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Thermodynamic Analysis of Transverse Momentum Spectra in Pb-Pb Collisions at 2.76 TeV: Centrality Dependence of Temperature, Freezeout Parameters and Non-Extensitivity

T0 review · 5 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Individual Tsallis fits to pion, kaon, and proton spectra in Pb-Pb collisions at 2.76 TeV yield mass-dependent freezeout parameters supporting multiple and volume-differential kinetic freezeout.

desk verdict Plausible Tsallis fits and a useful cross-system temperature/q compilation, but the freezeout decomposition is unvalidated and the derived parameters carry a series of fixable errors. read the letter →

arxiv 2507.07369 v1 pith:YJGB27FG submitted 2025-07-10 hep-ph nucl-th

M. Waqas , Hassan Ali Khan , Wolfgang Bietenholz , Muhammad Ajaz , Jihane Ben Slimane , Haifa I. Alrebdi , A. Haj Ismail This is my paper
classification hep-phnucl-th
keywords Quark-GluonPlasmaTsallisdistributioneffectivetemperaturenon-extensivityparameterkineticfreezeoutvolumetransverseflowvelocityPb-Pbcollisions
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

This paper asks whether the three most abundant positively charged hadrons—pions, kaons, and protons—freeze out together when the fireball created in Pb-Pb collisions at 2.76 TeV cools. Fitting each species' transverse momentum spectrum separately with a thermodynamically consistent Tsallis distribution, the authors extract the effective temperature $T_{\rm eff}$, the non-extensivity parameter $q$, and the kinetic freezeout volume $V$, and from those derive the mean kinetic freezeout temperature $\langle T_0\rangle$, the mean transverse flow velocity $\langle\beta_T\rangle$, the thermal temperature $T_{\rm th}$, and a multiplicity-fluctuation parameter $\zeta$. They find that heavier hadrons come out with higher $T_{\rm eff}$, smaller $q$, and smaller $V$, which they read as evidence that protons decouple from the fireball earlier and from a smaller freezeout surface than pions (multiple kinetic freezeout and volume-differential freezeout). They also find that all extracted quantities rise from peripheral to central collisions and with charged-particle multiplicity, except $q$, which falls, and that this pattern distinguishes large collision systems from small ones across beam energies.

What carries the argument

The engine is the thermodynamically consistent Tsallis distribution, Eq. (3): $$$d^{2}$N/(dp_T\,dy) = gV\,p_T m_T\$\cosh$ y/(2\pi)^2\left[1+(q-1)(m_T\$\cosh$ y-\mu)/T\right]^{q/(1-q)},$$ which converts each measured $p_T$ spectrum into fitted values of $T$, $q$, and $V$. To separate collective flow from temperature the paper uses the linear relation $T = T_0 + m\langle u_t\rangle^2$ (Eq. 7), where $u_t$ is the strength of the mean radial transverse flow, with $\langle\beta_T\rangle = \langle u_t\rangle/\sqrt{1+\langle u_t\rangle^2}$, and it computes the fluctuation parameter from $T = E/[\zeta^2 - (q-1)]$ (Eq. 6). The mass dependence of $T_{\rm eff}$, $q$, and $V$ is the observable that carries the freezeout-scenario conclusion.

What would settle it

Re-run the individual Tsallis fits for all three species over one identical $p_T$ window and compare them with a simultaneous blast-wave fit that fixes a common freezeout temperature and flow velocity; if the simultaneous fit matches the data as well as the individual fits do, the mass-dependent $T_{\rm eff}$, $q$, and $V$ do not by themselves establish multiple or volume-differential freezeout.

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Extended reading notes

Core claim

On the paper's own terms, each hadron species has its own freezeout moment and its own freezeout surface. Individual fits of the thermodynamically consistent Tsallis distribution to the measured $\pi^+$, $K^+$, and $p$ spectra give an effective temperature that increases with hadron mass (protons highest, pions lowest), a non-extensivity parameter $q$ that decreases with mass and lies closest to 1 for protons, and a kinetic freezeout volume $V$ that decreases with mass (pions largest, protons smallest). The paper interprets the mass ordering of $T_{\rm eff}$ and $V$ as support for multiple kinetic freezeout and volume-differential freezeout. It further reports that $T_{\rm eff}$, $V$, $\langle T_0\rangle$, $\langle\beta_T\rangle$, $T_{\rm th}$, and $\zeta$ all increase from peripheral to central collisions and with $\langle dN_{\rm ch}/d\eta\rangle$, while $q$ decreases, and that averaged $\langle T_{\rm eff}\rangle$ and $\langle q\rangle$ across collision systems show a systematic ordering by system size and energy. The authors note that global simultaneous fits of the same spectra (Refs. [24, 28, 30, 64]) produce the opposite centrality trend for $\langle T_0\rangle$, and they attribute the difference to the individual-fit method being able to test, rather than presume, a single freezeout scenario.

Load-bearing premise

The load-bearing premise is the linear formula $T = T_0 + m\langle u_t\rangle^2$ (Eq. 7), used with only three particle masses to split each fitted effective temperature into a common freezeout temperature plus a mass-proportional flow term; if that formula is not exact, the extracted $\langle T_0\rangle$ and $\langle\beta_T\rangle$ and the multiple-freezeout conclusion do not follow, and the companion $\zeta$ results additionally require an energy scale $E$ (Eq. 6) that the paper never defines.

Editorial extensions

If this is right

  • If the paper is right, a single common freezeout surface for all hadrons is ruled out at this collision energy: protons leave earlier and from a smaller volume than kaons, which leave earlier and from a smaller volume than pions.
  • The centrality and multiplicity trends imply that head-on collisions produce a hotter, larger, longer-lived fireball that stays closer to equilibrium, since $q$ approaches 1 as $\langle dN_{\rm ch}/d\eta\rangle$ grows.
  • The anti-correlation between $\langle T_{\rm eff}\rangle$ and $\langle q\rangle$ across systems would mean that thermalization in heavy-ion collisions is governed mainly by system size (participant number) and energy density, with small systems remaining non-thermal.
  • A volume-differential freezeout implies that direct measurements of the emitting source size should see the same ordering as the fitted volumes: pion emission regions larger than kaon regions, and kaon regions larger than proton regions.

Reading between the lines

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

  • Extending the paper's individual-fit logic to strange baryons such as $\Lambda$, $\Xi$, and $\Omega$ would test whether the mass ordering in $T_{\rm eff}$, $q$, and $V$ continues smoothly or saturates, giving a cleaner discriminator between mass-driven and quark-content-driven freezeout hierarchies.
  • Because the energy scale $E$ in Eq. (6) is never defined, the $\zeta$ results are not directly testable; fixing $E$ (for instance to the average transverse mass or total fireball energy per particle) would let experiment verify whether the claimed $\zeta$ centrality ordering matches measured event-by-event multiplicity fluctuations.
  • The turnaround of the proton's $q$ in the most peripheral bins suggests baryon stopping or projectile fragmentation; a baryon-number-conserving model that reproduces this non-monotonicity would be a stronger test of the freezeout picture than the mass-ordering argument alone.
  • Comparing individual and global fits on strictly identical $p_T$ windows would decide whether the two methods' opposite $T_0$ trends reflect physics or fitting convention, making that comparison a direct experimental extension of the paper.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The manuscript fits the ALICE transverse-momentum spectra of π+, K+, and p in ten centrality bins of Pb-Pb collisions at 2.76 TeV to the thermodynamically consistent Tsallis distribution (Eq. (3)), extracting per-species effective temperature T, non-extensivity parameter q, and kinetic freezeout volume V. It then uses Eqs. (7), (10), and (6) to derive the mean kinetic freezeout temperature ⟨T0⟩, mean transverse flow velocity ⟨βT⟩, thermal temperature Tth, and the fluctuation parameter ζ. The paper further compiles ⟨Teff⟩ and ⟨q⟩ for several collision systems and energies at RHIC and LHC. The central claim is that the mass ordering of Teff, q, and V supports multiple kinetic freezeout and volume differential freezeout, with all thermal parameters increasing from peripheral to central collisions while q decreases.

Significance. If the central claims were substantiated, the paper would provide a systematic centrality-dependent freezeout study and a useful cross-system survey of Tsallis-fit parameters. The analysis uses public ALICE data, displays fits with data/fit ratios, and makes the extracted parameters available in Table 1, which are useful elements. However, the main interpretive conclusion is not currently supported: the freezeout decomposition is not tested against a common-freezeout null model, and several load-bearing formulas are either algebraically inconsistent, dimensionally inconsistent, or depend on an undefined energy scale. The paper therefore has useful empirical content, but the advertised conclusion about multiple and volume differential freezeout requires substantial additional work.

major comments (5)
  1. [Table 1] Table 1 lists identical χ2/dof values for K+ and p in every centrality bin (e.g., 60/33, 92/33, 92/33, 84/33, ...). Since these are independent fits to different particle spectra with different numbers of degrees of freedom, identical values are impossible in genuine fits. This indicates an error in the table or in the fits themselves, and because Table 1 is the evidential basis for all extracted parameters, the table must be corrected and verified.
  2. [Section 3.1, Eq. (7)] The linear decomposition T = T0 + m⟨ut⟩² is used to extract ⟨T0⟩ and ⟨βT⟩ from the fitted temperatures of the three species. A weighted least-squares fit of Eq. (7) to the quoted 0–5% values (0.110±0.004, 0.170±0.005, 0.370±0.005 GeV) gives χ2/dof of order 100 for one degree of freedom. The quoted uncertainties are therefore irreconcilable with this ansatz, and the values of ⟨T0⟩, ⟨βT⟩, and Tth shown in Figures 3 and 4 are unsupported as they stand.
  3. [Section 2, Eqs. (4) and (6); Figure 7] The energy scale E in Eq. (6) is never defined, and Eq. (6) is not derivable from Eq. (4). Substituting ζ² = ΔN²/⟨N⟩² into Eq. (4) gives q = 1 − 1/⟨N⟩ + ζ², hence 1/⟨N⟩ = ζ² − (q−1) and T = E(ζ² − (q−1)), not T = E/(ζ² − (q−1)) as printed. Without a defined E and a consistent derivation, the parameter ζ displayed in Figure 7 cannot be interpreted as a physical multiplicity-fluctuation parameter.
  4. [Abstract; Section 3.1] The claim that the mass dependence of Teff, q, and V supports multiple kinetic freezeout and volume differential freezeout is not tested against a common-freezeout null model. Since T, q, and V are free for each species, differences between species are allowed by construction; no simultaneous fit enforcing a common T0 and flow, and no likelihood-ratio or χ2 comparison, is provided. The paper's own statement that global fits in Refs. [24,28,30,64] find the opposite T0 centrality trend underscores that the individual-fit differences are not self-interpreting. The accompanying sentence claiming that the trend does not change between individual and simultaneous fits is in direct tension with the disagreement just reported.
  5. [Section 3.1, Eq. (10)] The printed flow-correction formula Tth = T + sqrt((1+βT)/(1−βT)) is dimensionally inconsistent: it adds a quantity in GeV to a dimensionless ratio. If a multiplicative Doppler factor of the form T × sqrt((1+β)/(1−β)) was intended, the formula must be corrected and the Tth values recomputed; as printed, Eq. (10) cannot serve as the definition of the reported thermal temperature.
minor comments (4)
  1. [Title and text] The word 'Non-Extensitivity' in the title should be spelled 'Non-Extensivity'.
  2. [References] References [40] and [57] are duplicates of the same PHENIX paper (Phys. Rev. C 83, 064903 (2011), arXiv:1102.0753); one should be removed.
  3. [After Eq. (9)] The phrase 'N0 in 1 is the normalization constant' should refer to the relevant equation number, presumably Eq. (3).
  4. [Figure 2(b) discussion] The explanation for the proton q-dependence in the most peripheral bins in terms of baryon stopping and projectile fragmentation is speculative and should be marked as such rather than presented as an established interpretation.

Circularity Check

3 steps flagged · score 5.0 of 10

Central freezeout claim restates free per-species fit parameters; derived T0, beta_T, T_th, and zeta are closed functions of the fits with an undefined scale E, and the rebuttal of contradicting global fits relies on self-citations [67,68].

  1. fitted input called prediction [Abstract; Sec. 3.1 (Figure 2 discussion); Conclusions]
    "Heavy particles are more likely to freezeout quickly, because of their frequent interaction. Our results for the mass dependence of Teff reveal the multiple kinetic freezeout situation [70, 71]. ... V is largest for π+, followed by K+ and p. This enables a volume differential freezeout and shows that each particle has its own freezeout surface."

    In Eq. (3), T, q, and V are free parameters fitted independently per species, so any mass ordering of Teff and V is allowed by the fit construction. The central conclusion that the mass dependence of Teff and V 'supports' multiple kinetic freezeout and volume differential freezeout is read off from that same ordering: 'Our results for the mass dependence of Teff reveal the multiple kinetic freezeout situation.' No global fit enforcing a common freezeout temperature and flow is performed or shown to fail, so the scenario claim is never tested against a null model; the paper itself notes that global fits [24,28,30,64] yield the opposite T0 trend. The interpretive reading is anchored in the authors' own Refs. [70,71] rather than in a falsifiable cross-check.

  2. self definitional [Sec. 2, Eqs. (4)-(6); Sec. 3, Figure 7 discussion]
    "By referring to Eq. (4), we obtain T = E/[ζ² − (q − 1)]. ... The parameter ζ of Eq. (6) is shown in Figure 7 ... In the current work, it turns out that ζ informs us how strongly T and q − 1 are correlated."

    ζ is introduced via Eqs. (4)-(6) as a closed algebraic function of the fitted T and q: from T = E/(ζ² − (q−1)), ζ² = (q−1) + E/T. The energy scale E is never defined or assigned a value anywhere in the paper, so the plotted ζ values in Fig. 7 are not pinned down. The statements that 'ζ informs us how strongly T and q − 1 are correlated' and that ζ decreases from central to peripheral collisions restate the fitted T and q passed through the defining formula; no independent measurement of multiplicity fluctuations (Eq. 5) is made. The fluctuation interpretation of ζ is assumed from [18,45], not validated against the data, so the 'extracted' ζ and its claimed trends carry no information beyond the fit inputs.

1 more flagged steps
  1. self citation load bearing [Sec. 3.1, paragraph following Figure 3 discussion]
    "Our results disagree with [24, 28, 30, 64], where T0 is larger in peripheral collisions. However, Refs. [65,66,69] are in agreement with our work ... However, the trend of the parameters does not change whether the fit is individual or simultaneous, as shown in our previous studies [67, 68]."

    The paper's centrality trend of T0 (and βT) is asserted against four external global-fit analyses [24,28,30,64] that find the opposite trend. The only argument given for preferring the present result is a self-citation: 'the trend of the parameters does not change whether the fit is individual or simultaneous, as shown in our previous studies [67, 68]', where [67,68] are authored by members of the present group and use the same individual-fit method. The cited agreement is likewise partly self-referential ([69] is the authors' own work). The load-bearing justification for dismissing the contradicting independent global fits is therefore a chain of the authors' own analyses, not an independent demonstration against the quoted global fits.

full rationale

The three species' Tsallis fits to ALICE data are genuine extractions — T, q, V are data-grounded, which keeps the paper far from total circularity (not 8-10). The circularity is partial, concentrated in the interpretive layer. (1) The headline claim that the mass dependence of Teff, q, V supports multiple and volume-differential freezeout is the free per-species fit output restated as a conclusion: independent fits allow any mass dependence by construction, and the paper never fits or rejects a common-freezeout model, explicitly conceding the opposite central-peripheral T0 trend in global fits [24,28,30,64]. (2) All derived quantities are algebraic functions of the fitted T and q: ⟨T0⟩ and ⟨βT⟩ via the assumed linear ansatz Eq. (7) through only three masses; Tth via the dimensionally inconsistent Eq. (10); and ζ via Eq. (6), which depends on an energy scale E that is never defined — so Fig. 7 is an unfalsifiable restatement. (3) The dismissal of the contradicting global-fit results leans on self-citations [67,68] for the claim that the trend is fit-method independent. Additional internal-consistency problems (Table 1 lists identical χ²/dof for K+ and p in every centrality bin, which cannot occur for genuinely independent fits, indicating a transcription artifact) further undermine reliance on the fit outputs as evidence. Net: the fitted parameters are real, but the paper's central scenario conclusions substantially reduce to those parameters plus a self-citation-anchored interpretation; score 5.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The ledger is dominated by the fit parameters themselves (T, q, V per species and centrality), plus an assumed linear flow decomposition and an unstated energy scale for zeta. No genuinely new physical entities are postulated beyond the zeta re-parameterization, which has no independent evidence. The paper's interpretive claims, multiple freezeout, volume differential freezeout, and zeta as a fluctuation measure, rest on assumptions not benchmarked against any independent observable.

free parameters (5)
  • T, q, V per species and centrality (30 values each) = Table 1: T from 0.071 to 0.370 GeV, q from 1.021 to 1.171, V from 600 to 2380 fm^3
    These are the free parameters of the Tsallis fit (Eq. 3) to ALICE spectra; every conclusion in the paper is a trend of these fitted quantities.
  • N0 normalization constant = 6 to 12500 (Table 1)
    An additional per-fit normalization reported alongside V, even though V already sets the overall scale in Eq. (3); the redundancy is left unexplained.
  • E (energy scale) in Eq. (6) for zeta = not stated
    Required to convert fitted T and q into zeta^2 = (q - 1) + T/E; never defined or tabulated, so the zeta curves in Figure 7 cannot be reproduced.
  • u_t and T0 from the T-versus-mass line (Eq. 7) = <beta_T> about 0.20 to 0.32 and <T0> about 0.056 to 0.096 GeV (Figure 3)
    Two parameters per centrality obtained from a three-point linear fit through the fitted temperatures of pions, kaons, and protons; the linear mass dependence is the assumed flow model, not a measured relation.
  • The factor appended to sqrt((1 + beta_T)/(1 - beta_T)) in Eq. (10) = not stated
    The T_th formula is printed as T_th = T + s sqrt(...) with s never defined; as written, the term added to T is dimensionless, making the formula dimensionally inconsistent.
assumptions (5)
  • domain assumption Mu = 0 (negligible chemical potential) at LHC energies
    Stated in Section 2 before Eq. (1); standard at LHC energies, yet an assumption about hadrochemical conditions.
  • domain assumption The Cleymans-Worku thermodynamically consistent Tsallis form (Eq. 3) is the correct description over the entire measured pT range
    The whole extraction rests on this functional form; the paper cites Ref. [34] but provides no independent check against alternative forms.
  • ad hoc to paper Linear flow decomposition T = T0 + m<u_t>^2 (Eq. 7) alone accounts for the mass dependence of the fitted temperatures
    Invoked in Section 3 to produce <T0> and <beta_T>; no validation from independent observables, and the paper notes global fits yield a different T0 trend.
  • ad hoc to paper Interpretation of q via multiplicity fluctuations, Eqs. (4)-(6) from Refs. [18, 45], with a constant relative fluctuation scale
    The basis for the zeta observable; imported from prior literature and combined with an unspecified energy scale E.
  • domain assumption ALICE data as published in Ref. [28], including its quoted spectra and errors
    All fits depend on these published data points and their stated errors; no independent data handling or cross-check is described.
invented entities (1)
  • zeta (multiplicity-fluctuation-driving parameter)
    purpose: Claims to characterize the fluctuating number of generated particles and to connect q with temperature fluctuations through Eq. (6)
    Its computation is never specified because the energy scale E is undefined, and per Eqs. (4)-(6) it is a deterministic function of the already fitted T and q, i.e., a re-parameterization of fit outputs rather than an independently measurable quantity.

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Cite this review

Pith. "Pith review of Thermodynamic Analysis of Transverse Momentum Spectra in Pb-Pb Collisions at 2.76 TeV: Centrality Dependence of Temperature, Freezeout Parameters and Non-Extensitivity." pith.science (2026). https://pith.science/paper/YJGB27FG

@misc{pith2026250707369,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic Analysis of Transverse Momentum Spectra in Pb-Pb Collisions at 2.76 TeV: Centrality Dependence of Temperature, Freezeout Parameters and Non-Extensitivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJGB27FG}},
  note         = {Machine review of arXiv:2507.07369}
}
abstract

We study properties of Pb-Pb collisions at 2.76 TeV in mid-rapidity, $|y|<0.5$, based on data by the ALICE Collaboration. In particular, we examine the transverse momentum ($p_T$) spectra of positively charged (identified) hadrons, $\pi^+$, $K^+$ and $p$, generated in various centrality intervals. We perform individual fits using the thermodynamically consistent Tsallis distribution to extract the following quantities: the non-extensitivity parameter, $q$, the effective temperature, $T_{\rm eff}$, the kinetic freezeout volume, $V$, the mean transverse flow velocity, $\beta_T$, the mean kinetic freezeout temperature, $\langle T_0\rangle$, the thermal temperature, $T_{\rm th}$, and the parameter $\zeta$, which characterizes the fluctuating number of generated particles. From peripheral to central collision, and from lower to higher charged particle multiplicity per pseudorapidity unit, $\langle dN_{\rm ch}/d\eta \rangle$, all these quantities are observed to increase, with the exception of $q$, which has the opposite behavior. The parameters $T_{\rm eff}$, $q$, and $V$ depend on the hadron mass in a way that supports the scenarios of volume differential freezeout and multiple kinetic freezeout. Furthermore, we extracted $\langle T_{\rm eff}\rangle$ and $\langle q\rangle$ for different collisions and energies at LHC and RHIC, and compare their dependencies on $\langle dN_{\rm ch}/d\eta \rangle$ and $\langle N_{\rm part} \rangle$.

Figures

Figures reproduced from arXiv: 2507.07369 by the authors.

Figure 1
Figure 1. The pT spectra of π +, K+, and p, generated in Pb-Pb collisions at √ sNN = 2.76 TeV and mid-rapidity, |y| < 0.5, with various centralities. The experimental data recorded by the ALICE Collaboration is represented by consistent symbols in each panel, which display distinct particle pT spectra dispersed in different centrality classes [28], cf [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Effects of centrality and the particle mass on (a) the effective temperature [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Effects of centrality and particle mass on (a) the mean kinetic freezeout temperature, [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The dependence of various quantities on the multiplicity of charged particles, [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: The results of ⟨Teff⟩ versus (a) ⟨dNch/dη⟩, and (b) ⟨Npart⟩. 0 250 500 750 1 000 1 250 1 500 1 750 1 . 02 1 . 05 1 . 08 1 . 1 1 1 . 1 4 1 . 1 7 1 . 20 1 . 23 Pb-Pb 2. 76 TeV Xe-Xe 5. 44 TeV Au -Au 200 GeV Au -Au 62. 4 GeV p-Pb 5. 0 TeV p-p 1 3 TeV p-p 7 TeV Lower Mu l …
Figure 6
Figure 6. Figure 6: The results of ⟨q⟩ versus (a) ⟨dNch/dη⟩, and (b) ⟨Npart⟩. 0 20 40 60 80 1 00 0. 24 0. 26 0. 28 0. 30 0. 32 0. 34 0. 36 0. 38 π + K + p P b-P b (a) 2. 76TeV ζ Cen tral i ty (%) 0 200 400 600 800 1 000 1 200 1 400 1 600 1 800 0. 24 0. 26 0. 28 0. 30 0. 32 0. 34 0. 36 0. …
Figure 7
Figure 7. Figure 7: The result for the parameter ζ as a function of (a) centrality, and (b) of ⟨dNch/dη⟩. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

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