arxiv: 2604.27473 · v2 · submitted 2026-04-30 · ⚛️ nucl-th · hep-ph· nucl-ex

Recognition: 2 theorem links

· Lean Theorem

Quarkonium p_{rm T} spectra in heavy--ion collisions at LHC energies within a hydrodynamic core--corona framework

Biswarup Paul

Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:45 UTC · model grok-4.3

classification ⚛️ nucl-th hep-phnucl-ex

keywords quarkoniumheavy-ion collisionshydrodynamic modelcore-coronapT spectraLHC energiesPb-Pb collisionscharmonium bottomonium

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The pith

An analytical hydrodynamic model with core-corona separation reproduces quarkonium transverse momentum spectra in lead-lead collisions at LHC energies.

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

The paper applies an analytical relativistic hydrodynamics framework to the medium created in heavy-ion collisions, assuming cylindrical symmetry, boost-invariant longitudinal expansion, and Hubble-like transverse flow. It evaluates quarkonium production via the Cooper-Frye formalism on a constant-temperature freeze-out hypersurface while using a core-corona decomposition to account for both thermal yields inside the dense region and non-thermal contributions at the periphery. The resulting pT spectra for J/ψ, ψ(2S), and Υ(nS) states are compared directly to ALICE and CMS measurements in Pb-Pb collisions at 5.02 TeV, reproducing the data over a broad momentum range and the observed yield ratios. This establishes that the combined approach supplies a transparent description of both charmonium and bottomonium production without requiring full numerical hydrodynamics.

Core claim

Within the hydrodynamic core-corona framework, the transverse momentum spectra of charmonium (J/ψ, ψ(2S)) and bottomonium (Υ(nS)) states are calculated by integrating the thermal distribution over a constant-temperature freeze-out hypersurface in a cylindrically symmetric, boost-invariant medium with Hubble-like transverse flow; the core-corona separation incorporates thermal production in the dense interior and non-thermal contributions from the corona, yielding spectra and ratios that match ALICE and CMS data across a wide pT range while indicating additional hard processes at high pT for J/ψ and sequential suppression for bottomonia.

What carries the argument

Analytical relativistic hydrodynamics under cylindrical symmetry with boost-invariant longitudinal expansion and Hubble-like transverse flow, evaluated on a constant-temperature freeze-out hypersurface via the Cooper-Frye formalism and combined with core-corona separation of thermal and non-thermal quarkonium contributions.

If this is right

The model accounts for high-pT deviations in J/ψ spectra by invoking additional hard production mechanisms outside the thermal core.
Bottomonium spectra and their yield ratios follow the expected sequential suppression ordered by binding energy.
The ψ(2S)/J/ψ ratio is reproduced across the measured pT range, confirming the role of the core-corona distinction.
The same framework applies uniformly to both charmonium and bottomonium sectors at LHC energies.

Where Pith is reading between the lines

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

If the analytical flow profile suffices here, similar closed-form hydrodynamics could be tested on other heavy-flavor observables without full numerical codes.
The success at 5.02 TeV suggests the assumed symmetry captures the dominant expansion features for quarkonium decoupling, but deviations at lower energies would test the Hubble-like flow assumption.
Extending the constant-temperature freeze-out to a temperature-dependent surface could quantify how sensitive the spectra are to the precise decoupling condition.

Load-bearing premise

The expanding medium is taken to obey cylindrical symmetry, boost-invariant longitudinal flow, and Hubble-like transverse flow, with all quarkonia decoupling at one fixed temperature on the freeze-out surface.

What would settle it

A new measurement of the pT spectra or ψ(2S)/J/ψ ratio in a different centrality bin or at a lower collision energy that shows large deviations from the model's predictions while the assumed flow symmetry remains intact would falsify the claim.

read the original abstract

We present a systematic study of the transverse momentum ($p_{\rm T}$) spectra of charmonium (J/$\psi$, $\psi(2S)$) and bottomonium ($\Upsilon(nS)$) states in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV within an analytical relativistic hydrodynamics framework. The medium evolution is described assuming cylindrical symmetry with boost-invariant longitudinal expansion and Hubble-like transverse flow. Quarkonium spectra are evaluated using the Cooper-Frye formalism on a constant-temperature freeze-out hypersurface, supplemented by a core--corona approach to include both thermal and non-thermal contributions. The model describes the measurements from ALICE and CMS over a broad $p_{\rm T}$ range. For charmonium, both the spectra and the $\psi(2S)$/J/$\psi$ ratio are well reproduced, while deviations at high $p_{\rm T}$ for J/$\psi$ indicate additional hard production mechanisms. In the bottomonium sector, the $\Upsilon(nS)$ spectra and their yield ratios are successfully described, consistent with the expected sequential suppression pattern. These results demonstrate that an analytical hydrodynamic approach combined with a core-corona framework provides a unified and transparent description of quarkonium production in heavy--ion collisions at LHC energies.

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.

Desk Editor's Note private letter to a colleague

This analytical core-corona hydro model reproduces quarkonium spectra at LHC energies reasonably well, though the idealized flow assumptions warrant checking.

read the letter

The one thing to know is that this paper takes a simple analytical hydrodynamic setup with a core-corona decomposition and shows it can describe the pT spectra of J/ψ, ψ(2S), and Υ(nS) in 5.02 TeV heavy-ion collisions, matching data from ALICE and CMS for most of the measured range. The new element is the systematic treatment of both charmonium and bottomonium states together at this LHC energy. It includes direct comparisons of spectra and ratios, highlighting the sequential suppression pattern in the bottomonium sector and the reproduction of the ψ(2S) to J/ψ ratio. The framework assumes cylindrical symmetry with boost-invariant longitudinal expansion and a Hubble-like transverse flow velocity that grows linearly with radius. Spectra come from the Cooper-Frye prescription on a constant-temperature hypersurface, with the corona part added to account for non-thermal production. This approach has the advantage of being transparent and computationally light. It captures the overall shape of the spectra at low to intermediate pT and gets the ratios right without needing complex numerical hydrodynamics. The note on high-pT deviations for J/ψ pointing to additional hard processes is reasonable and consistent with other observations in the field. Where it gets softer is in the flow modeling. The specific choice of Hubble-like flow and a single freeze-out temperature creates a rigid relation between the radial velocity profile and the resulting blue-shift in the spectra. For particles as heavy as quarkonia, small changes in the actual space-time evolution could alter the predicted pT distributions noticeably. Since the core-corona separation parameters are adjusted along with the flow strength and temperature, there is a risk that the corona component is tuned to compensate for any shortcomings in the thermal part rather than reflecting independent physics. The abstract mentions good agreement but does not provide details on fit quality or how the parameters were constrained independently. This paper is for specialists in heavy-ion collisions who are interested in analytical models for quarkonium production and suppression. It offers a clear way to think about the interplay between thermal and non-thermal contributions. A reader looking for quick estimates or a baseline for more detailed studies would get value from it. The work is solid enough on its own terms to merit a serious referee, who could probe the sensitivity to the flow assumptions and ask for more quantitative comparisons. I recommend sending it for peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a systematic study of the transverse momentum spectra of charmonium (J/ψ, ψ(2S)) and bottomonium (Υ(nS)) states in Pb-Pb collisions at √s_NN = 5.02 TeV. It employs an analytical relativistic hydrodynamics framework assuming cylindrical symmetry, boost-invariant longitudinal expansion, and Hubble-like transverse flow. Spectra are evaluated with the Cooper-Frye formalism on a constant-temperature freeze-out hypersurface, combined with a core-corona approach to include thermal and non-thermal contributions. The authors report that the model describes ALICE and CMS data for spectra and yield ratios over a broad p_T range, with deviations at high p_T for J/ψ and consistency with sequential suppression for bottomonia.

Significance. If the central results hold, the work offers a transparent, computationally simple framework that unifies thermal and hard production mechanisms for quarkonia, potentially useful as a benchmark or for quick estimates in heavy-ion phenomenology. The analytical hydro plus core-corona split makes the separation of contributions explicit. However, the significance is reduced by the reliance on three adjustable parameters and the idealized flow assumptions, which may limit the extent to which the agreement constitutes a genuine test of the hydrodynamic picture rather than a fit.

major comments (2)

The hydrodynamic evolution assumes cylindrical symmetry with boost-invariant longitudinal expansion and Hubble-like transverse flow (v_r ∝ r), with spectra obtained via Cooper-Frye on a single constant-T hypersurface. This rigidly fixes the radial boost and effective temperature. For states whose mass is comparable to the flow velocity scale, mismatches with realistic viscous evolution, longitudinal gradients, or fluctuating initial conditions would directly affect the low-to-intermediate p_T shape once the core-corona split is applied. The manuscript should quantify the sensitivity of the spectra to the flow profile (e.g., by varying the Hubble parameter or comparing to numerical hydro) to establish that the thermal component is physically motivated rather than absorbed into the corona weights.
The abstract states that the model describes the data for most observables while noting deviations at high p_T for J/ψ. Because the Hubble flow parameter, freeze-out temperature, and core-corona separation weights are free parameters typically tuned to spectra, the reported agreement may largely reflect the fit rather than independent predictive power. The paper should report quantitative fit metrics (e.g., χ² per degree of freedom) and show results with the corona component turned off to demonstrate that the analytical hydro component supplies the correct shape on its own.

minor comments (1)

The abstract mentions deviations at high p_T for J/ψ but does not quantify the p_T threshold or the size of the discrepancy; adding this information would clarify the range of validity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the positive overall assessment of our work. We address each major comment point by point below, indicating the revisions we will implement in the manuscript.

read point-by-point responses

Referee: The hydrodynamic evolution assumes cylindrical symmetry with boost-invariant longitudinal expansion and Hubble-like transverse flow (v_r ∝ r), with spectra obtained via Cooper-Frye on a single constant-T hypersurface. This rigidly fixes the radial boost and effective temperature. For states whose mass is comparable to the flow velocity scale, mismatches with realistic viscous evolution, longitudinal gradients, or fluctuating initial conditions would directly affect the low-to-intermediate p_T shape once the core-corona split is applied. The manuscript should quantify the sensitivity of the spectra to the flow profile (e.g., by varying the Hubble parameter or comparing to numerical hydro) to establish that the thermal component is physically motivated rather than absorbed into the corona weights.

Authors: We agree that a quantitative assessment of sensitivity to the flow profile strengthens the physical interpretation. In the revised manuscript we will add a dedicated subsection (or appendix) presenting results obtained by varying the Hubble parameter by ±20% around its default value while readjusting only the overall normalization. These calculations show that the low-to-intermediate p_T spectral shapes remain robust, with the main variations absorbed into the core-corona weights; this supports that the thermal component is not an artifact of parameter tuning. A full comparison against numerical viscous hydrodynamics lies outside the scope of the present analytical study, whose purpose is to supply a transparent benchmark; we will nevertheless add a brief discussion of this limitation and its implications for the model assumptions. revision: partial
Referee: The abstract states that the model describes the data for most observables while noting deviations at high p_T for J/ψ. Because the Hubble flow parameter, freeze-out temperature, and core-corona separation weights are free parameters typically tuned to spectra, the reported agreement may largely reflect the fit rather than independent predictive power. The paper should report quantitative fit metrics (e.g., χ² per degree of freedom) and show results with the corona component turned off to demonstrate that the analytical hydro component supplies the correct shape on its own.

Authors: We accept that quantitative goodness-of-fit measures and an explicit separation of components would improve clarity. In the revised version we will report χ² per degree of freedom for the p_T spectra fits to both ALICE and CMS data. We have also prepared additional plots (to be included as a new figure or in the main text) that isolate the pure thermal (core) contribution. These show that the hydrodynamic component alone reproduces the low-p_T rise and the characteristic flow-induced hardening, while the corona term is required only for the high-p_T tail; this explicitly demonstrates the independent role of the analytical hydro part. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents an analytical hydrodynamic model (cylindrical symmetry, boost-invariant expansion, Hubble-like transverse flow) combined with Cooper-Frye freeze-out and a core-corona split to compute quarkonium pT spectra. The provided text states that the model 'describes the measurements' over a broad pT range and reproduces ratios, but contains no quoted equations or sections showing that any central result (e.g., spectra shapes or ratios) is obtained by fitting a parameter to the very same data and then relabeling the output as a prediction. No self-citation chains are invoked as load-bearing uniqueness theorems, and the derivation chain rests on explicit physical assumptions rather than reducing to its inputs by construction. This is the normal case of a model with adjustable parameters being compared to data.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard hydrodynamic assumptions for the medium and on a core-corona decomposition whose relative weights are not independently constrained in the abstract.

free parameters (3)

Hubble-like transverse flow parameter
Controls the strength of transverse expansion in the hydrodynamic evolution.
Freeze-out temperature
Defines the constant-temperature hypersurface used in the Cooper-Frye emission.
Core-corona separation parameters
Determine the relative thermal and non-thermal contributions to the spectra.

axioms (2)

domain assumption Cylindrical symmetry with boost-invariant longitudinal expansion
Assumed to simplify the relativistic hydrodynamic description of the medium.
domain assumption Cooper-Frye formalism on a constant-temperature freeze-out hypersurface
Used to convert fluid flow into quarkonium momentum distributions.

pith-pipeline@v0.9.0 · 5544 in / 1550 out tokens · 67266 ms · 2026-05-11T01:45:07.746798+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The medium evolution is described assuming cylindrical symmetry with boost-invariant longitudinal expansion and Hubble-like transverse flow. Quarkonium spectra are evaluated using the Cooper-Frye formalism on a constant-temperature freeze-out hypersurface, supplemented by a core-corona approach
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel contradicts

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contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

The model parameters are determined via a χ² minimization procedure... T≈160 MeV and ⟨v_T⟩≈0.60... T≈224 MeV and ⟨v_T⟩≈0.56... with three parameters total

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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