Entropy and heat capacity of the transverse momentum distribution for $pp$ collisions at RHIC and LHC energies

We investigate the transverse momentum distribution (TMD) statistics from three different theoretical approaches. In particular, we explore the framework used for string models, wherein the particle production is given by the Schwinger mechanism. The thermal distribution arises from the Gaussian fluctuations of the string tension. The hard part of the TMD can be reproduced by considering heavy tailed string tension fluctuations, for instance, the Tsallis $q$-Gaussian function, giving rise to a confluent hypergeometric function that fits the entire experimental TMD data. We also discuss the QCD-based Hagerdon function, another family of fitting functions frequently used to describe the spectrum. We analyze the experimental data of minimum bias $pp$ collisions reported by the BNL Relativistic Heavy Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC) experiments (from $\sqrt{s}=0.2$ TeV to $\sqrt{s}=13$ TeV). We extracted the corresponding temperature by studying the behavior of the spectra at low transverse momentum values. For the three approaches, we compute all moments, highlighting the average, variance, and kurtosis. Finally, we compute the Shannon entropy and the heat capacity through the entropy derivative with respect to the temperature. We found that the $q$-Gaussian string tension fluctuations lead to a monotonically increasing heat capacity as a function of the center-of-mass energy, which is also observed for the Hagedorn fitting function. This behavior is consistent with the experimental observation that the temperature slowly rises with increments of the collision energy.

Figure 1

Gaussian and $q$-Gaussian string tension fluctuations for $pp$ collisions at different center-of-mass energies. The tail of the $q$-Gaussian distribution becomes less tilted as the center-of-mass energy increases, which means that the probability of observing a string with a higher tension rises.

Figure 2

Fits (lines) to the experimental data (figures) for minimum bias $pp$ collisions at different center-of-mass energies by using (a) the thermal distribution, (b) the hypergeometric confluent $U$ function, and (c) the Hagedorn function. Shaded regions correspond to the 95% uncertainty propagation.

Figure 3

Values of the parameters (a) $q$ and (b) $\sigma$ obtained from the confluent hypergeometric $U$ fits as a function of center-of-mass energy for minimum bias $pp$ collisions. The scaling law dependences (18) are shown in solid lines. Shaded regions correspond to the uncertainty propagation.

Figure 4

Parameters (a) $m$ and (b) $p_0$ obtained from Hagedorn fits (13) to minimum bias $pp$ collisions data as a function of the center-of-mass energy. Dashed lines are the parameters described by (19). The shaded regions correspond to the propagation of uncertainty.

Figure 5

Temperature extracted from (squares) thermal, (circles) confluent hypergeometric $U$, and (triangles) Hagedorn fitting functions. Lines are the trend of the (dotted) $T_{\text{th}}$, (solid) $T_U$, and (dashed) $T_{\text{Hag}}$ described by Eq. (20). Shaded regions correspond to the uncertainty propagation.

Figure 6

(a) $\langle {\mathcal{P}}_T\rangle$ and (b) $\langle p_T\rangle$ as a function of the center-of-mass energy for $pp$ collisions for the three different fitting functions with their corresponding parameter dependence. Lines, figures, and colors are the same as in Fig. 5.

Figure 7

Absolute percentile deviations of (a) $\langle {\mathcal{P}}_T\rangle$ and (b) $\langle p_T\rangle$ for the confluent hypergeometric $U$ (circles), Hagedorn (triangles) and thermal (squares) fitting functions. Figures and colors are the same as in Fig. 5.

Figure 8

Variance of the TMD as a function of the center-of-mass energy for minimum bias $pp$ collisions for the three different approaches. Lines, figures, and colors are the same as in Fig. 5.

Figure 9

Excess of kurtosis calculated with respect to the thermal distribution as a function of the center-of-mass energy for $pp$ collisions. Lines, figures, and colors for the Hagedorn and $U$ fitting functions are the same as in Fig. 5.

Figure 10

Shannon entropy dependence on the (a) center-of-mass energy and (b) temperature. Heat capacity as a function of the (c) center-of-mass energy and (d) temperature scaled by $a_T$. The shaded regions correspond to the uncertainty propagation of the corresponding equations. Lines, figures, and colors for the thermal, Hagedorn, and $U$ fitting functions are the same as in Fig. 5.