Event multiplicity, transverse momentum and energy dependence of charged particle production, and system thermodynamics in $pp$ collisions at the Large Hadron Collider

Arvind Khuntia; Jean Cleymans; Raghunath Sahoo; Rutuparna Rath

REVIEW 3 major objections 4 minor 28 references

Event multiplicity, transverse momentum and energy dependence of charged particle production, and system thermodynamics in $pp$ collisions at the Large Hadron Collider

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper reports that a thermodynamically consistent Tsallis distribution describes the full transverse-momentum spectra of charged particles in proton-proton collisions at 5.02 and 13 TeV, while a blast-wave model captures only the…

desk verdict Competent Tsallis and BGBW fits to new ALICE pp data, but the thermodynamic trends are only as solid as the arbitrary fit range and the missing error bars. read the letter →

arxiv 1908.04208 v2 pith:ZVQFGGDU submitted 2019-08-12 hep-ph hep-exnucl-exnucl-th

Rutuparna Rath , Arvind Khuntia , Raghunath Sahoo , Jean Cleymans This is my paper · ORCID

classification hep-phhep-exnucl-exnucl-th

keywords Tsallisnon-extensivestatisticstransversemomentumspectrappcollisionsLHCkineticfreeze-outtemperatureradialflowblast-wavemodelchargedparticlemultiplicity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Using measured charged-particle transverse-momentum ($p_{T}$) spectra in $pp$ collisions at $\sqrt{s}=5.02$ and 13 TeV, this paper asks which statistical description governs the produced hadronic system: a non-extensive Tsallis distribution or a Boltzmann-Gibbs blast-wave model. It reports that the thermodynamically consistent Tsallis form fits the complete $p_{T}$ spectra up to 20 GeV/$c$, whereas the blast-wave model covers only the bulk region below about 2.5 GeV/$c$. The fitted non-extensive parameter $q$ rises with event multiplicity and saturates above $\mathrm{d}N_{\rm ch}/\mathrm{d}\eta \sim 15$, with larger values at 13 TeV; the fitted temperature also grows with multiplicity. The blast-wave fits show radial flow that is essentially independent of collision energy and multiplicity, alongside a kinetic freeze-out temperature that clearly increases with multiplicity. This matters because it separates genuine freeze-out trends in small collision systems from fitting-function artifacts, and it identifies empirical thresholds ($p_{T}\sim 8$ GeV/$c$; $\mathrm{d}N_{\rm ch}/\mathrm{d}\eta\sim15$) in particle production.

What carries the argument

The load-bearing object is the thermodynamically consistent Tsallis distribution, Eq. (11): a two-parameter $(T,q)$ invariant yield for charged particles in which the power-law exponent carries an extra factor of $q$ so that entropy, energy, pressure, and particle number satisfy standard thermodynamic relations. Fitting this form to the $p_{T}$ spectra yields the Tsallis temperature $T$ and the non-extensive parameter $q$, interpreted as kinetic freeze-out temperature and distance from equilibrium. The comparison mechanism is the BGBW model, Eq. (14), using modified Bessel functions $K_{1}$ and $I_{0}$ with a linear flow profile, which gives the average radial flow $\langle\beta\rangle$ and $T_{\rm kin}$ from the bulk $p_{T}$ region. The paper's central systematic tool is a scan of the upper $p_{T}$ fitting boundary from 1.2 to 20 GeV/$c$, showing opposite $q$ and $T$ trends for the highest and lowest multiplicity classes and a saturation threshold near 8 GeV/$c$.

What would settle it

Re-fit the $pp$ spectra at $\sqrt{s}=5.02$ and 13 TeV with a fixed upper boundary, say $p_{T,\max}=3$ GeV/$c$, for every multiplicity class; if $q$ no longer saturates above $\mathrm{d}N_{\rm ch}/\mathrm{d}\eta\sim15$ or $T_{\rm kin}$ no longer rises with multiplicity, the reported trends are consequences of the changing fitting range rather than properties of the collision system.

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Extended reading notes

Core claim

The paper's central claim is that the charged-particle $p_{T}$ spectra in $pp$ collisions at $\sqrt{s}=5.02$ and 13 TeV separate into two regimes: a bulk component below $\sim 2.5$ GeV/$c$ captured by the Boltzmann-Gibbs blast-wave model, and a full-range component captured by Tsallis non-extensive statistics. Using the thermodynamically consistent Tsallis form (with an extra power of $q$ in the exponent), the fitted $q$ rises from about 1.12 to 1.18 with multiplicity and saturates above $\mathrm{d}N_{\rm ch}/\mathrm{d}\eta \sim 15$, while the fitted temperature $T$ rises into the $\sim 0.1$ GeV range and is higher at 13 TeV. The BGBW fits yield a radial-flow velocity $\langle\beta\rangle$ that is essentially independent of collision energy and multiplicity, and a kinetic freeze-out temperature $T_{\rm kin}$ that increases with multiplicity. The paper further claims that $p_{T}\sim 8$ GeV/$c$ is a threshold: above it, $q$, $T$ and $\chi^{2}/{\rm ndf}$ saturate, and high-multiplicity fits worsen when the upper fit boundary exceeds $\sim 2$ GeV/$c$.

Load-bearing premise

The results stand or fall on the choice of the thermodynamically consistent Tsallis formula as the correct statistics for the system: if that functional form is wrong, the fitted temperatures and $q$-values are properties of the fitting function, not of the collisions.

Editorial extensions

If this is right

At both collision energies, the Tsallis distribution tracks the measured charged-particle spectra over the full measured $p_{T}$ range in all multiplicity classes, while the BGBW model underpredicts the low-$p_{T}$ region below about 0.3 GeV/$c$, attributed to resonance decays.
The non-extensive parameter $q$ and the Tsallis temperature $T$ both increase with charged-particle multiplicity, with $q$ saturating above $\mathrm{d}N_{\rm ch}/\mathrm{d}\eta\sim15$ and taking higher values at 13 TeV than at 5.02 TeV.
The BGBW analysis gives a radial-flow velocity $\langle\beta\rangle$ that is almost constant across collision energy and multiplicity, and a kinetic freeze-out temperature $T_{\rm kin}$ that rises with multiplicity.
Fitting-range scans show that $q$ and $T$ move in opposite directions for the highest (V0M1) and lowest (V0M10) multiplicity classes as the upper $p_{T}$ cutoff grows, with $p_{T}\sim8$ GeV/$c$ marking a saturation threshold; for V0M1, $\chi^{2}/{\rm ndf}$ degrades once the fit extends beyond about 2 GeV/$c$.
Together these results support a two-component picture: the bulk of the system behaves thermally with a multiplicity-dependent freeze-out temperature, while the high-$p_{T}$ tail requires a non-extensive power-law description.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Re-fitting the same spectra with an event-shape or jet-veto selection, as the paper itself suggests, would test whether the $q$ rise at high multiplicity comes from jet fragmentation rather than from genuine non-extensive equilibration.
The demonstrated sensitivity of $q$ and $T$ to the upper $p_{T}$ cutoff implies that freeze-out parameters extracted with different fitting windows should not be compared directly; a fixed, pre-registered window would make cross-energy and cross-experiment comparisons cleaner.
If the saturation threshold near $\mathrm{d}N_{\rm ch}/\mathrm{d}\eta\sim15$ is real, it gives a concrete target for multi-parton-interaction models and for future LHC Run 3 $pp$ measurements at higher multiplicities.
Applying the BGBW fit separately to identified pions, kaons, and protons would test whether the multiplicity-independent radial flow and multiplicity-dependent $T_{\rm kin}$ persist per species, rather than only in the charged-particle sum with fixed weight factors.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

For an internal take: this is a solid, conventional fit analysis, not a discovery. It extends established Tsallis and BGBW descriptions to the 2019 ALICE pp spectra at 5.02 and 13 TeV across ten V0M multiplicity classes, and it adds a systematic scan over the upper pT fit boundary. That fit-range study is the most useful part of the paper.

What is actually new: the application to these data, plus the explicit demonstration that q and T move with pT,max and move in opposite directions for high and low multiplicity. The fits are honestly presented with ratio panels and chi2/ndf, and the authors are transparent that the high-multiplicity Tsallis fit degrades above about 2 GeV/c. Credit is due for showing that rather than hiding it.

The soft spots are real but not fatal. First, the extracted parameters have no error bars anywhere, so the multiplicity and energy trends in Figs. 3, 4, 11, and 12 cannot be assessed statistically. Second, the headline values come from a single pT < 20 GeV/c choice for all classes, while the paper's own Fig. 5 shows that this is not a safe range for the high-multiplicity class. The 8 GeV/c threshold is promoted to a summary bullet even though the text says it needs a closer look. Third, the BGBW comparison fits a region where the model visibly underpredicts low-pT data, so claiming is multiplicity-independent without errors is too strong. On the thermodynamics: the q-dual form is cited to the authors' own papers, not re-derived, but this is a literature assumption rather than a hidden circular step.

Does the central argument hold? The descriptive claim does: Tsallis describes the full spectra, BGBW only the bulk part. The physical interpretation of q and T as freeze-out thermodynamics is conditional, not demonstrated. The paper is transparent enough that a careful reader can see the distinction.

This is for practitioners who do Tsallis fits of LHC small-system spectra. It is a useful reference on fitting-range behavior and parameter dependences, less useful for physics conclusions. I would send it to peer review; a decent referee can require uncertainty estimates, a fit-range robustness statement, and softer wording on the threshold and on the BGBW conclusions.

Referee Report

3 major / 4 minor

Summary. The paper fits the ALICE inclusive charged-particle transverse momentum spectra in pp collisions at sqrt(s) = 5.02 and 13 TeV, in ten V0M multiplicity classes, using two functional forms: a 'thermodynamically consistent' Tsallis distribution (Sec. II A) and the Boltzmann-Gibbs Blast Wave (BGBW) model (Sec. II B). From the Tsallis fits the authors extract the non-extensive parameter q and the Tsallis temperature T as functions of charged-particle multiplicity and collision energy, and they carry out a fit-range scan for the highest and lowest multiplicity classes. From the BGBW fits to the bulk region (pT up to about 2.5 GeV/c) they extract the average radial flow <beta> and the kinetic freeze-out temperature Tkin. The main claims are that the Tsallis form describes the complete pT range better than BGBW, that q increases with multiplicity and saturates above dNch/deta around 15, that q is larger at 13 TeV than at 5.02 TeV, that T increases with multiplicity, that <beta> is almost independent of multiplicity and collision energy, and that Tkin depends clearly on multiplicity class.

Significance. If the claimed trends were quantitatively established, this would be a useful phenomenological map of freeze-out parameters across multiplicity classes in small systems, directly relevant to current discussions of collectivity and equilibration in high-multiplicity pp collisions. The paper has real strengths: it uses recent ALICE data, shows data/fit ratios in the lower panels, reports chi2/ndf values, and includes a fit-range study that is often missing from such analyses. Its main weaknesses are that the extracted parameters are presented without uncertainties and that the paper's own fit-range scan shows strong variation of q and T with the upper pT cutoff for the high-multiplicity class. The transition from 'we observe a trend' to 'the data demonstrate a property of the collision system' therefore requires error propagation and a defended choice of fit range. The paper does not provide code, parameter tables with uncertainties, or independent validation of the thermodynamic form, which limits reproducibility.

major comments (3)

[Sec. III A, Figs. 3-7] The central multiplicity and energy trends of q and T are read from fits performed with a single upper limit of pT,max = 20 GeV/c for all classes, while the paper's own fit-range scan shows that for the high-multiplicity V0M1 class the extracted q decreases and T increases as the upper boundary is extended (Figs. 6 and 7), and chi2/ndf grows strongly above roughly pT = 2 GeV/c (Fig. 5). Because Figs. 3 and 4 show no parameter uncertainties, the reported ordering of q and T across multiplicity classes could be an artifact of the chosen fit range rather than a property of the collision system. Please report fit uncertainties and either use a common fit range in which the chi2/ndf is acceptable for all classes or demonstrate explicitly that the trends in Figs. 3 and 4 persist when the same physically motivated pT,max is used consistently.
[Sec. II A, Eqs. (1)-(6) and (11)] The thermodynamic interpretation of q and T rests on the 'addition power of q' in Eq. (1), which is imported from the authors' own earlier papers (refs. [3,4,6]) rather than derived or independently verified here. If that form is not the correct thermostatistical normalization for the system, the fitted q and T are only shape parameters of an ad-hoc function and the abstract's thermodynamic claims do not follow. As a concrete test, please provide the derivation of Eq. (6) or a precise citation to a derivation, and perform a robustness check by fitting the same data with the standard Tsallis form (without the extra power of q) and showing whether the q and T trends and the fit-range behavior survive.
[Sec. III B, Figs. 10-12] The BGBW conclusions that <beta> is almost independent of multiplicity and energy while Tkin depends clearly on multiplicity are stated without parameter uncertainties, so 'almost independent' cannot be distinguished from 'consistent within large errors'. In addition, the bottom panels of Figs. 8 and 9 show a systematic low-pT deviation, and the interpretation that this is due to resonance decays is not demonstrated. Please report the fit uncertainties for beta_s, <beta>, and Tkin, provide per-class chi2/ndf values, and state whether the observed Tkin variation is statistically significant relative to those uncertainties.

minor comments (4)

[Sec. II A, after Eq. (2)] The phrase 'an addition power of q' should read 'an additional power of q'.
[Sec. III A, Fig. 5] The ordinate label chi2/NDF is not defined; stating whether NDF is the number of data points minus the number of fit parameters would make the goodness-of-fit values interpretable.
[Sec. IV, summary bullets] The bullet 'pT ~ 8 GeV/c ... needs a closure look' contains a typo ('closure' should be 'closer') and the claimed threshold at 8 GeV/c is not supported by a quantitative test of, for example, a change in slope or saturation.
[Sec. II A, Eq. (8)] The pion, kaon, and proton weights (0.8, 0.12, 0.08) are stated to follow experimental yields, but no uncertainties or reference values are given; a short table of weights and the result of the pion-fraction sensitivity test would make this statement more concrete.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the paper's claims are fit extractions against ALICE data, and the self-cited Tsallis form is a modeling ansatz rather than a prediction derived from the data.

full rationale

The paper makes no first-principles claim: it fits ALICE pT spectra with the Tsallis form and the BGBW model, and the abstract and summary describe q, T, <beta>, and Tkin as extracted ("extract", "observed"), not predicted. The thermodynamic interpretation rests on the q-weighted Tsallis form, whose consistency is supported by refs. [3,4,6]; these are self-citations, but they provide a mathematical normalization, and the empirical fit quality and parameter trends are judged against the external ALICE data, so the central fit results do not reduce to the citations. The paper's own fit-range scan (Figs. 5-7) shows that q and T depend on the upper pT cutoff and that chi2/ndf degrades for the high-multiplicity class, which is a robustness limitation on the interpretation of T and q as physical freeze-out parameters, not a circularity. No fitted parameter is renamed as an independent prediction, and the observed T-q correlation is a property of the fit, not a prediction claimed from the ansatz. The self-citations to prior Tsallis analyses are relevant background, not an unverified uniqueness theorem, so the score is low rather than high. A higher circularity score would require the paper to claim it derives the Tsallis thermodynamics from data or to predict a quantity that is identical to a fitted parameter; neither occurs here.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The observable content of the paper is entirely in parameters fitted with two models: q and T for Tsallis, T_kin and beta_s for BGBW, plus normalizations, with the particle composition weights fixed by hand. The thermodynamic meaning of these parameters rests on the self-cited Tsallis form and standard blast-wave assumptions. No new entities are introduced and no falsifiable handle outside the fitted data is provided.

free parameters (6)

q, Tsallis non-extensive parameter = roughly 1.12 to 1.18 depending on energy and multiplicity (Figs. 3 and 6)
Fitted per spectrum at both energies and all ten V0M classes; it is the paper's measure of departure from equilibrium and one of its two headline observables.
T, Tsallis temperature = roughly 0.04 to 0.14 GeV depending on multiplicity (Figs. 4 and 7)
Fitted together with q and the normalization A; interpreted as kinetic freeze-out temperature.
A, normalization of the Tsallis yield = not tabulated; fitted per spectrum
Replaces the freeze-out volume via Eq. (7) using the measured dN/dy, so the fit has one more adjustable parameter per spectrum.
T_kin, BGBW kinetic freeze-out temperature = roughly 0.03 to 0.10 GeV depending on multiplicity (Fig. 12)
Fitted with the BGBW model over the bulk pT range up to 2.5 GeV/c.
beta_s, BGBW maximum surface velocity (with n = 1 fixed) = reported through <beta> around 0.4 to 0.9 (Fig. 11)
Fitted per class; the average flow is derived from beta_s via Eq. (15) with the linear profile n = 1 fixed by hand.
Particle weights w_pi = 0.8, w_K = 0.12, w_p = 0.08 = fixed constants
Chosen from experimental identified-particle yields (ref [22]) and assumed pT-independent; only the pion fraction was varied in a limited sensitivity test.

assumptions (5)

domain assumption The thermodynamically consistent Tsallis distribution requires an additional power of q in the distribution function (Eqs. (1)-(3))
Adopted from the authors' own papers (refs [3,4,6]); the central modeling premise that makes fitted T and q the paper's thermodynamic observables.
domain assumption Chemical potential mu is zero at LHC energies
Invoked to reduce Eq. (5) to Eq. (6); standard at LHC but not verified here.
domain assumption Charged particle spectra are dominated by pions, kaons and protons with fixed, pT-independent weights
Used in Eqs. (8) and (11); the composition actually changes with pT and multiplicity, and only the pion fraction was varied in the sensitivity check.
domain assumption Bjorken correlation y = eta holds for the BGBW integration
Standard assumption from ref [24], invoked before Eq. (14).
domain assumption Linear radial flow profile n = 1 in BGBW
Fixed by hand rather than fitted; with Eq. (15) it sets the conversion from beta_s to average flow.

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Cite this review

Pith. "Pith review of Event multiplicity, transverse momentum and energy dependence of charged particle production, and system thermodynamics in $pp$ collisions at the Large Hadron Collider." pith.science (2026). https://pith.science/paper/ZVQFGGDU

@misc{pith2026190804208,
  author       = {Pith},
  title        = {Pith review of: Event multiplicity, transverse momentum and energy dependence of charged particle production, and system thermodynamics in $pp$ collisions at the Large Hadron Collider},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZVQFGGDU}},
  note         = {Machine review of arXiv:1908.04208}
}

abstract

In the present work, we study the recent collision energy and multiplicity dependence of the charged particle transverse momentum spectra as measured by the ALICE collaboration in $pp$ collisions at $\sqrt{s}$ = 5.02 and 13 TeV using the non-extensive Tsallis distribution and the Boltzmann-Gibbs Blast Wave (BGBW) model. A thermodynamically consistent form of the Tsallis distribution is used to extract the kinetic freeze-out parameters from the transverse momentum spectra of charged particles at mid-rapidity. In addition, a comprehensive study of fitting range dependence of transverse momentum spectra on the freeze-out parameters is done using Tsallis statistics. The applicability of BGBW model is verified by fitting the transverse momentum spectra of the bulk part ($\sim 2.5~ {\rm GeV}/c$)for both 5.02 and 13 TeV energies and also in different multiplicity classes. The radial flow, $<\beta>$ is almost independent of collision energy and multiplicity whereas the behavior of kinetic freeze-out temperature significantly depends on multiplicity classes. It is found that the Tsallis distribution generally leads to a better description for the complete transverse momentum spectra whereas the BGBW model explains the bulk part of the system.

Figures

Figures reproduced from arXiv: 1908.04208 by the authors.

**Figure 3.** FIG. 3: (Color online) Non-extensive parameter as a function [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** FIG. 4: (Color online) Tsallis temperature as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (5 more)

**Figure 8.** Figure 8: FIG. 8: (Color online) Charged particle spectra fit with [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

**Figure 9.** Figure 9: FIG. 9: (Color online) Charged particle spectra fit with [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]

**Figure 10.** Figure 10: FIG. 10: (Color online) The values of [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]

**Figure 11.** Figure 11: FIG. 11: (Color online) Radial flow velocity parameter as a [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]

**Figure 12.** Figure 12: FIG. 12: (Color online) Temperature as a function of charged [PITH_FULL_IMAGE:figures/full_fig_p007_12.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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