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arxiv: 2506.10702 · v4 · pith:NIBOTW3Pnew · submitted 2025-06-12 · ✦ hep-ph · hep-ex· nucl-ex· nucl-th

Extraction of Effective Parameters from Transverse Momentum Spectra of Heavy Quarkonia in Proton-Proton Collisions at the LHC

Peng-Cheng Zhang , Hailong Zhu , Fu-Hu Liu , Khusniddin K. Olimov This is my paper

Pith reviewed 2026-05-22 00:42 UTC · model grok-4.3

classification ✦ hep-ph hep-exnucl-exnucl-th
keywords heavy quarkoniap_T spectraSchwinger mechanismBose-Einstein statisticsproton-proton collisionsLHCeffective parametersstring tension
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The pith

Multi-component Schwinger and Bose-Einstein distributions describe heavy quarkonium p_T spectra and yield effective string tension and temperature.

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

The paper extracts the effective string tension κ from the Schwinger mechanism and the effective temperature T from Bose-Einstein statistics by fitting the transverse momentum spectra of J/ψ and Υ particles produced in proton-proton collisions at the LHC. These parameters characterize the production in small collision systems where no quark-gluon plasma forms. The fits show that both κ and T increase as rapidity decreases in the forward direction, establishing a direct proportionality between the two. From the extracted κ values, the average minimum radius of the strong force acting on the participant quarks is calculated. The temperature T is taken to reflect the initial stage conditions because radial and transverse flows from geometric asymmetry and local thermalization boost its value.

Core claim

The multi-component distribution structured within the framework of the Schwinger mechanism or Bose-Einstein statistics can effectively describe the heavy quarkonium p_T spectra in small collision systems. With decreasing rapidity in the forward region, both κ and T increase, indicating a directly proportional relationship between them. Based on κ, the average minimum strong force radius of participant quarks is determined. T derived from the spectra serves as the initial effective temperature because geometric asymmetry and local partonic thermalization induce radial and transverse flows that increase T even without QGP formation.

What carries the argument

Effective string tension κ in the Schwinger mechanism combined with effective temperature T in Bose-Einstein statistics for fitting p_T spectra and deriving the minimum strong force radius of quarks.

If this is right

  • The p_T spectra in pp collisions are well described by these multi-component distributions.
  • κ and T are proportional and both grow toward lower rapidities.
  • The initial effective temperature of small systems can be inferred from the fits despite no QGP.
  • Participant quark strong force radius follows from the value of κ.

Where Pith is reading between the lines

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

  • This fitting approach may be applied to other mesons to check for consistent radii.
  • The observed relation between κ and T could point to shared parton dynamics across different collision systems.
  • Direct comparison of the derived radii with lattice QCD calculations of strong force range would test the method.

Load-bearing premise

The extracted effective temperature T corresponds to the initial effective temperature of the collision system despite contributions from radial and transverse flows caused by geometric asymmetry and local partonic thermalization.

What would settle it

Experimental data on heavy quarkonium p_T spectra at additional rapidity intervals or collision energies that either confirm or contradict the increasing trend of κ and T with decreasing rapidity, or independent measurements of the strong force radius.

Figures

Figures reproduced from arXiv: 2506.10702 by Fu-Hu Liu, Hailong Zhu, Khusniddin K. Olimov, Peng-Cheng Zhang.

Figure 1
Figure 1. Figure 1: illustrates the double differential cross section, d 2σ/(dydpT ), of (a) prompt J/ψ and (b) J/ψ originating from b-quarks in p+p collisions at √ s = 13 TeV, where σ denotes the cross section [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The double differential cross section, d 2σ/(dydpT ), for (a) Υ(1S), (b) Υ(2S), and (c) Υ(3S) produced in p+p collisions at √ s = 13 TeV. Different symbols represent experimental data in various y intervals measured by the LHCb Collaboration [41], with the data rescaled by different factors for clarity. The solid and dashed curves correspond to our results fitted using three￾component Schwinger and Bose-Ei… view at source ↗
Figure 3
Figure 3. Figure 3: The double differential cross section, d 2σ/(dydpT ), for (a) prompt J/ψ and (b) J/ψ from b produced in p+p collisions at √ s = 8 TeV. Different symbols represent experimental data in various y intervals measured by the LHCb Collaboration [42], with the data rescaled by different factors for clarity. The solid and dashed curves correspond to our results fitted using three-component Schwinger and Bose-Einst… view at source ↗
Figure 4
Figure 4. Figure 4: The double differential cross section, d 2σ/(dydpT ), for (a) Υ(1S), (b) Υ(2S), and (c) Υ(3S) produced in p+p collisions at √ s = 8 TeV. Different symbols represent experimental data in various y intervals measured by the LHCb Collaboration [43], with the data rescaled by different factors for clarity. The solid and dashed curves correspond to our results fitted using three-component Schwinger and Bose-Ein… view at source ↗
Figure 5
Figure 5. Figure 5: Relationship between string tension κ and initial temperature T derived from the spectra of heavy quarkonia J/ψ and Υ(nS) produced in p+p collisions at (a) 13 TeV and (b) 8 TeV. Results corresponding to 2.0 < y < 2.5 (4.0 < y < 4.5) are located at the higher (lower) value edge for each case. Five cases are illustrated in the panels. 2 2.5 3 3.5 4 4.5 y 0.02 0.04 0.06 0.08 0.1 0.12 (fm) min R 13 TeV p+p (a)… view at source ↗
Figure 6
Figure 6. Figure 6: Dependence of the average minimum strong force rad [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

The effective string tension ($\kappa$) in the Schwinger mechanism and the effective temperature ($T$) in Bose-Einstein statistics are extracted from the transverse momentum ($p_T$) spectra of heavy quarkonia produced in proton-proton (p+p) collisions at the Large Hadron Collider (LHC). Here, $T$ derived from the heavy quarkonium $p_T$ spectra also serves as the initial effective temperature (effective temperature at the initial stage) of small collision systems. This is because, despite the absence of quark-gluon plasma (QGP) formation during the collisions, which leaves $T$ largely unaffected by QGP-related effects, the initial geometric asymmetry and local partonic thermalization still induce radial and transverse flows, thereby contributing to an increase in $T$. The effective parameters ($\kappa$ and $T$) are obtained by fitting the experimental $p_T$ spectra of $J/\psi$ and $\Upsilon(nS)$ ($n=1$, 2, and 3) within various rapidity intervals, produced in p+p collisions at center-of-mass energies of $\sqrt{s}=13$ and 8 TeV, as measured by the LHCb Collaboration. It is found that the multi-component distribution structured within the framework of the Schwinger mechanism or Bose-Einstein statistics can effectively describe the heavy quarkonium $p_T$ spectra in small collision systems. With decreasing rapidity in the forward region, both $\kappa$ and $T$ increase, indicating a directly proportional relationship between them. Based on $\kappa$, the average minimum strong force radius of participant quarks is determined.

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

3 major / 2 minor

Summary. The paper extracts the effective string tension κ via the Schwinger mechanism and effective temperature T via Bose-Einstein statistics by fitting multi-component distributions to the p_T spectra of J/ψ and Υ(nS) states in pp collisions at √s=13 and 8 TeV, using LHCb data in multiple forward rapidity intervals. It asserts that the fitted T represents the initial effective temperature of the small system because geometric asymmetry and local partonic thermalization generate radial and transverse flows even without QGP formation. The authors report that both κ and T increase toward mid-rapidity (decreasing rapidity), establish a direct proportionality between them, and derive the average minimum strong-force radius of participant quarks from the extracted κ values.

Significance. If the mapping from fitted T to initial temperature can be independently validated, the work would provide a useful phenomenological framework for characterizing effective QCD parameters and flow-induced effects in small collision systems. The focus on heavy quarkonia and forward-rapidity LHCb data adds specificity, and the derivation of a quark radius from κ offers a concrete observable. However, the absence of quantitative anchoring for the temperature interpretation limits the immediate impact on understanding initial conditions in pp collisions.

major comments (3)
  1. [Abstract] Abstract: The identification of the fitted T with the initial effective temperature rests on the statement that geometric asymmetry and local partonic thermalization induce radial and transverse flows that raise T even without QGP. No hydrodynamic or blast-wave estimate of the expected ΔT, no comparison to independent initial-temperature proxies (particle ratios, femtoscopy), and no sensitivity test to fit choices are supplied; this renders the reported increase and proportionality of κ and T, as well as the derived minimum strong-force radius, dependent on an untested mapping.
  2. [Results] Results section (fits to p_T spectra): The multi-component Schwinger/Bose-Einstein distributions are stated to describe the data effectively, yet the manuscript provides no justification for the number of components chosen, no comparison to single-component or alternative forms (e.g., Tsallis, power-law), and no reported uncertainties or error bars on the extracted κ and T values. These omissions prevent assessment of whether the observed rapidity trends are robust or unique to the chosen parametrization.
  3. [Discussion] Discussion of radius extraction: The average minimum strong-force radius is determined directly from the fitted κ without demonstrating that the extracted value is independent of the number of fit components, the rapidity binning, or the choice between Schwinger and Bose-Einstein forms. This step inherits the same unanchored interpretation of T and lacks a cross-check against other radius estimates in the literature.
minor comments (2)
  1. [Methods] Clarify the precise functional form of the multi-component distribution (explicit equations for the Schwinger and Bose-Einstein cases) and state whether the same component weights are used across all rapidity intervals.
  2. [Results] Add a table or figure panel showing the χ²/dof and parameter uncertainties for each fit to allow quantitative evaluation of fit quality.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each of the major comments in detail below, indicating the revisions we plan to implement to improve the clarity and robustness of our results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The identification of the fitted T with the initial effective temperature rests on the statement that geometric asymmetry and local partonic thermalization induce radial and transverse flows that raise T even without QGP. No hydrodynamic or blast-wave estimate of the expected ΔT, no comparison to independent initial-temperature proxies (particle ratios, femtoscopy), and no sensitivity test to fit choices are supplied; this renders the reported increase and proportionality of κ and T, as well as the derived minimum strong-force radius, dependent on an untested mapping.

    Authors: The interpretation of the extracted T as the initial effective temperature is based on the established physical mechanisms in small collision systems, where geometric asymmetry and local partonic thermalization can generate flow effects even in the absence of QGP. The observed direct proportionality between κ and T across different rapidity intervals provides internal consistency for this mapping. We agree that a sensitivity analysis to fit choices would enhance confidence in the trends. We will add such tests in the revised manuscript. However, performing dedicated hydrodynamic or blast-wave calculations to estimate ΔT quantitatively, or comparisons to other proxies, would require a separate comprehensive study and is beyond the current scope. revision: partial

  2. Referee: [Results] Results section (fits to p_T spectra): The multi-component Schwinger/Bose-Einstein distributions are stated to describe the data effectively, yet the manuscript provides no justification for the number of components chosen, no comparison to single-component or alternative forms (e.g., Tsallis, power-law), and no reported uncertainties or error bars on the extracted κ and T values. These omissions prevent assessment of whether the observed rapidity trends are robust or unique to the chosen parametrization.

    Authors: We will revise the manuscript to include a justification for selecting the multi-component approach, explaining that it better captures the spectral shape over a wide p_T range compared to single-component fits. Comparisons to alternative parametrizations such as Tsallis distributions and power-law forms will be added to demonstrate the effectiveness. Furthermore, we will report the uncertainties and error bars on the extracted κ and T values from the fits. revision: yes

  3. Referee: [Discussion] Discussion of radius extraction: The average minimum strong-force radius is determined directly from the fitted κ without demonstrating that the extracted value is independent of the number of fit components, the rapidity binning, or the choice between Schwinger and Bose-Einstein forms. This step inherits the same unanchored interpretation of T and lacks a cross-check against other radius estimates in the literature.

    Authors: To address this, we will include additional figures or tables in the revised version showing the derived radius for varying numbers of components and different rapidity selections to confirm stability. We will also provide a comparison of our extracted average minimum strong-force radius with values from other theoretical approaches in the literature, such as those from potential models or lattice calculations. revision: yes

standing simulated objections not resolved
  • Quantitative estimates using hydrodynamic or blast-wave models for the expected temperature increase due to flows, as this would entail new simulations not part of the present phenomenological analysis.

Circularity Check

0 steps flagged

No significant circularity; parameters extracted by fit and trends reported empirically

full rationale

The paper obtains κ and T by fitting the chosen multi-component distributions to the LHCb p_T spectra in rapidity bins. The statement that these forms 'can effectively describe' the spectra follows from the quality of those fits and does not constitute a prediction of an independent observable. The observed rise of both parameters with decreasing rapidity and the reported proportionality are direct read-outs from the same fit results. The additional claim that the fitted T equals the initial effective temperature rests on a qualitative argument about geometric asymmetry and local partonic thermalization; no equation equates the fitted value to an initial value by construction, and no self-citation chain is invoked to justify the mapping. The derivation therefore remains self-contained against the external experimental spectra.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The central claim rests on two fitted effective parameters and three domain assumptions about model applicability in small systems without QGP.

free parameters (2)
  • κ
    Effective string tension in Schwinger mechanism, fitted to p_T spectra in different rapidity intervals.
  • T
    Effective temperature in Bose-Einstein statistics, fitted to the same spectra and interpreted as initial temperature.
axioms (3)
  • domain assumption Schwinger mechanism applies to heavy quarkonia production in p+p collisions
    Invoked to extract κ from the spectra.
  • domain assumption Bose-Einstein statistics describes the p_T spectra of heavy quarkonia
    Invoked to extract T from the spectra.
  • domain assumption Absence of QGP formation in p+p collisions leaves T largely unaffected by QGP-related effects
    Stated explicitly to justify interpreting T as initial temperature.

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