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REVIEW 2 major objections 4 minor 62 references

Critical fluctuations placed on hydrodynamic freeze-out surfaces produce a non-monotonic net-proton kurtosis versus beam energy that the non-critical baseline does not.

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2026-07-30 17:14 UTC pith:YPAEKSPU

load-bearing objection Solid methods paper that puts fRG critical susceptibilities on calibrated hydro freeze-out surfaces and gets a cleaner non-monotonic C4/C2 than the old freeze-out-curve approach; the (anti)baryon split is the real soft spot but does not erase the advance. the 2 major comments →

arxiv 2607.26912 v1 pith:YPAEKSPU submitted 2026-07-29 nucl-th hep-phnucl-ex

Critical net-proton number fluctuations with hydrodynamics

Rui-zhe Zhao , Shi Yin , Shanjin Wu , Lipei Du , Xiaofeng Luo , Wei-jie Fu This is my paper
classification nucl-th hep-phnucl-ex
keywords net-proton fluctuationsQCD critical end pointhydrodynamicsfunctional renormalization groupcumulant ratiosbaryon conservationheavy-ion collisions
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 asks whether a QCD critical end point can still leave a visible mark in net-proton number fluctuations once those fluctuations are embedded in a realistic heavy-ion collision. The authors take baryon-number susceptibilities from the functional renormalization group, including both ordinary and critical contributions, and assign them cell by cell to the particlization hypersurface of calibrated viscous hydrodynamics at nine RHIC energies. Experimental momentum and rapidity cuts, isospin randomization from baryons to protons, and exact global baryon conservation are applied. Low-order cumulant ratios barely change when critical fluctuations are switched on, but the kurtosis ratio C4/C2 develops a non-monotonic energy dependence at low beam energy that is absent in the pure hadron-resonance-gas baseline and is closer to STAR measurements than earlier calculations that used a single chemical freeze-out curve. A sympathetic reader cares because this is a concrete bridge between first-principles critical physics and the actual observables used to hunt for the critical end point.

Core claim

When fRG net-baryon susceptibilities that include critical end-point fluctuations are evaluated on the hydrodynamic freeze-out hypersurface, then filtered by STAR acceptance, isospin randomization and subensemble baryon conservation, the net-proton ratio C4/C2 shows a non-monotonic dependence on collision energy at low √s_NN; the identical pipeline without critical fluctuations does not, and the critical result is more comparable to STAR BES-II data than fRG evaluated on a simple freeze-out curve.

What carries the argument

Cell-by-cell encoding of fRG susceptibilities on the hydro particlization hypersurface: each fluid element contributes a local volume times χ_n(T,µ_B), followed by binomial acceptance, isospin filtering to protons, hypersurface summation, and canonical correction via the subensemble acceptance method.

Load-bearing premise

Critical fluctuations for baryons and antibaryons separately are obtained by rescaling the theoretically known net-baryon susceptibilities with ordinary hadron-gas ratios, and baryon–antibaryon correlations are neglected.

What would settle it

High-statistics net-proton C4/C2 at the lowest BES and fixed-target energies that remains monotonic, or the same hydro-plus-fRG pipeline with a freeze-out energy density or equation of state still consistent with bulk hadrons that erases the non-monotonicity, would falsify the claimed critical imprint.

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

If this is right

  • A critical end point can still imprint non-monotonic C4/C2 after realistic hydrodynamics, acceptance cuts and baryon conservation.
  • Evaluating critical susceptibilities on a single freeze-out curve overstates the non-monotonic signal relative to a full hypersurface.
  • Low-order ratios such as C2/C1 remain dominated by non-critical physics, acceptance and conservation.
  • The same pipeline extended below 7.7 GeV is a direct next test of the critical contribution.

Where Pith is reading between the lines

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

  • Once fixed-target points are added, residual data tension can jointly constrain both the critical-end-point location and the freeze-out energy density.
  • Because the critical excess grows with cumulant order, sixth-order ratios computed the same way would be a sharper experimental discriminant if measured.
  • The cell-by-cell hydro encoding is a reusable template for inserting any other first-principles susceptibility set into the same dynamical background.

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 / 4 minor

Summary. The paper computes net-proton cumulant ratios C2/C1, C3/C2 and C4/C2 on (3+1)D hydrodynamic particlization hypersurfaces for Au+Au collisions at nine beam energies √sNN=7.7–200 GeV. Net-baryon susceptibilities that include both regular and CEP-driven critical fluctuations are taken from prior fRG work and assigned to each fluid cell; STAR pT and rapidity acceptance, isospin randomization (q=1/2), and global baryon conservation via the subensemble acceptance method are then applied. Results are compared with an HRG+hydro baseline on the same surfaces and with earlier fRG evaluations along a single Andronic freeze-out curve. The central claim is that critical input produces a non-monotonic collision-energy dependence in C4/C2 that is absent in the non-critical baseline and is closer to STAR BES-II data than the single-curve calculation.

Significance. If the mapping from net-baryon critical susceptibilities to proton cumulants is reliable, the work supplies a concrete, experimentally comparable prediction that critical fluctuations on a realistic multi-cell freeze-out surface generate a non-monotonic C4/C2 signal. Combining calibrated hydrodynamics with first-principles-inspired susceptibilities, acceptance cuts, isospin filtering and SAM conservation is a clear methodological advance over single-point freeze-out evaluations. The HRG baseline on identical surfaces cleanly isolates the critical contribution, and the acceptance-dependence study in the supplement strengthens the qualitative claim. These elements make the paper a useful step toward quantitative CEP searches even if further validation of intermediate assumptions is required.

major comments (2)
  1. [Critical fluctuations encoded on the freeze-out hypersurface (Eqs. 4, 10)] Eqs. (4) and (10) constitute the load-bearing step that converts fRG net-baryon χ n^B into (anti)baryon and then net-proton cumulants. Critical (anti)baryon susceptibilities are defined by rescaling the fRG net-baryon result with HRG (anti)baryon-to-net ratios, and baryon–antibaryon correlations are set to zero. The manuscript itself notes that exactness holds only when all acceptance cuts are removed. At √sNN=7.7 GeV, where the non-monotonicity appears, the hypersurface spans a wide μ B range (Fig. 1) and the critical excess in χ4^B is large; the partition of that excess between B+ and B− before isospin filtering therefore directly controls the reported proton C4/C2. No alternate partition, no critical-regime validation of the HRG-ratio ansatz, and no theory band on this choice are provided. A sensitivity test (or an explicit statement that the qualitative non-monotonicity survives reas
  2. [Method paragraph before Eq. (3); Fig. 1] Each fluid element is assumed to be independently equilibrated at its local T and μ B (text preceding Eq. 3). Near the CEP the correlation length can become comparable to or larger than typical cell sizes, so the independent-cell grand-canonical assignment may overestimate the critical contribution that survives after acceptance and SAM. The paper does not estimate the correlation length relative to the hypersurface granularity or discuss how finite-size/critical slowing-down effects would modify the mapped cumulants. Even a qualitative argument or a reference to existing estimates would clarify whether this approximation is under control at the lowest energies.
minor comments (4)
  1. [Results and discussion] Results at √sNN=9.2 and 11.5 GeV are omitted because the hydrodynamic background is less well constrained; a short quantitative remark on residual uncertainty at the neighboring 7.7 and 14.5 GeV points would help the reader judge the robustness of the non-monotonic feature.
  2. [Results and discussion; Fig. 2 caption] The constant switching density ε fo=0.26 GeV/fm3 is fixed for all energies. A brief check (or citation to prior work) showing that modest variations of ε fo do not erase the non-monotonicity in C4/C2 would strengthen the presentation.
  3. [Fig. 1 and surrounding text] Figure 1 gray region (χ4^B/χ2^B not computed) overlaps part of the low-energy hypersurface; a sentence on how cells falling into that region are treated would remove ambiguity.
  4. [Fig. 2 and supplement figures] Typographical inconsistencies appear in the figure labels (e.g., C2=C1 versus C2/C1) and in the arXiv header date; these should be standardized.

Circularity Check

1 steps flagged

No significant circularity: non-monotonic C4/C2 is a genuine output of embedding independent fRG susceptibilities on calibrated hydro surfaces, not forced by definition or by fitting to the target data.

specific steps
  1. self citation load bearing [Introduction / Results; citation [35]; Eqs. (1)–(4)]
    "based on the fluctuations obtained from the functional renormalization group (fRG) approach, where both the regular and the critical fluctuations arising from the critical end point (CEP) are included. ... the grand canonical kurtosis of net-baryon fluctuations χ_B^4/χ_B^2 calculated in fRG [35] ... (T_CEP, μ_BCEP)=(98,643) MeV."

    Critical susceptibilities and CEP location are taken entirely from the authors' prior fRG paper [35] (overlapping authors Fu, Luo, Yin). This is upstream self-citation of the critical input. It is not load-bearing circularity for the present claim: [35] does not already compute STAR-cut net-proton C4/C2 on hydro hypersurfaces, and the non-monotonicity vs HRG+hydro is produced by the new embedding calculation, not by renaming [35].

full rationale

The derivation chain is: (i) net-baryon susceptibilities χ_n^B including CEP critical structure from prior fRG work [35]; (ii) placement on MUSIC/iS3D particlization hypersurfaces whose parameters were calibrated to bulk hadronic observables, not to net-proton cumulants; (iii) binomial acceptance, isospin randomization (q=1/2), and SAM canonical corrections; (iv) comparison to an HRG+hydro baseline on the same surfaces and to STAR BES-II. None of these steps makes C4/C2 equal to an input by construction. Hydro is not fitted to the fluctuation ratios being reported. The fRG input does not already contain STAR-acceptance net-proton C4/C2. The HRG baseline on identical surfaces is an internal control that isolates the critical excess. Eq. (4) and Eq. (10) are load-bearing modeling assumptions (rescaling net-baryon critical χ by HRG baryon/antibaryon ratios and neglecting B–B̄ correlations), but an untested map is a correctness risk, not circularity: the output is not definitionally identical to the input. Mild author overlap with [35] and the hydro calibration papers is ordinary upstream dependence and does not force the non-monotonic signal. Score 1 only for that minor self-citation of the critical input source; central claim remains independently computed.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The result inherits the fRG equation of state and CEP, a fully calibrated multi-stage hydro model, and several modeling maps that convert net-baryon critical susceptibilities into accepted net-proton cumulants. Free parameters are mostly upstream (ε_fo, hydro transport, CEP coordinates). The distinctive ad hoc content is the HRG-ratio split of critical fluctuations and the independent-cell equilibrium assumption.

free parameters (4)
  • ε_fo (particlization energy density) = 0.26 GeV/fm³
    Fixed at 0.26 GeV/fm³ for all energies by fitting bulk hadronic observables in the hydro framework; sets the entire freeze-out surface used for fluctuations.
  • CEP location (T_CEP, μ_B,CEP) from fRG = (98, 643) MeV
    Taken from prior fRG work as (98, 643) MeV; controls where critical enhancement sits relative to the hydro freeze-out bands.
  • Hydro initial-condition and transport parameters
    Calibrated in cited works to yields, spectra, mean pT, and rapidity distributions at the nine energies; not refit here but fully determine cell (T, μ_B, u^μ) distributions.
  • Isospin proton fraction q = 1/2
    Set to 1/2 for all cells rather than computed cell-by-cell from HRG with weak decays.
axioms (5)
  • domain assumption Each freeze-out fluid element is an independent grand-canonical thermal source whose baryon cumulants are δV T³ χ_n(T, μ_B).
    Stated in the method section before Eq. (3); ignores spatial correlations and critical dynamical evolution.
  • ad hoc to paper Critical (anti)baryon susceptibilities equal fRG net-baryon χ_n^B rescaled by HRG (anti)baryon-to-net ratios (Eq. 4); baryon–antibaryon correlations neglected in Eq. (10).
    Only net-baryon critical fluctuations are available from fRG/lattice; this map is required to feed isospin randomization and proton cumulants.
  • domain assumption Binomial acceptance and isospin randomization correctly convert cell-level baryon cumulants into accepted proton cumulants.
    Standard in the fluctuation literature (Kitazawa–Asakawa, Savchuk et al.); applied via Bell polynomials in Eqs. (5)–(8).
  • domain assumption Global baryon conservation is captured by the subensemble acceptance method mapping S of Vovchenko.
    Used in Eq. (11) and expanded in the supplement; assumes the SAM applicability conditions.
  • domain assumption fRG generalized susceptibilities χ_n^B correctly encode both regular and CEP critical fluctuations in the relevant (T, μ_B) region.
    Imported from cited fRG papers; not recomputed here.

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read the original abstract

We compute the net-proton number fluctuations and their ratios $C_2/C_1$, $C_3/C_2$ and $C_4/C_2$ on the hydrodynamic freeze-out hypersurface of particlization at nine collision energies, $\sqrt{s_{\mathrm{NN}}}=7.7-200$ GeV, based on the fluctuations obtained from the functional renormalization group (fRG) approach, where both the regular and the critical fluctuations arising from the critical end point (CEP) are included. The transverse momentum and rapidity acceptance windows as same as the experimental measurements, the isospin randomization for the proton number fluctuations, and the global baryon conservation effect are implemented in the calculations. The results are also compared with the baseline results without critical fluctuations. It is found that for the low-order cumulants, e.g., $C_2/C_1$ the difference between the critical and non-critical results is small, while the difference increases with the increasing order of cumulants in the region of low collision energy. A non-monotonic dependence on the collision energy is observed in $C_4/C_2$ with critical fluctuations, which is absent in the results without critical fluctuations.

Figures

Figures reproduced from arXiv: 2607.26912 by Lipei Du, Rui-zhe Zhao, Shanjin Wu, Shi Yin, Wei-jie Fu, Xiaofeng Luo.

Figure 1
Figure 1. Figure 1: FIG. 1. Counts of fluid elements distributed on the QCD [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Collision energy dependence of net-proton cumulant ratios [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Collision energy dependence of net-proton cumulant ratios [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Collision energy dependence of net-proton cumulant ratios [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

discussion (0)

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Reference graph

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