating Eq. (10) below the range where it was demonstrated.

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read the original abstract

Temperature fluctuations of the matter produced in relativistic heavy-ion collisions are related to event-by-event fluctuations of the mean transverse momentum and, through the specific heat, to the QCD equation of state. We calculate the temperature-fluctuation cumulants $c_2$, $c_3$, and $c_4$ in the ideal hadron resonance gas (HRG), in an excluded-volume HRG consistent with lattice QCD constraints on baryon repulsion, and in a van der Waals HRG that reproduces the nuclear ground state. At zero baryon density all three models agree with lattice QCD thermodynamics up to the chiral crossover and separate above it. Along the chemical freeze-out curve the models remain close for $\sqrt{s_{NN}} \gtrsim 20$ GeV and separate strongly at lower energies, where baryonic interactions dominate the thermal response. In the van der Waals model the variance $c_2$ develops a minimum along the Widom line of the nuclear liquid-gas transition and $c_3$ changes sign across it. Temperature cumulants thus connect mean-transverse-momentum fluctuations to thermodynamic structures in two regions of the QCD phase diagram.

Figures

Figures reproduced from arXiv: 2607.24616 by Debasish Mallick.

Figure 1
Figure 1. Figure 1: Equation of state at µB = 0: (a) pressure, (b) trace anomaly, (c) specific heat, and (d) speed of sound. Points with error bars are HotQCD data [32]; the blue band is the WB data [34]; dotted curves are the HotQCD and WB analytic parametrizations [32, 33]. The model curves show the IHRG, baryon-only EVHRG interval b = 0.4–1.0 fm3 [29], and nuclear-matter VDW-HRG [28]. 4. Results and Discussion 4.1. Equatio… view at source ↗
Figure 2
Figure 2. Figure 2: Temperature-fluctuation cumulants (a) c2, (b) c3, and (c) c4 at µB = 0. Black points in panel (a) show c2 = 1/χ2 from the published HotQCD specific heat [32], with propagated uncertainties. The grey band spans results derived from the HotQCD and WB parametrizations [32, 33]. The HRG interactions follow Refs. [28, 29]. The vertical band marks TCFO = 158.4 ± 1.4 MeV [35]. 2.4 5 10 20 50 100 200 sNN [GeV] 0.0… view at source ↗
Figure 3
Figure 3. Figure 3: Temperature-fluctuation cumulants (a) c2, (b) c3, and (c) c4 along the chemical freeze-out curve of Eq. (10) [35], with nS = 0 and nQ = 0.4nB. The curves show the IHRG, baryon-only EVHRG with b = 1.0 fm3 , and nuclear-matter VDW-HRG [28, 29]. tainties. The c3 and c4 bands, in contrast, are obtained by differ￾entiating the analytic HotQCD and WB parametrizations, and their spread reflects the analytic repre… view at source ↗
Figure 4
Figure 4. Figure 4: Nuclear liquid–gas structure in the VDW-HRG [ [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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    A. Andronic, P. Braun-Munzinger, K. Redlich, J. Stachel, Decoding the phase structure of QCD via particle pro- duction at high energy, Nature 561 (2018) 321.arXiv: 1710.09425. 7 Appendix A. Particle-sector diagnostics and higher orders Appendix A.1. Meson and baryon sectors To identify the origin of the model hierarchy, we evaluate the cumulants from sect...

    Pith/arXiv arXiv 2018