[Paper Review] Possible Implication of a Single Nonextensive $p_T$ Distribution for Hadron Production in High-Energy $pp$ Collisions

Cheuk-Yin Wong, G. Wilk+2 more|OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information)|Dec 1, 2014

Statistical Mechanics and Entropy4 citations

TL;DR

This paper proposes that a single nonextensive Tsallis distribution with a nonextensivity parameter $ q $ can describe the full transverse momentum ($ p_T $) spectrum of hadrons in high-energy $ pp $ collisions at central rapidity, replacing the traditional two-component soft/hard model. By showing that the hard-scattering integral approximates a quasi-power-law form tied to nonextensive statistics, the authors demonstrate that the entire $ p_T $ range—spanning 14 orders of magnitude—can be fitted with one distribution, implying the dominance of hard scattering and a 'no-hair' statistical description at high energies.

ABSTRACT

Multiparticle production processes in $pp$ collisions at the central rapidity region are usually considered to be divided into independent "soft" and "hard" components. The first is described by exponential (thermal-like) transverse momentum spectra in the low-$p_T$ region with a scale parameter $T$ associated with the temperature of the hadronizing system. The second is governed by a power-like distributions of transverse momenta with power index $n$ at high-$p_T$ associated with the hard scattering between partons. We show that the hard-scattering integral can be approximated as a nonextensive distribution of a quasi-power-law containing a scale parameter $T$ and a power index $n=1/(q -1)$, where $q$ is the nonextensivity parameter. We demonstrate that the whole region of transverse momenta presently measurable at LHC experiments at central rapidity (in which the observed cross sections varies by $14$ orders of magnitude down to the low $p_T$ region) can be adequately described by a single nonextensive distribution. These results suggest the dominance of the hard-scattering hadron-production process and the approximate validity of a "no-hair" statistical-mechanical description of the $p_T$ spectra for the whole $p_T$ region at central rapidity for $pp$ collisions at high-energies.

Motivation & Objective

• To investigate whether a single nonextensive statistical distribution can describe the entire $ p_T $ spectrum in high-energy $ pp $ collisions at central rapidity.
• To challenge the conventional two-component model (soft and hard processes) by showing that a single distribution can adequately fit the full $ p_T $ range.
• To explore the theoretical justification for this unified description by approximating the hard-scattering integral as a nonextensive distribution.
• To demonstrate that the effective temperature $ T = 0.13\,\text{GeV} $ remains consistent across the entire $ p_T $ spectrum, supporting a universal statistical description.

Proposed method

• Use of the Tsallis nonextensive statistical mechanics framework, parameterized by $ q $ and $ T $, to model the $ p_T $ spectrum via the distribution $ F(p_T) = A[1 - (1-q)\frac{p_T}{T}]^{1/(1-q)} $.
• Equivalence of this form to the phenomenological interpolating formula $ F(p_T) = A(1 + \frac{p_T}{p_0})^{-n} $, with $ n = 1/(q-1) $ and $ p_0 = nT $.
• Analytical approximation of the hard-scattering integral to show it naturally yields a form consistent with the nonextensive distribution.
• Comparison of the single nonextensive distribution with LHC experimental data (CMS, ATLAS, ALICE) across the full $ p_T $ range.
• Use of the parameter $ q $ to unify the description of both low- and high-$ p_T $ regions, with $ q \to 1 $ recovering the standard Boltzmann-Gibbs exponential.
• Analysis of residual log-periodic oscillations in data to suggest possible fractal-like hierarchical structures in the quark-gluon system.

Experimental results

Research questions

• RQ1 Can a single nonextensive distribution describe the entire $ p_T $ spectrum in high-energy $ pp $ collisions at central rapidity, spanning 14 orders of magnitude in cross-section?
• RQ2 Is the hard-scattering process the dominant mechanism for hadron production across all $ p_T $ values, including low $ p_T $, at high energies?
• RQ3 Does the observed consistency of the effective temperature $ T = 0.13\,\text{GeV} $ across the spectrum support a universal statistical-mechanical description?
• RQ4 Can the hard-scattering integral be approximated as a nonextensive distribution, providing a theoretical basis for the unified description?
• RQ5 What is the origin of the small log-periodic oscillations observed in the $ p_T $ spectra, and do they indicate fractal-like structures in the partonic system?

Key findings

• The entire $ p_T $ spectrum in $ pp $ collisions at LHC energies, spanning approximately 14 orders of magnitude in cross-section, is well described by a single nonextensive Tsallis distribution with $ q $ and $ T $.
• The effective temperature $ T = 0.13\,\text{GeV} $ is consistent across the full $ p_T $ range, indicating a universal scale in the system.
• The hard-scattering integral can be approximated as a nonextensive distribution, providing a theoretical justification for the single-distribution description.
• The dominance of the hard-scattering process is supported by two-particle correlation data showing back-to-back correlations at $ \Delta\phi \sim \pi $, indicating partonic collisions even at low $ p_T $.
• Residual log-periodic oscillations in the data suggest possible fractal-like hierarchical structures in the quark-gluon system, possibly linked to complex $ q $ or fluctuating scale parameters.
• The results support a 'no-hair' statistical-mechanical description of $ p_T $ spectra, where only three parameters—$ q $, $ T $, and normalization—suffice to describe the full spectrum.