Simulation Studies of Track-Based Analysis of Charged Particles in Symmetric Hadron–Hadron Collisions at 7 TeV

Abstract

This manuscript presents a simulation study of a track-based analysis of the multiplicity distributions of the primary charged particle compared to experimental measurements in symmetric hadron–hadron collisions acquiring maximum energy for the new particle production. The data are compared to the simulations of EPOS, PYTHIA8, Sibyll, and QGSJET under the same conditions. The event generators in the current study are simple parton-based models that incorporate the Reggie–Gribov theory. The latter is a field theory based on the QCD that uses the mechanism of multiple parton interactions. It has been found that the PYTHIA8 model chases the data well in most of the distributions but depends on the momentum and the requirement of charged particles in a given track, due to its feature-like color reshuffling of quarks and gluons through the color re-connection modes and initial and final state radiations by incorporating the parton showers. The EPOS model could also reproduce some spectral regions and presents a good comparison after the PYTHIA8. All the other models could not produce most of the spectra except for the limited region, which also depends on the analysis’s cuts. Besides the model’s prediction, we used Tsallis–Pareto and Hagedorn functions to fit the aforementioned spectra of the charged particles. The fit is applied to the data and models, and their results are compared. We extract the temperature parameter T₀₁ (effective temperature (T_eff)) from the Tsallis–Pareto-kind function and T₀₂ (kinetic freezeout temperature) from the Hagedorn function. The temperatures are affected by p_T as well N_ch cuts.

1. Introduction

The soft and hard processes are two primary mechanisms responsible for particle production in collision experiments. The soft process is involved mainly in the low momentum side of the spectra, also known as the underlying events. In contrast, the hard process takes part in the high momentum side of the spectra. The multi-parton interactions are thought to be responsible for the soft processes, whereas the hard processes are the result of a few or single-parton interactions. These two regions of low and high transverse momentum ($p_T$) can be understood by the exponential and power-law equations [1,2,3,4]. At the same time, a single distribution nicely depicts both the regions, called the Tsallis distributions [5,6]. It has been widely used in experiments such as STAR at RHIC [7], and CMS, ALICE, and LHCb at the LHC [8].

In the current manuscript, we analyzed inclusively charged particles in symmetric hadron-hadron collisions at 7 TeV based on tracks (requiring minimally charged particles in an event) with an inclusive and reduced phase space $p_T$ of the produced particles. The aim of this work is to tune the models’ parameters, as these are being used by all colliders and fixed target physics experiments to design and tune their detectors and analysis strategies. Thus, this is of the utmost importance to develop the models that guide particle physics experimentalists and phenomenologists. Similar studies, where identified or inclusively charged particles are used, can be found in [9,10,11,12,13,14]. The Hadronic collisions are carried out at the LHC to understand the particle’s yield and spectra and obtain insight into more complex interactions of heavy ions. Both symmetric (pp and AA) [15,16] and anti-symmetric (pA and AB) [17] collisions are employed at different center of mass (cms) energies. At different cms energies, the distributions of the inclusively charged particles have been carried out in nucleon–nucleon collisions [9,10,18,19,20]. These studies offer explanations about some features of the low-energy strong interactions. Many Monte Carlo (MC) QCD-inspired models are available to support these explanations. The models are built in such a way that they use MC simulations with free parameters that spectral studies of particles can constrain. Such studies are essential because they contribute to knowing about soft QCD. Furthermore, they are essential to determine high $-p_T$ biases arising from the event pileup effects and underlying events. Therefore, they are necessary for the physics of the future LHC.

The soft processes or underlying events, as a result of the hadronization and particle collisions, can be investigated and well understood by the Standard odel. Still, as the strong forces are non-linear, the precise numerical computations of such processes are challenging. Different models and event generators are tuned to predict the behavior of SM physics, and hints at any new physics at the LHC to achieve and reproduce the experimental findings, as there is no theoretical understanding to conduct the complete QCD calculations.

We organize the remaining manuscript as follows. After the introduction in Section 1, “The Method and Models” in Section 2 is dedicated to briefly describe the MC models. The main results of the manuscript are given in Section 3, “Results and Discussion”, where the model’s predictions are shown side-by-side against the experimental data taken from [21]; furthermore, the conclusions and summary of the work are placed at the end in Section 4.

2. The Method and Models

Measurements of primary charged particle at 7 TeV reported by the ATLAS experiment [21] are taken from the paper [22] of the LHC [10]. The spectra obtained by experimental data were compared to the simulations of the four MC models, EPOS [23], PYTHIA8 [24,25,26], Sibyll [27], and QGSJET [28], at the same energy. The simple parton model is the basis of all these models, using the Reggie–Gribov theory [29]. The theory is on the QCD effective field theory, which uses multi-parton interactions [29]. In the following paragraphs, a short overview of each one of these is given below:

• The EPOS model is a multi-scattering quantum mechanical approach using the off-shell remnants of partons and their ladders’ splitting [23]. It is a well-known MC package that simulates the interactions of hadrons and nuclei to reproduce high-energy cosmic ray showers. The Reggie–Gribov’s theory [29] is the model base, and the Pomerons’ swap describe the interactions in hard, semi-hard, or soft scatterings. The particle production process in EPOS assumes two kinds of sources: remnant decays and cut Pomerons [30]. To account for the energies of the LHC, the model was recently updated to a newer version named EPOS-LHC [23], which covers all the minimum-bias results for the particle production in the $p_T$ range from 0 to several GeV/c. Lastly, different parameters to incorporate the flow effects have been proposed for high multiplicity $pp$ collisions in small volumes and heavy-ion collisions in large volumes. The EPOS-LHC version is used in the presented simulation in this analysis, but we used EPOS throughout the paper for simplicity.

• PYTHIA8 is an MC event generator based on partons’ interactions that produce showers. The model uses the Lund string fragmentation model for hadronization [31]. It is widely used in particle collision at high energy, mainly emphasizing the $pp$ interaction [24]. The events of $pp$ collisions are represented by different processes such as diffractive, elastic, electroweak, hard, and soft QCD scattering, and top quark production. The EPOS model also includes the Higgs mechanism within and beyond the standard model, as well as supersymmetry and the implementation of the hidden valley, and incorporates several models on dark matter. Several other exotic processes are also added to the PYTHIA8 model described in [26]. The model includes multiple parton interactions, final and initial state interactions, and beam remnants [25]. There is a possibility of three different modules for parton showers, where we used the simple/default model in our case. The Pythia8.307 version is used in this analysis, but for simplicity and consistency, we will use PYTHIA8 in the text throughout the manuscript.

• Sibyll is an event generator to simulate cosmic rays and describe hadronic high energy interactions. Sibyll uses a dual-parton model [32], mini-jet model [33,34], and Lund MC algorithm [35,36] to adopt the hadronization of particles. The DPM treats the soft interactions, while the Lund string fragmentation model is used for hard interactions. To determine the first point of the interaction in hadronic, semi-hadronic, and nuclear collisions, the Sibyll model uses a semi-superposition model. The exact cut-off value is used to separate hard and soft Pomeron in the EPOS model. In addition, Sibyll can be used to generate high-energy cosmic rays up to ${10}^{11}$ GeV. We used the last freely available version of the Sibyll model, Sibyll2.3d, but again, for convenience, only Sibyll will be used throughout the text.

• The QGSJET model uses Pomeron exchanges to represent a simple scattering process [28]. The model uses the Quark–Gluon Sting model [37] as a starting point, which in turn uses the Regge–Gribov theory. The model distinguishes between nonperturbative and perturbative (also termed as soft interactions and hard interactions, respectively) processes by a cut-off value. Nevertheless, both processes simultaneously contribute to the case of the QGSJET model. All soft interactions resulting in elastic scattering represent a pure nonperturbative parton cascade. In contrast, exchanging two soft Pomerons produces a “semi-hard” Pomeron merged by a parton-ladder [38]. The semi-hard Pomeron’s contribution exceeds the soft ones at high energies, resulting in many such scatterings.

In addition to presenting the models’ prediction as compared to the experimental data, we use a Tsallis–Pareto-kind function and a Hagedorn function to describe the $p_T$ spectra of the charged particles produced in p-p collisions at 7 TeV. The fit is applied to the data and models, and their results are compared. The Tsallis–Pareto-kind function [5,39,40,41] has a very good description of many of the spectral measurements [42,43,44]. The function has the following form:

(1)$\frac{{\mathrm{d}}^{2}N}{\mathrm{d}y\mathrm{d}{p}_T}=\frac{\mathrm{d}N}{\mathrm{d}y}C{p}_T{\left[1+\frac{m_T-m_0}{n_1{T}_{01}}\right]}^{-n_1},$

where

$C=\frac{(n_1-1)(n_1-2)}{n_1{T}_{01}[n_1{T}_{01}+(n_1-2)m_0]},$

$m_T=\sqrt{m_0^2+p_T^2}$, $m_0$, N, y, and $dN/dy$ are, respectively, the transverse mass, particle’s rest mass, particles’ number, particles’ rapidity, and particles’ integrated yield. As elaborated in some of the non-extensive thermodynamics models [40], $T_{01}$ is linked with the particles’ average energy and represents the $<T_{eff}>$ of the medium produced in such collisions or an estimate of the kinetic freeze-out temperature. The $n_1$ gives the “non-extensivity” of the process.

According to the Refs. [45,46,47,48,49], the Hagedorn function with the flow parameter can be expressed as:

(2)$\frac{{\mathrm{d}}^{2}N}{2\pi N_{ev}{p}_T\mathrm{d}y\mathrm{d}{p}_T}=C_0{\left[+⟨{\gamma }_T⟩(\frac{m_T-p_T⟨{\beta }_T⟩}{n_2{T}_{02}})\right]}^{-n_2},$

where

$⟨{\gamma }_T⟩=\frac{1}{\sqrt{1-{⟨{\beta }_T⟩}^2}},$

$C_0$ and $⟨{\beta }_T⟩$ are, respectively, the fitting constant and the mean value of the transverse flow velocity, and $T_{02}$ and $n_2$ are similar to $T_{01}$ and $n_1$ in the Tsallis–Pareto-kind function.

3. Results and Discussions

The pseudorapidity, $p_T$ and charged particle distributions, and the relationship between $<p_T>$ and $N_{ch}$ in $p-p$ collisions at 7 TeV are shown comparing the models’ prediction with the ATLAS experimental data [21]. The measurements reported by the experimental data are compared to the simulations of EPOS [23], PYTHIA8 [26], Sibyll [27], and QGSJETII-04 [28] at the same energy as the data.

Simulation and Data Comparison

Figure 1 is the $\eta$ distributions of the inclusive charged particles in events of different characteristics. Figure 1a is the $\eta$ distribution, with events having $N_{ch}$ ≥ 1 and $p_T$ > 0.5 GeV. The PYTHIA8 reproduces the data while the EPOS underpredicts the data at $\eta$ around 0 while reproducing the $\eta$ distribution at the two extremes. The QGSJET model overpredicts while the Sibyll underpredicts the data. In the next event class, where (b) $N_{ch}$ ≥ 2, $p_T$ > 0.1 GeV, the PYTHIA8 predicts the data well, followed by the EPOS and QGSJET models, which slightly underpredict the data at mid-rapidity only, whereas the Sibyll model underpredicts over the entire $\eta$ range. For (c) $N_{ch}$ ≥ 6, $p_T$ > 0.5 GeV, the PYTHIA8 and QGSJET models explain the data, while the other two underpredict. All models reproduce the data well except EPOS, which underpredict at mid-rapidity only for the event class (d) $N_{ch}$ ≥ 20, $p_T$ > 0.1 GeV. Finally, for the case of the event class (e) $N_{ch}$ ≥ 1, $p_T$ > 2.5 GeV, the PYTHIA8 and Sibyll have a good prediction of the data while EPOS and QGSJET underpredict. From the above observation, one can conclude that the model prediction became better for higher cuts in the number of charged particles, with loosening cuts in momentum.

Figure 2 is the simulation results of the charged particle’s $p_T$ distribution compared with the experimental data. The same event classes as before are used to categorize the distributions: (a) $p_T$ > 0.5 GeV, $N_{ch}$ ≥ 1, (b) $p_T$ > 0.1 GeV, $N_{ch}$ ≥ 2, (c) $p_T$ > 0.5 GeV, $N_{ch}$ ≥ 6, (d) $p_T$ > 0.1 GeV, $N_{ch}$ ≥ 20, and (e) $p_T$ > 2.5 GeV, $N_{ch}$ ≥ 1. All models have a good prediction of the data at the low momentum region of the distribution, while only PYTHIA8 has a good prediction of the data until the higher ends, followed by EPOS, which is up to 40 %. The Sibyll largely overpredicts while the QGSJEt underpredicts. Furthermore, the model’s prediction is independent of the distinctions used for the event classes and produces similar results in all the cases. The Pythia8 model is preferred over other hadron production models, as it is tested on the LHC data and has an additional feature, including the color reshuffling of quarks and gluons through the colour re-connection modes. It has an advantage over the others, as it also includes the initial and final state radiations by incorporating the parton showers. It better explains the Standard Model prediction by taking into account the perturbative and non perturbative QCD effects.

Figure 3 shows the result of the fit functions (Equations (1) and (2)) on the $p_T$ spectra of charged particles (given in Figure 2) in events for which (a) $p_T$ > 0.5 GeV, $N_{ch}$ ≥ 1, (b) $p_T$ > 0.1 GeV, $N_{ch}$ ≥ 2, (c) $p_T$ > 0.5 GeV, $N_{ch}$ ≥ 6, (d) $p_T$ > 0.1 GeV, $N_{ch}$ ≥ 20, and (e) $p_T$ > 2.5 GeV, $N_{ch}$ ≥ 1 at 7 TeV. The black square, black circle, black up triangle, black diamond, and black star symbols, respectively, represent the experimental data [21] and the predictions of four MC models of EPOS-LHC [23], PYTHIA8 [25], Sibyll [27], and QGSJETII-04 [28]. For clarity, the spectra of the four MC models are scaled by suitable factors shown in Figure 3a. The errors shown here are statistical only. The red and blue dash curves are fitting results with the Tsallis–Pareto-kind and Hagedorn functions, respectively. The values of free parameters $T_{01}$ and $n_1$ for the Tsallis–Pareto-kind function, $⟨{\beta }_T⟩$, $T_{02}$, and $n_2$ for the Hagedorn function, and corresponding ${\chi }^2$ per degree of freedom (${\chi }^2$/dof) are listed in Table 1. One can observe that both the Tsallis–Pareto-kind and Hagedorn functions can well describe the experimental data and the prediction results of the four models of the charged particles in p-p collisions at 7 TeV for the events mentioned above. We present the $p_T$ distribution of the charged particles with the help of the Tsallis–Pareto-type function and Modified Hagedorn model with embedded flow. The temperature in the former is the effective temperature, which is affected by the flow velocity, while the temperature in the latter is the kinetic freeze-out temperature, which excludes the flow velocity. The entropy parameter in the former distribution is represented by n; meanwhile, it is denoted by $n_1$ in the latter. ${\beta }_T$ is also extracted from the Hagedorn function. We observed that the entropy parameter extracted from both the fit models is approximately the same; however, the temperatures extracted from them are non-homogeneuos. The effective temperature from the Tsallis–Pareto function is larger than the kinetic freeze-out temperature extracted from the Hagedorn function, which is natural because the latter excludes them, while the former includes the flow effect. We also observed that there is a slight dependence of the temperature (the temperature refers to $T_{01}$ and $T_{02}$) on $p_T$ and $N_{ch}$ cuts. The temperature increases when the $N_{ch}$ cut increases, which can be observed from the results obtained in Table 1 (a), (b), and (c). However, in Table 1 (d) and (e), the $N_{ch}$ cut is larger but the value of temperature decreases, which indicates that the temperature is also affected by the $p_T$ cut. However, to make sure that the $p_T$ and $N_{ch}$ have an effect on the temperature, more data has to be analyzed properly with different distributions, which we will consider conducting in the future.

The values of the parameters give us the information of the equilibrium state of the system by using the two fit models. For instance, for the system in Table 1.a, $T_{01}$ =0.138 GeV and $n_1$ =6.4 (obtained from the Tsallis–Pareto-type function); meanwhile, $T_{02}$ = 0.101 GeV, $n_2$ = 6.4, and ${\beta }_T$ = 0.120 c (obtained from the Hagedorn-type function) for the experimental data are the implemented physical restrictions, and the system attains the equilibrium approach under restrictions.

For the experimental data with $p_T$ > 0.5 GeV and $N_{ch}$ ≥ 1, $T_{01}=$ 0.138 ± 0.01, $T_{02}=$ 0.101 ± 0.008, while $⟨{\beta }_T⟩$ = 0.12 ± 0.06; whereas, the $n_1=n_2$ = 6.4 ± 0.02. When increasing the cut from 1 to 2 on $N_{c}h$ and decreasing the $p_T$ from 0.5 GeV to 0.1 GeV, the values of $T_{01}$ and $⟨{\beta }_T⟩$ increase, while the rest of the parameters remain the same. A further increase in $N_{ch}$ to 6 with $p_T$ > 0.5 GeV, increases $T_{01}$, and $T_{02}$ to 0.162 ± 0.1 and 0.122 ± 0.008, respectively, while $<{\beta }_T>$ decreases to 0.13 ± 0.06. For $p_T>$ 0.1 GeV, $N_{ch}$ ≥ 20, $T_{01}$ and $T_{02}$ are 0.156 ± 0.01 and 0.111 ± 0.008, respectively, (decreasing behavior) while $⟨{\beta }_T⟩=$ 0.15 ± 0.06 (increase behavior); whereas, the values of the $n_1$ and $n_2$ are the same within the errors. Finally, with special track cuts, $p_T>$ 2.5 GeV, $N_{ch}\ge$ 1, $T_{01}$, and $T_{02}$ are 0.120 ± 0.01, and 0.094 ± 0.008, respectively, (decreasing behavior) while $<{\beta }_T>=$ 0.12 ± 0.07 (decrease behavior). In the case of the parameters extracted from fitting the simulations, EPOS and Sibyll have smaller values for $T_{01}$ and $T_{02}$, while PYTHIA8 and QGSJET have higher values. The values of $<{\beta }_T>$, $n_1$, and $n_2$ for the models are similar to the data in all cases except for QGSJET, which yields higher values for the last two parameters. From this observation, we conclude that increasing the $N_{ch}$ increases the temperature ($T_{01}$ and $T_{02}$) values, while tightening the cut in $p_T$ will decrease the $<{\beta }_T>$. The values of n ($n_1$ and $n_2$) are independent of the properties of the tracks used in the current study.

Figure 4 shows the charged particles’ multiplicity distributions with the following characteristics: charged particles’ multiplicities with a reduced phase space in the momentum of $p_T>$ 500 and (a) $N_{ch}\ge$ 1, (c) $N_{ch}\ge$ 6, and (e) $p_T>$ 2500, $N_{ch}\ge$ 1. In addition to the above, an inclusive phase space in the momentum of $p_T>$ 100 MeV is used with (b) $N_{ch}\ge$ 2 and (d) $N_{ch}\ge$ 20. A black-filled dot represents the experimental data, and the models’ predictions are represented by the lines of different colors shown in the plot legends. Statistical uncertainties are represented by the vertical bars in the data points. The ratio of Monte Carlo results in the data is shown in the bottom insets.

PYTHIA8 has an excellent description of the data at low and high $N_{ch}$; however, a slight overshoot at 40 $\le N_{ch}\le$ 70 for the reduced phase space in $p_T$ is observed. Similarly, the EPOS model predicts the data for the inclusive phase space of $p_T$ > 0.1 GeV at the higher end of $N_{ch}$. At low $N_{ch}$, the model predicts more events than the data compensated by the prediction of fewer events at the middle $N_{ch}$. The latter is due to the normalization ($1/N_{ev}$); a higher prediction in one region is balanced by predicting fewer events in another region. Furthermore, both the models predict better for the higher cut in $N_{ch}$ = 20 than the $N_{ch}$ = 2. The other two models, i.e., Sibyll and QGSJET, have an oscillatory prediction for the number of events compared to the experimental data. As before, the lower prediction of events in one region is compensated by a higher prediction in the other and vice versa. For $p_T$ > 2.5 GeV, the latter two models largely underpredict the data, with QGSJET being the worst.

Figure 5 shows the track-based relation of the average $p_T$ to the number of charged particles in an event. The first plot (top right) is the $<p_T>$ versus $N_{ch}$ for $p_T$ > 500 and $N_{ch}$ > 1, while the second plot (top left) is the $<p_T>$ versus $N_{ch}$ for $p_T$ > 100 and $N_{ch}$ > 2. Finally, the bottom-centered plot is the $<p_T>$ versus $N_{ch}$ for $p_T$ > 2500 and $N_{ch}$ > 1.

The PYTHIA8 model has an excellent prediction for the case when $N_{ch}$ > 1 became better with a higher cut in $p_T$. For $N_{ch}$ > 2 and $p_T$ > 100 (loose cut), PYTHIA8 overpredicts the experimental data by about 10 %. The other models overshoot at lower $N_{ch}$ while undershooting at higher $N_{ch}$ for the first two plots. For $<p_T>$ versus $N_{ch}$ for $p_T$ > 2500 and $N_{ch}$ > 1, QGSJET predicts slightly higher values for $N_{ch}$ up to 10 range but overshoots afterward. On the other hand, the Sibyll model reproduces good results at low and high $N_{ch}$ but overpredicts at median values. Overall, the PYTHIA8 model reproduces the experimental data well.

The measurements made in [21] are also compared with the PHOJET and different tunes of the PYTHIA8 model, where the models underpredicted the data at most of the distributions. In our case, PYTHIA8 reproduced the distributions very well. We mostly used default parameters except for the $pTHatMin$ parameter, whose value is set to 4.3. The parameter is found to scale the data. We found the best result of the PYTHIA8 model at the above-quoted value.

4. Summary and Conclusions

In the current study, we presented the properties of the track-based events in different phase space regions. The multiplicity distributions of primarily charged particles are studied at 7 TeV and measured by the ATLAS in comparison with the prediction of several MC event generators. The spectra obtained by the experimental data were compared to the simulations EPOS, PYTHIA8, Sibyll, and QGSJET under the same condition. It has been found that the PYTHIA8 model reproduces the measurements well in most of the distributions but depends on the momentum and the requirement of charged particles in a given track. The EPOS model could also reproduce some spectra regions and presents a good comparison after PYTHIA8. All other models could not produce most of the spectra, but only in a limited region that also depends on the analysis’s cuts. The Pythia8 model has a better prediction due to its additional feature, including the color reshuffling of quarks and gluons through the colour re-connection modes and the possiblity of initial and final state radiations by incorporating the parton showers.

In addition to the comparison of simulation models to the experimental measurements, we used fit functions to describe the track-based $p_T$ spectra of the particles under study. The fit is applied to the data and models, and their results are compared.

We extract the parameter $T_{01}$ from the fit function using Equation (1), which represents the $<T_{eff}>$ of the interacting system and is affected by the thermal motion and transverse directed motion; thus, $T_{01}$ is an estimate of the kinetic freeze-out temperature, which is mainly contributed by the thermal motion. The parameter $T_{02}$ extracted by the Hagedorn function is primarily affected by the thermal motion, where the motion in the transverse direction is shown by the average transverse flow velocity $⟨{\beta }_T⟩$; thus, $T_{02}$ is a more accurate estimate of the kinetic freeze-out temperature than $T_{01}$. It can be observed from Table 1 that $T_{02}$ has smaller values than the corresponding values of $T_{01}$, as $T_{02}$ is only affected by the thermal motion. The parameter values for PYTHIA8 and EPOS are nearer to the values extracted from the experimental data, supporting a better prediction of these models than others.

Furthermore, it is worth mentioning that a clear difference between the measurements and models’ predictions has also been reported in the source paper [21]. In our presented results, a good prediction was obtained from the comparison of the PYHTIA8 model and the experimental data, which is due to the model features explained above.

Author Contributions

Formal analysis, M.W., M.A., A.A.K.H.I., M.U.M., M.A.K., A.M.K., A.-H.A.-A., H.-R.W. and H.I.A.; Funding acquisition, A.A.K.H.I.; Investigation, M.A., M.W., A.A. and M.A.K.; Project administration, M.W., M.A. and A.A.K.H.I.; Resources, M.W., M.A. and A.A.K.H.I.; Software, A.M.K. and H.-R.W.; Supervision, A.A., M.A. and A.A.K.H.I.; Validation, M.W., M.A. and A.A.K.H.I.; Writing—original draft, M.A., A.-H.A.-A., H.I.A., A.A.K.H.I. and A.A.; Writing—review and editing, M.W., M.A., A.A.K.H.I. and M.U.M. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

We would like to acknowledge the support of Ajman University, Internal Research Grant No: [DGSR Ref. 2022-IRG-HBS-9]. This work is supported by the Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R106), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. Moreover, we would like to acknowledge the Deanship of Scientific Research of Jazan University Jazan for supporting the research.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

We have used the following abbreviations in this manuscript:

$p_T$	Transverse momentum
$T_{eff}$	Kinetic freeze-out temperature
$<T_{eff}>$	Effective temperature
${\beta }_T$	Transverse flow velocity
$<{\beta }_T>$	Average transverse flow velocity
MC	Monte Carlo
LHC	Large Hadron Collider
$N_{ch}$	Number of charged particles

Footnotes

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Figures and Table

Figure 1. [Forumla omitted. See PDF.] dependence of the charged particles’ multiplicity for events (a) [Forumla omitted. See PDF.] 0.5 GeV with minimum of one charged particle in an event, (b) [Forumla omitted. See PDF.] 0.1 GeV with minimum of two particles in an event, (c) [Forumla omitted. See PDF.] 0.5 GeV with minimum of six charged particles in an event, (d) [Forumla omitted. See PDF.] 0.1 GeV with minimum of twenty charged particles in an event, and (e) [Forumla omitted. See PDF.] 2.5 GeV with minimum of one charged particle in an event. The data measurements are represented by the dots in the figure, while the curves represent the results of the simulation models. Statistical uncertainties are represented by the vertical bars. The ratio of MC results in the data is shown in each panel.

Figure 2. [Forumla omitted. See PDF.] distributions of charged particle multiplicities for events (a) [Forumla omitted. See PDF.] 0.5 GeV with minimum of one charged particle in an event, (b) [Forumla omitted. See PDF.] 0.1 GeV with minimum of two particles in an event, (c) [Forumla omitted. See PDF.] 0.5 GeV with minimum of six charged particles in an event, (d) [Forumla omitted. See PDF.] 0.1 GeV with minimum of twenty charged particles in an event, (e) [Forumla omitted. See PDF.] 2.5 GeV with minimum of one charged particle in an event. The data points are shown by dots in the plot, while the lines having different colors are used to show the models’ predictions. Statistical errors are shown by the vertical bar. The ratio of Monte Carlo results to the data is given in each panel.

Figure 3. Results of the fit function for the [Forumla omitted. See PDF.] spectra of charged particles in events for which (a) [Forumla omitted. See PDF.] 0.5 GeV with minimum of one charged particle in an event, (b) [Forumla omitted. See PDF.] 0.1 GeV with minimum of two particles in an event, (c) [Forumla omitted. See PDF.] 0.5 GeV with minimum of six charged particles in an event, (d) [Forumla omitted. See PDF.] 0.1 GeV with minimum of twenty charged particles in an event, and (e) [Forumla omitted. See PDF.] 2.5 GeV with minimum of one charged particle in an event. The different symbols represent the experimental data [21] and the simulation results of four MC models. For clarity, each result is multiplied with a suitable factor. The red and blue dash curves show the Tsallis–Pareto-kind and Hagedorn function fitting the spectra, respectively.

Figure 4. Distributions of charged particle multiplicities for (a) [Forumla omitted. See PDF.] GeV with minimum of one charged particle in an event, (b) [Forumla omitted. See PDF.] GeV with minimum of two particles in an event, (c) [Forumla omitted. See PDF.] GeV with minimum of six charged particles in an event, (d) [Forumla omitted. See PDF.] GeV with minimum of twenty charged particles in an event, and (e) [Forumla omitted. See PDF.] GeV with minimum of one charged particle in an event. The black dot shows the data in the plot, while the colored lines represent the results of the simulation models. Statistical uncertainties are represented by the vertical bars. The ratio of Monte Carlo results in the data is shown in the bottom insets.

Figure 5. The track-based relation of the average [Forumla omitted. See PDF.] to the number of charged particles in an event is shown here. (b) is the [Forumla omitted. See PDF.] versus [Forumla omitted. See PDF.] for [Forumla omitted. See PDF.] > 0.5 GeV with minimum of one charged particle in an event, while (a) is the [Forumla omitted. See PDF.] versus [Forumla omitted. See PDF.] for [Forumla omitted. See PDF.] > 100 with minimum of two charged particles in an event. Finally, (c) is the [Forumla omitted. See PDF.] versus [Forumla omitted. See PDF.] for [Forumla omitted. See PDF.] > 2500 and with minimum of one charged particle in an event. Black dots show the data, whereas colored lines show the model’s prediction, as given in the plot legend. Each plot is followed by the ratio of the MC by the data (MC/Data).

Table 1

The values extracted by the fit functions using Equations (1) and (2) and ${\chi }^2$/dof for the charged particles’ $p_T$ distributions for different events, as given in Figure 3.

Figure	Type	${\mathit{T}}_{01}$	${\mathit{n}}_1$	${\mathit{\chi }}_1^2$/dof	$⟨{\mathit{\beta }}_{\mathit{T}}⟩$	${\mathit{T}}_{02}$	${\mathit{n}}_2$	${\mathit{\chi }}_2^2$/dof
		(GeV)			(c)	(GeV)
	Data	0.138 ± 0.01	6.4 ± 0.2	12.052/33	0.12 ± 0.06	0.101 ± 0.008	6.4 ± 0.2	12.443/32
	EPOS-LHC	0.134 ± 0.01	6.2 ± 0.2	33.979/33	0.12 ± 0.06	0.099 ± 0.008	6.2 ± 0.2	37.297/32
Figure 3a	PYTHIA8.307	0.145 ± 0.01	6.4 ± 0.2	26.594/33	0.12 ± 0.06	0.109 ± 0.008	6.4 ± 0.2	25.143/32
	Sibyll2.3d	0.125 ± 0.01	5.6 ± 0.2	103.796/33	0.12 ± 0.06	0.090 ± 0.008	5.6 ± 0.2	153.478/32
	QGSJETII-04	0.151 ± 0.01	7.0 ± 0.2	221.407/33	0.12 ± 0.06	0.110 ± 0.008	6.9 ± 0.2	266.848/32
	Data	0.146 ± 0.01	6.4 ± 0.2	24.256/41	0.15 ± 0.06	0.101 ± 0.008	6.3 ± 0.2	22.227/40
	EPOS-LHC	0.145 ± 0.01	6.3 ± 0.2	139.942/41	0.15 ± 0.06	0.102 ± 0.008	6.3 ± 0.2	137.161/40
Figure 3b	PYTHIA8.307	0.158 ± 0.01	6.5 ± 0.2	92.206/41	0.14 ± 0.06	0.110 ± 0.008	6.4 ± 0.2	88.329/40
	Sibyll2.3d	0.150 ± 0.01	6.0 ± 0.2	230.685/41	0.14 ± 0.06	0.110 ± 0.008	6.0 ± 0.2	251.161/40
	QGSJETII-04	0.170 ± 0.01	7.2 ± 0.3	221.959/41	0.14 ± 0.06	0.132 ± 0.008	7.2 ± 0.3	147.431/40
	Data	0.162 ± 0.01	6.5 ± 0.2	35.926/33	0.13 ± 0.06	0.122 ± 0.008	6.5 ± 0.2	33.346/32
	EPOS-LHC	0.147 ± 0.01	6.3 ± 0.2	68.705/33	0.11 ± 0.06	0.113 ± 0.008	6.3 ± 0.2	82.078/32
Figure 3c	PYTHIA8.307	0.160 ± 0.01	6.5 ± 0.2	25.926/33	0.11 ± 0.06	0.122 ± 0.008	6.5 ± 0.2	25.215/32
	Sibyll2.3d	0.142 ± 0.01	5.8 ± 0.2	88.605/33	0.11 ± 0.06	0.108 ± 0.008	5.8 ± 0.2	127.931/32
	QGSJETII-04	0.167 ± 0.01	7.1 ± 0.3	76.731/33	0.11 ± 0.06	0.130 ± 0.008	7.1 ± 0.3	87.593/32
	Data	0.156 ± 0.01	6.6 ± 0.2	23.171/41	0.15 ± 0.06	0.111 ± 0.008	6.5 ± 0.2	28.566/40
	EPOS-LHC	0.150 ± 0.01	6.3 ± 0.2	169.631/41	0.14 ± 0.06	0.110 ± 0.008	6.3 ± 0.2	150.190/40
Figure 3d	PYTHIA8.307	0.165 ± 0.01	6.5 ± 0.2	59.013/41	0.14 ± 0.07	0.123 ± 0.008	6.5 ± 0.3	48.746/40
	Sibyll2.3d	0.155 ± 0.01	5.9 ± 0.2	204.494/41	0.14 ± 0.06	0.113 ± 0.008	5.9 ± 0.2	195.757/40
	QGSJETII-04	0.164 ± 0.01	6.8 ± 0.2	226.862/41	0.14 ± 0.07	0.123 ± 0.008	6.8 ± 0.2	205.024/40
	Data	0.120 ± 0.01	6.4 ± 0.2	8.526/13	0.12 ± 0.07	0.094 ± 0.008	6.5 ± 0.2	6.841/12
	EPOS-LHC	0.122 ± 0.01	6.1 ± 0.2	4.573/13	0.10 ± 0.07	0.090 ± 0.008	6.1 ± 0.2	5.272/12
Figure 3e	PYTHIA8.307	0.122 ± 0.01	6.4 ± 0.2	2.646/13	0.11 ± 0.07	0.092 ± 0.008	6.4 ± 0.2	2.778/12
	Sibyll2.3d	0.107 ± 0.01	5.6 ± 0.2	11.510/13	0.10 ± 0.07	0.080 ± 0.008	5.6 ± 0.2	14.031/12
	QGSJETII-04	0.128 ± 0.01	7.0 ± 0.2	12.894/13	0.10 ± 0.07	0.100 ± 0.008	7.0 ± 0.2	13.911/12

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