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Multiplicity dependence of the entropy and heat capacity for pp collisions at LHC energies

C. E. Munguía López, D. Rosales Herrera, J. R. Alvarado García, A. Fernández Téllez, J. E. Ramírez

TL;DR

This work analyzes how the transverse momentum spectrum in pp collisions at LHC energies depends on event multiplicity within a nonextensive, heavy-tailed string-tension framework. The authors fit the $p_T$ spectra using a confluent hypergeometric (Tricomi) form and a Tsallis-like $q$-exponential, establishing correlations between the nonextensivity parameters and a temperature-like scale $T_U$. They compute moments, Shannon entropy, and heat capacity for spectra categorized by ALICE V0M and SPD multiplicity estimators, finding classifier-dependent trends: SPD exhibits stronger hardening, larger entropy, and increasing heat capacity with multiplicity, while V0M shows signs of saturation and sometimes decreasing heat capacity. The results highlight that event-selection biases significantly affect inferred observables and suggest extending the analysis to additional event-shape selectors and collision systems to test the robustness of nonextensive descriptions in high-energy collisions.

Abstract

We investigate the multiplicity dependence of the transverse momentum spectrum of the charged particle production in pp collisions at LHC energies. To this end, we consider the experimental data sets classified with different multiplicity estimators, defined by the ALICE Collaboration, that are analyzed within the framework of nonextensive particle production. We compute the variance, kurtosis, Shannon entropy, and heat capacity of the $p_T$ spectrum to study the hardening process as a function of the multiplicity and temperature under the different event classifiers. We found that both the Shannon entropy and the heat capacity show different responses for the triggers at the forward-backward and midrapidity regions. We emphasize that the selection of event biases may induce different responses in estimating theoretical and phenomenological observables that could lead to misleading conclusions.

Multiplicity dependence of the entropy and heat capacity for pp collisions at LHC energies

TL;DR

This work analyzes how the transverse momentum spectrum in pp collisions at LHC energies depends on event multiplicity within a nonextensive, heavy-tailed string-tension framework. The authors fit the spectra using a confluent hypergeometric (Tricomi) form and a Tsallis-like -exponential, establishing correlations between the nonextensivity parameters and a temperature-like scale . They compute moments, Shannon entropy, and heat capacity for spectra categorized by ALICE V0M and SPD multiplicity estimators, finding classifier-dependent trends: SPD exhibits stronger hardening, larger entropy, and increasing heat capacity with multiplicity, while V0M shows signs of saturation and sometimes decreasing heat capacity. The results highlight that event-selection biases significantly affect inferred observables and suggest extending the analysis to additional event-shape selectors and collision systems to test the robustness of nonextensive descriptions in high-energy collisions.

Abstract

We investigate the multiplicity dependence of the transverse momentum spectrum of the charged particle production in pp collisions at LHC energies. To this end, we consider the experimental data sets classified with different multiplicity estimators, defined by the ALICE Collaboration, that are analyzed within the framework of nonextensive particle production. We compute the variance, kurtosis, Shannon entropy, and heat capacity of the spectrum to study the hardening process as a function of the multiplicity and temperature under the different event classifiers. We found that both the Shannon entropy and the heat capacity show different responses for the triggers at the forward-backward and midrapidity regions. We emphasize that the selection of event biases may induce different responses in estimating theoretical and phenomenological observables that could lead to misleading conclusions.

Paper Structure

This paper contains 9 sections, 21 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Nonextensive origin of the transverse momentum spectrum
  3. Data analisys
  4. Moments and statistics of the transverse momentum spectrum
  5. Average transverse momentum
  6. Variance and kurtosis of the transverse momentum spectrum
  7. Dependence of the Shannon entropy and heat capacity on the multiplicity classifiers
  8. Discussion
  9. Summary and outlook

Figures (10)

  • Figure 1: Experimental data of $p_T$ spectrum (markers) of the charged particle production in pp collisions at different center of mass energies under the V0M and SPD classifiers defined by the ALICE Collaboration: (a.1)-(a.2) $\sqrt{s}=$ 5.02 TeV (V0M), (b.1)-(b.2) $\sqrt{s}=$ 7 TeV (V0M), (c.1)-(c.2) $\sqrt{s}=$ 13 TeV (V0M), (d.1)-(d.2) $\sqrt{s}=$ 5.02 TeV (SPD), and (e.1)-(e.2) $\sqrt{s}=$ 13 TeV (SPD). See Table \ref{['tab:dsets']} for detailed information on the kinematic cuts of the data sets. The solid lines correspond to the fits to data performed via Eq. \ref{['eq:TMDU']} (top panels) and the dashed lines correspond to the fits using Eq. \ref{['eq:qexp']} (bottom panels). In all cases, the shaded region corresponds to the 2-$\sigma$ uncertainty propagation.
  • Figure 2: Correlations between the model parameters (a) $q_e$ as a function of $q$ and (b) $T_e$ as a function of $T_U$ for all the analyzed cases. The solid lines correspond to the linear trends.
  • Figure 3: (a) Temperature as a function of the average multiplicity and (b) $\sigma$ as a function of temperature for all the analyzed data sets. The solid line corresponds to Eq. \ref{['eq:sT']}. The shaded region corresponds to the uncertainty propagation.
  • Figure 4: Dependence of the $q$ on the temperature for: (a) $\sqrt{s}=$ 5.02 TeV (V0M), (b) $\sqrt{s}=$ 7 TeV (V0M), (c) $\sqrt{s}=$ 13 TeV (V0M), (d) $\sqrt{s}=$ 5.02 TeV (SPD), and (e) $\sqrt{s}=$ 13 TeV (SPD). The markers correspond to the fitted value of each data set. The solid lines correspond to the parametrization in Eq. \ref{['eq:qT']}. The shaded regions correspond to the uncertainty propagation.
  • Figure 5: Temperature dependence of the mean $p_T$ (Eq. \ref{['eq:avgpTe']}) of the charged particles produced in pp collisions at (a) $\sqrt{s}=$ 5.02 TeV (V0M classes), (b) $\sqrt{s}=$ 7 TeV (V0M classes), (c) $\sqrt{s}=$ 13 TeV (V0M classes), (d) $\sqrt{s}=$ 5.02 TeV (SPD classes), and (e) $\sqrt{s}=$ 13 TeV (SPD Classes). The markers correspond to the computed values using the values of $q$ and $\sigma$ for each data set. The solid lines correspond to the parametrization considering the temperature dependence of $q$ and $\sigma$ via Eqs. \ref{['eq:sT']} and \ref{['eq:qT']}). The shaded regions correspond to the uncertainty propagation.
  • ...and 5 more figures