Entropy and Heat Capacity of the transverse momentum distribution for pp
collisions at RHIC and LHC energies

The study of high-energy ion collisions has been a significant area of research in nuclear and particle physics, providing insights into the properties of strongly interacting matter under extreme conditions Bjorken (1983). One relevant experimental measurement is the transverse momentum distribution (TMD), which is a histogram built with the transverse momentum ($p_{T}$) of the produced charged particles per momentum space unit and contains information on the processes involved in all scales of events, leading to the final state of produced particles Vogt (2007). The importance of the TMD renders the study of theoretical models and empirical fitting functions that adequately describe part or all the spectrum necessary. Earlier efforts to achieve this assume that the TMD follows an exponential distribution, where the inverse of the exponential decay is frequently associated with the temperature of the collision system Becattini (1996, 1997). This fitting function reasonably described the experimental data at the lower center of mass energies but deviated as experiments reached higher energies, revealing a non-exponential tail Bialas (2015); Feal et al. (2021). This approach is valid when most of the contribution to the spectra comes from soft scattering processes, leading to a soft thermal-like $p_{T}$ distribution Braun-Munzinger et al. (1995, 2004); Feal et al. (2021).

In the early ‘80s, Hagedorn introduced a QCD-based fitting function described by a power law of the transverse momentum shifted by a threshold that comes from the elastic scattering momentum scale Hagedorn (1965); Rafelski (2016); Hagedorn (1983). Interestingly, this proposal reproduced both behaviors of the TMD, thermal and a power law tail at low and high $p_{T}$ values, respectively. Later, the high-energy community presented a new fitting function based on the Tsallis $q$-Exponential function, which generalizes the thermal distribution by introducing a certain non-extensivity degree of the systems formed in high energy collisions Wilk and Włodarczyk (2000); Bíró et al. (2020). However, these fitting functions are shown to be equivalent Saraswat et al. (2018).

On the other hand, for string models, the production of charged particles is described by creating neutral color pairs through the breaking of the strings stretched between the partons. These subsequently decay, producing the observed hadrons Andersson (1998). In these cases, the transverse momentum distribution is governed by the Schwinger mechanism Schwinger (1962).

In the latter ‘90s, Bialas reconciled this approach with the thermal distribution by considering the string tension undergoes Gaussian fluctuations with zero average and variance proportional to the string tension Bialas (1999). Later, C. Pajares resumed this idea to incorporate a temperature-like parameter on the Color String Percolation Model, and thus, he provided a way to compute the string density from experimental data Dias de Deus and Pajares (2006); Braun et al. (2015); Bautista et al. (2019). Recently, in Refs. Pajares and Ramírez (2023); Alvarado García et al. (2023), the authors retake the original Bialas idea to extend the string tension fluctuations to a heavy tailed distribution. In particular, if the tensions fluctuate according to a $q$-Gaussian distribution, then the TMD becomes a confluent hypergeometric function that correctly fits the spectrum, thermal and power law behaviors at low and high $p_{T}$ values, respectively Pajares and Ramírez (2023); Alvarado García et al. (2023).

In this manuscript, we analyze the TMD data of minimum bias pp collisions reported by the experiments at the Relativistic Heavy-Ion Collider (RHIC), BNL and the Large Hadron Collider (LHC), CERN under the three schemes discussed above, namely, the thermal distribution, the Hagedorn, and the confluent hypergeometric fitting functions. In this way, we discussed the thermal temperature estimated in each scenario as a function of the center of mass energy. Since each fitting function has a different degree of accuracy in reproducing the spectrum, we compute some statistics to compare them, such as the average of transverse momentum, variance, and kurtosis. Additionally, we compute the Shannon entropy for each fitting function and estimate the heat capacity. The latter determination helps estimate the energy increment necessary to heat the collision systems.

The plan of the paper is as follows. In Sec. II, we comment on different approaches to describe the TMD and their main features. In Sec. III, we show the fits to the TMD of minimum bias pp collisions and give a description of fit parameters as a function of the center of mass energy. In Sec. IV, we compute the moments of the TMD in the approaches discussed in Sec. II. Section V contains our computations of the Shannon entropy and the heat capacity of the analyzed TMD data. Finally, in Sec. VI, we write our final comments, conclusions, and perspectives.

In this section, we discuss the particularities of the transverse momentum distribution, which can be obtained from different approaches. In particular, we are interested in discussing the cases of the TMD for color string systems and the QCD-based fitting function proposed by Hagedorn. In what follows, the TMD is denoted as $dN/dp_{T}^{2}$, meaning the invariant yield of produced particles.

We analyze the experimental transverse momentum spectra of charged particles of minimum bias pp collisions at different center of mass energies. By using Eqs. (3), (8), and (13), we fit over the experimental data reported on Refs. STAR Collaboration (2003); ALICE Collaboration (2010, 2023) using the ROOT 6 software. The fits were performed by using different $p_{T}$ ranges. For instance, we adjust the $p_{T}$ range for the thermal fits, finding the minimization of $\chi^{2}$ (see Table 1). However, for the Hagedorn and $U$ functions, the fit was done for the entire $p_{T}$ range reported by the experiments STAR Collaboration (2003); ALICE Collaboration (2010, 2023). In all cases, the value of the quotient $\chi^{2}$/NDF does not exceed 1 for the fits performed to the TMD data. Nevertheless, $\chi^{2}$/NDF$\gg$1 in the case of the thermal fits extrapolated to the entire range of $p_{T}$. This means that the three functions can provide a good description of the experimental data in the appropriate $p_{T}$ range. As seen in Fig. 2, each fitting function describes part of the spectrum: the thermal fit reproduces only the low $p_{T}$ region. Meanwhile, the Hagedorn was proposed to describe the high $p_{T}$ region of the spectrum (0.3 GeV $<p_{T}<$ 10 GeV Hagedorn (1983)) but is capable of reproducing the complete range of experimental data. Finally, the confluent hypergeometric confluent function successfully describes the behavior of the whole TMD for the data sets.

It is found that the $q$-Gaussian parameters rise as the center of mass energy increases. We propose the following scaling laws to describe these behaviors

with $\sqrt{s_{0}}=1$ GeV, $a_{q}=1.253(3)$, $c_{q}=0.0154(14)$, $a_{\sigma}=0.388(8)$ GeV, and $c_{\sigma}=0.037(11)$. In Fig. 3 we can see the $q$, and $\sigma$ dependence with $\sqrt{s}$ described by Eqs. (18).

For the Hagedorn function, the fitting parameters $m$ and $p_{0}$ are described by:

with $a_{m}=8.45(17)$, $c_{m}=-0.082(11)$, $a_{p_{0}}=1.22(5)$ GeV, and $c_{p_{0}}=-0.03(3)$. Fig. 4 shows the Hagedorn parameters dependence with $\sqrt{s}$. The parameter $q_{e}=(1+m)/m$ is also shown in Fig. 4(a) in the y-axis on the right side.

We recall that the three different approaches to analyzing the TMD provide their estimation of the thermal temperature. We found that $T_{\text{th}}$, $T_{U}$, and $T_{\text{Hag}}$ scale with the center of mass energy as Alvarado García et al. (2023)

The obtained parameters of Eq. (20) for the three approaches are shown in Table 2, and they are plotted in Fig. 5.

We recall that the temperature parameter is given by the TMD behavior at low $p_{T}$. The experimental TMD data exhibit a thermal behavior at low $p_{T}$ for identified species of produced particles, including the Higgs boson Pajares and Ramírez (2023). The higher the masses, the higher temperature values are expected. In the collisions are also produced heavy resonances which decay in lower mass particles, enhancing the low $p_{T}$ spectrum, but they can not be considered as formerly produced by the fragmentation of color string clusters. Notice that the parameters $q$ and $m$ control the tail of the TMD, exhibiting a monotonic behavior with the center of mass energy, as shown in Fig. 3(a) and 4(a). Similar behaviors are expected as a function of the multiplicity. Additionally, in the heavy-tailed string tension fluctuations approach, the high $p_{T}$ particle production can be considered as rare events, including jets. This information is implicitly incorporated in the tail of the $q$-Gaussian fluctuations. Nevertheless, this approach is not able to distinguish the longitudinal and transverse jet structure.

We compute the $n$-th moment of the transverse momentum spectra in the standard form

for all the fitting functions discussed in Sec. II. Here, we have introduced the notation $\left\langle\mathcal{P}_{T}^{n}\right\rangle$ to avoid misinterpretation with the computation of the moments reported by the HEP community. In those cases, the TMD must be integrated by considering the differential contributions of the longitudinal momentum component Hagedorn (1983); Bylinkin and Rostovtsev (2014). Then

The latter definition is also equivalent to considering $p_{T}^{2}$ as the random variable.

The calculation of (21) is immediate for the thermal distribution, which gives $\langle\mathcal{P}_{T}^{n}\rangle_{\text{exp}}=n!T^{n}$.

Let us explain the computation of $\langle\mathcal{P}_{T}^{n}\rangle$ for the Hagedorn and confluent hypergeometric function $U$ in detail. In both cases, we define the auxiliary function

In this way, $\langle\mathcal{P}_{T}^{n}\rangle=I_{n}/I_{0}$. For the Hagedorn function (13), we found

where $B$ is the Beta function, which is well defined for $m>n+1$. Therefore

Similarly, for the $U$ function, we need to compute the integral

where $a$, $b$, and $z$ corresponds to Eq. (7). By following the definition of the confluent hypergeometric function (6), we rewrite $I_{n}$ as

To simplify notation, we prescinded writing the factor $1/\Gamma(a)$, which appears in the denominator and numerator of Eq. (21). Note that the integral over $p_{T}$ in Eq. (25) is a Gaussian integral

By plugging the later on Eq. (25) and performing the change of variable $y=(t+1)^{-1}$, the remaining integral becomes

Finally, the $I_{n}$ integrals are given by

which are well defined if $q<({4+n})/({2+n}).$ So, the moments of the distribution are expressed as

It is worth mentioning that some experiments report the TMD without the functional normalization by dividing by $p_{T}$. In those cases, the function describing the transverse momentum spectra is $p_{T}(dN/dp_{T}^{2})$, and the moments of the distribution are calculated as discussed above. Notice that the moments (22) can also be expressed in terms of the $I_{n}$ integrals as $I_{n+1}/I_{1}$. Moreover, the ratio $\left\langle\mathcal{P}_{T}^{n}\right\rangle/\left\langle p_{T}^{n}\right\rangle$ is given by

for the thermal distribution, Hagedorn, and the confluent hypergeometric function, respectively. Nevertheless, in what follows, we will continue to discuss the computation of observables considering $p_{T}$ as the random variable.

Let us delve into a fundamental concept in information theory, the Shannon entropy. It provides a way for quantifying the uncertainty and information of the TMD Shannon (1948). This observable can shed light on the characteristics of the final state particles of collision systems. Since the temperature-like parameter $T$ is extracted from the TMD in each approach, the natural way of computing the Shannon entropy is by considering the normalized TMD as the probability density function of the random variable $p_{T}$, as usually done in the generalized ensemble theory Niven and Andresen (2010); Langen et al. (2015). Then, the Shannon entropy is computed as Shannon (1948)

where $I_{0}$ is the normalization constant given by Eq. (23).

The Shannon entropy (38) can be expressed in terms of elementary functions for the thermal and Hagedorn fitting functions. For these cases, we obtain

respectively.

On the other hand, the Shannon entropy of the confluent hypergeometric function is explicitly given by

with

which can be rewritten as

where

As far as we know, the Eq. (43) cannot be solved analytically. Then, we computed $\mathcal{H}_{1}$ by using numerical methods. Figure 10 shows our estimations of the Shannon entropy for minimum bias pp collisions as a function of the center of mass energy and the corresponding temperature for each approach.

We also compute the heat capacity using its thermodynamic definition Mandal (2013)

In this context, Eq. (44) is a measure of how much “heat” is necessary to “warm” the TMD. “Heating” the TMD must be understood as a global change of the TMD shape, flattening the soft part together with an enhancement of the TMD tail.

To compute Eq. (44), we must take into account that the fitting parameters may depend on the temperature. In these cases, the computation of the heat capacity must be done using the chain rule. In particular, for the thermal case, we found $C_{\text{th}}=1$. In the case of the Hagedorn fitting function, the heat capacity is given by

where the derivative $dm/dT_{\text{Hag}}$ for minimum bias pp collisions is computed through Eq. (19a) and using the inverse relation of the temperature with the center of mass energy in Eq. (20), which reads

Plugging Eq. (46) into (45), the heat capacity is

with $m(T_{\text{Hag}})=a_{m}(T_{\text{Hag}}/a_{T})^{c_{m}/c_{T}}$.

On the other hand, for the calculation of the heat capacity of the confluent hypergeometric $U$ function, we start replacing $\sigma$ in favor of $T_{U}$ through Eq. (10). Thus, the normalization constant $I_{0}$ and the $z$ parameter (see (7)) in the third argument of the $U$ functions are rewritten as follows

Therefore, the heat capacity for the $U$ fitting function is

In Eq. (50), the remaining integral is

where $\psi^{(0)}(x)$ is the zeroth order polygamma function. We also added the superscript to the $U$ function to denote their first derivative with respect to the first argument. Notice that there are two remaining integrals, which are done by means of numerical methods. Similarly to the case of the Hagedorn fitting function, $q$ is expressed as a power law as a function of $T_{U}$, i.e., $q(T_{U})=a_{q}(T_{U}/a_{T})^{c_{q}/c_{T}}$, and its derivative with respect to the temperature is

The heat capacity of the three different schemes as a function of the center of mass energy and the temperature is plotted in Fig. 10.

In this work, we discussed the statistics of three different fitting functions that describe the TMD data, namely, the thermal $p_{T}$-exponential, the confluent hypergeometric, and the Hagedorn. The former arises from the string tension fluctuation in the QCD color string picture, meanwhile, the latter is a power law inspired by the foundations of QCD. All of them predict a temperature parameter at the low $p_{T}$ regime.

The temperature estimated for minimum bias pp collisions as a function of the center of mass energies reflects the physical motivation of the different approaches. The thermal distribution adequately describes the soft part because it assumes an exponential decay (similar to a Boltzmann distribution), resulting in overestimating the temperature for the complete TMD. On the contrary, the power law proposed by Hagedorn establishes a description of the hard processes, leading to the heavy tail of the spectrum. This means that the thermal temperature may not precisely incorporate the soft part of the TMD. In fact, Hagedorn suggests that their fit must be performed in the $p_{T}$ interval from $0.3$ to $10$ GeV Hagedorn (1983). On the other hand, we must emphasize that the confluent hypergeometric $U$ adequately combines the information of the soft and hard scales to predict the temperature.

We also discuss the statistics of the normalized TMD by computing the moments of the distribution and, thus, the variance and kurtosis for the three different approaches. This analysis lets us distinguish the particularities of each fitting function. For example, the Hagedorn and $U$ functions reveal more dispersion than the thermal one because of the information coming from the heavy tail. In all cases, we found an increasing trend on the first moment and variance with the center of mass energy as seen in Figs. 6 and 8. Moreover, the heavy tail absence in the thermal distribution leads to a constant kurtosis, which was taken as a reference to measure the excess of kurtosis in the Hagedorn and $U$ distributions, both increase as the center of mass energy rises, highlighting that the $U$ grows more substantially (see Fig. 9). This means the TMD derived from the $q$-Gaussian string tension fluctuations contains more information in the tail than the Hagedorn approach. This is important because the $U$ fitting function adequately reproduces the power law behavior associated with the QCD hard processes from the color string picture.

In addition, we compare the average of the transverse momentum estimated from the fitting function and the computed from the experimental data. It was found that the Hagedorn and the $U$ functions precisely reproduce the value of $\left\langle p_{T}\right\rangle^{\text{hist}}$, but the predictions of the thermal distribution considerably deviate (15%-25%).

Other observables that we computed are the Shannon entropy and the heat capacity. The Shannon entropy increases as the center of mass energy grows. This is consistent with the TMD variance, which exhibits a similar behavior. This happens because the probability of observing particles with high $p_{T}$ rises with an increment on the center of mass energy. Then, the TMD suffers a global widening and an enhancement of its tail. Moreover, we observed quite differences between the entropy computed for the Hagedorn and hypergeometric confluent $U$ functions. These subtle deviations may come from the shape of the TMD at very low $p_{T}$, which can be inferred from the temperature estimated by each model.

Moreover, the computation of the heat capacity for the thermal fitting function reveals that the system does not change its requirements to heat up. From a classical thermodynamics point of view, this means that collision systems described by a thermal distribution resemble an ideal gas of monoatomic (or rigid diatomic) molecules. On the contrary, for the Hagedorn and $U$ functions, the heat capacity grows as the temperature does, similar to a thermodynamic system that can manifest other degrees of freedom when heating. This implies that, to heat up the collision system, it is necessary to reach an increasingly higher center of mass energies. This is a direct consequence of the heavy tailed TMD, since it requires not only heating the thermal part but also the hard one for the discussed minimum bias pp collisions. From our results, we infer that the higher the TMD temperature, the more energetic collision is required. This observation is consistent with the analysis of the experimental data of the temperature saturation as a function of the center of mass energy (see Fig. 5).

This work can be extended in several ways. For instance, it would be interesting to analyze the TMD and compute the Shannon entropy and heat capacity of pp collisions as a function of the multiplicity, heavy ion collisions, production of identified particles, and other processes. Part of these results are currently under discussion, and we will report our findings in a future paper.

This work has been funded by the projects PID2020-119632GB-100 of the Spanish Research Agency, Centro Singular de Galicia 2019-2022 of Xunta de Galicia and the ERDF of the European Union. This work was funded by Consejo Nacional de Humanidades, Ciencias y Tecnologías (CONAHCYT-México) under the project CF-2019/2042, graduated fellowship grant number 1140160, and postdoctoral fellowship grant numbers 289198 and 645654. J. R. A. G. acknowledges financial support from Vicerrectoría de Investigación y Estudios de Posgrado (VIEP-BUAP).