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arxiv: 2411.04969 · v2 · submitted 2024-11-07 · ⚛️ nucl-th · hep-ex· hep-ph· nucl-ex

Systematic study of flow of protons and light clusters in intermediate-energy heavy-ion collisions with momentum-dependent potentials

Viktar Kireyeu , Vadim Voronyuk , Michael Winn , Susanne Gl\"a{\ss}el , J\"org Aichelin , Christoph Blume , Elena Bratkovskaya , Gabriele Coci
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Jiaxing Zhao
This is my paper

Pith reviewed 2026-05-23 17:56 UTC · model grok-4.3

classification ⚛️ nucl-th hep-exhep-phnucl-ex
keywords nuclear equation of stateheavy-ion collisionscollective flowlight clustersmomentum-dependent potentialPHQMDdirected flowelliptic flow
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The pith

The soft momentum-dependent nuclear equation of state matches measured proton and light-cluster flows in GeV heavy-ion collisions while the static soft equation of state does not.

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

This paper compares three versions of the nuclear equation of state inside the PHQMD transport model to see how they shape the directed and elliptic flow of protons and light clusters produced in intermediate-energy heavy-ion collisions. Adding a momentum-dependent nucleon potential calibrated to proton-nucleus scattering data creates a soft momentum-dependent scenario alongside the usual static soft and hard options. The soft momentum-dependent version produces particle yields and flow patterns that line up with HADES and FOPI measurements, whereas the static soft version deviates. The work also finds that elliptic flow scales with cluster mass number at low transverse momentum and that different cluster-formation prescriptions inside the model generate distinct flow signals.

Core claim

The soft momentum-dependent equation of state yields an overall good agreement with experimental data on proton and cluster rapidity distributions, transverse-momentum spectra, directed flow v1 and elliptic flow v2 from the HADES and FOPI collaborations, while the static soft equation of state does not. Softening the equation of state reduces midrapidity proton yields and increases light-cluster production. For protons the momentum-dependent soft potential slightly raises the magnitudes of v1 and v2 relative to the hard equation of state; cluster flows remain comparable across the soft variants. Elliptic flow v2 scales with cluster mass number A at midrapidity and low pT but breaks at higher

What carries the argument

The momentum-dependent nucleon potential added to the static Skyrme interaction inside the PHQMD model, which distinguishes the effects of momentum dependence from pure density dependence on particle propagation and clustering.

If this is right

  • Softening the equation of state reduces proton yields at midrapidity while increasing light-cluster production.
  • Proton directed and elliptic flow magnitudes increase slightly when the soft momentum-dependent potential is used instead of the hard equation of state.
  • Elliptic flow scales with cluster mass number A at midrapidity and low transverse momentum but the scaling breaks at higher pT.
  • Deuterons formed by MST clustering inside the model show different flow patterns from those formed by coalescence at freeze-out.

Where Pith is reading between the lines

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

  • The observed difference in flow between the two cluster-formation prescriptions suggests that flow harmonics could be used experimentally to test which mechanism dominates in real collisions.
  • If the mass-number scaling of v2 persists in other collision systems, it may offer a model-independent signature of collective motion at the cluster level.
  • The requirement for momentum dependence at these energies implies that static Skyrme forces alone are insufficient for quantitative predictions of cluster observables.

Load-bearing premise

The momentum-dependent potential parameters taken from external proton-nucleus scattering data and the PHQMD clustering or coalescence rules correctly capture the dominant cluster formation physics.

What would settle it

New measurements of light-cluster elliptic flow at moderate to high transverse momentum that deviate from the soft momentum-dependent predictions while agreeing with the static soft equation of state.

Figures

Figures reproduced from arXiv: 2411.04969 by Christoph Blume, Elena Bratkovskaya, Gabriele Coci, Jiaxing Zhao, J\"org Aichelin, Michael Winn, Susanne Gl\"a{\ss}el, Vadim Voronyuk, Viktar Kireyeu.

Figure 1
Figure 1. Figure 1: FIG. 1. Schr¨odinger equivalent optical potential [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Equation-of-state for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Time evolution of the normalized baryon density [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Rapidity distribution [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The directed flow [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p018_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p019_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19 [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. The elliptic flow scaled with atomic number [PITH_FULL_IMAGE:figures/full_fig_p022_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21 [PITH_FULL_IMAGE:figures/full_fig_p022_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22 [PITH_FULL_IMAGE:figures/full_fig_p023_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23 [PITH_FULL_IMAGE:figures/full_fig_p024_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24 [PITH_FULL_IMAGE:figures/full_fig_p025_24.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26. Upper plot: Comparison of the [PITH_FULL_IMAGE:figures/full_fig_p026_26.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. Upper plot: Comparison of the [PITH_FULL_IMAGE:figures/full_fig_p026_25.png] view at source ↗
Figure 27
Figure 27. Figure 27: FIG. 27. Upper plot: Comparison of [PITH_FULL_IMAGE:figures/full_fig_p027_27.png] view at source ↗
Figure 29
Figure 29. Figure 29: FIG. 29. Schematic presentation of the particle motion in the [PITH_FULL_IMAGE:figures/full_fig_p029_29.png] view at source ↗
read the original abstract

We study the influence of the nuclear equation-of-state (EoS) on collective observables -- the directed ($v_1$) and elliptic flow ($v_2$) of nucleons and light clusters -- in heavy-ion collisions at GeV energies using the Parton-Hadron-Quantum-Molecular Dynamics (PHQMD) approach. A novel development in this work is the inclusion of a momentum-dependent nucleon potential in the PHQMD in addition to the static, density-dependent Skyrme interaction. This enables three distinct EoS scenarios: two static ("soft" and "hard", differing in compressibility) and a soft, momentum-dependent EoS calibrated to $pA$ elastic scattering data. We find a strong EoS sensitivity in proton and cluster rapidity and $p_T$ distributions: soft and soft momentum-dependent EoS yield similar results, markedly different from the hard EoS. Softening the EoS reduces proton yields at midrapidity while enhancing light-cluster production. The EoS also affects flow observables differently for nucleons and clusters. For protons, a soft momentum-dependent potential increases slightly the magnitude of $v_1$ and $v_2$ relative to the hard EoS, whereas cluster flows are nearly similar. The soft momentum-dependent EoS provides an overall good agreement with experimental data from HADES and FOPI Collaborations while the soft EOS is not in line with the data. A scaling of $v_2$ with cluster mass number $A$ is observed at midrapidity for low $p_T$, which breaks at higher $p_T$. Finally, we examine the sensitivity of flow observables to deuteron production mechanisms. Deuterons formed via MST clustering exhibit different flow patterns from those produced by coalescence at freeze-out, indicating that flow harmonics may help discriminate between cluster formation scenarios.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript uses the PHQMD transport model to study the sensitivity of directed (v1) and elliptic (v2) flows of protons and light clusters to the nuclear EoS in GeV-energy heavy-ion collisions. Three EoS variants are compared: static soft, static hard, and a soft momentum-dependent EoS whose parameters are fixed from independent pA elastic scattering. The central claim is that soft and soft-md variants produce similar rapidity and pT distributions (distinct from hard), with the soft-md case providing overall good agreement with HADES and FOPI data while the static soft EoS does not; additional results address v2 mass scaling and sensitivity to deuteron production mechanisms (MST vs. coalescence).

Significance. If the quantitative distinction between static soft and soft-md holds, the work demonstrates that momentum dependence in the mean-field potential improves description of collective flows at intermediate densities, offering a route to tighter EoS constraints. The external calibration of the momentum-dependent parameters to pA data is a positive feature that limits circularity with the flow observables.

major comments (2)
  1. [Abstract] Abstract: The claim that the soft momentum-dependent EoS 'provides an overall good agreement with experimental data' while 'the soft EOS is not in line with the data' is not supported by any quantitative metric (chi-squared, pull distribution, or explicit acceptance criterion). The text states that soft and soft-md produce 'similar results' for rapidity/pT distributions and only a 'slight increase' in proton |v1| and |v2| under the momentum-dependent case, leaving the decisive distinction unquantified relative to HADES/FOPI uncertainties.
  2. [Results (flow observables)] Flow observables section: The load-bearing distinction between static soft and soft-md rests on small proton flow differences whose statistical significance is not assessed; cluster flows are described as 'nearly similar' between the two, so the overall data-agreement claim depends on whether the proton shift exceeds experimental errors—an assessment absent from the manuscript.
minor comments (1)
  1. [Abstract] Abstract: The reported v2 scaling with cluster mass number A at low pT is stated without the numerical exponent or the precise pT cutoff at which the scaling breaks.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. The concerns about lack of quantitative support for the data-agreement claims are valid, and we address them point by point with proposed revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the soft momentum-dependent EoS 'provides an overall good agreement with experimental data' while 'the soft EOS is not in line with the data' is not supported by any quantitative metric (chi-squared, pull distribution, or explicit acceptance criterion). The text states that soft and soft-md produce 'similar results' for rapidity/pT distributions and only a 'slight increase' in proton |v1| and |v2| under the momentum-dependent case, leaving the decisive distinction unquantified relative to HADES/FOPI uncertainties.

    Authors: We agree the abstract claim would benefit from qualification. The distinction relies on visual comparison in the figures, where soft-md proton flows align better with data than static soft despite the small shift. We will revise the abstract to state that soft-md 'provides improved agreement with the data' based on the presented comparisons, and add a sentence referencing the figures for the assessment. This avoids overstatement while preserving the central finding. revision: yes

  2. Referee: [Results (flow observables)] Flow observables section: The load-bearing distinction between static soft and soft-md rests on small proton flow differences whose statistical significance is not assessed; cluster flows are described as 'nearly similar' between the two, so the overall data-agreement claim depends on whether the proton shift exceeds experimental errors—an assessment absent from the manuscript.

    Authors: The referee is correct that no formal significance test or error comparison is provided. The text notes the small proton differences and near-similarity for clusters. We will revise the flow observables section to include a qualitative discussion of the proton |v1| and |v2| shift magnitude relative to the experimental error bars shown in the figures, noting that it improves agreement while the static soft remains offset. A full chi-squared is not added, as model systematics complicate it, but the visual assessment will be explicitly tied to uncertainties. revision: partial

Circularity Check

0 steps flagged

No significant circularity; EoS parameters from independent pA data and flows compared to separate experiments

full rationale

The paper calibrates the momentum-dependent nucleon potential to external pA elastic scattering data (distinct from the AA collision flows under study) and compares resulting v1/v2 observables to independent HADES/FOPI measurements. No equations or claims reduce a prediction to a fit on the same dataset by construction, and no self-citation load-bearing steps appear in the abstract or described text. The distinction between soft and soft-md EoS rests on direct simulation outputs versus external data, which is self-contained against benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the external calibration of the momentum-dependent potential to pA data and on standard assumptions of the Skyrme force and PHQMD transport framework; no new entities are postulated.

free parameters (1)
  • momentum-dependent potential strength and range parameters
    Calibrated to pA elastic scattering data to define the soft momentum-dependent EoS
axioms (1)
  • domain assumption Skyrme interaction provides the static density-dependent part of the nucleon potential
    Standard background assumption in nuclear transport models invoked for all three EoS variants

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Proton and kaon production in Au+Au collisions at $\sqrt{s_{\rm NN}}=3$ GeV

    nucl-th 2026-04 unverdicted novelty 3.0

    Momentum-dependent nuclear mean field with K0=230 MeV provides a good description of proton and kaon production and collective flows in 3 GeV Au+Au collisions, outperforming momentum-independent fields.

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