#### Abstract

The strangeness enhancement signature of QGP formation at LHC energies is carefully tackled in the present study. Based on HRG, the particle ratios of mainly strange and multistrange particles are studied at energies from lower up to 13 TeV. The strangeness enhancement clearly appeared at more high energies, and the ratios are confronted to the available experimental data. The particle ratios are also studied using the Cosmic Ray Monte Carlo (CRMC) interface model with its two different event generators, namely, EPOS 1.99 and EPOSlhc, which show a good agreement with the model calculations at the whole range of the energy. We utilize them to produce some particles ratios. EPOS 1.99 is used to estimate particle ratios at lower energies from AGS up to the Relativistic Heavy Ion Collider (RHIC) while EPOSlhc is used at LHC energies. The production of kaons and lambda particles is studied in terms of the mean multiplicity in p-p collisions at energies ranging from 4 to 26 GeV. We find that both HRG model and the used event generators, EPOS 1.99 and EPOSlhc, can describe the particle ratios very well. Additionally, the freeze-out parameters are estimated for different collision systems, such as p-p and Pb-Pb, at LHC energies using both models.

#### 1. Introduction

One of the most important signatures of the phase transition between the hadronic matter “confined phase” to the quark-gluon plasma (QGP) “deconfined phase” is the strangeness enhancement, in other words, the production of strange particles [1]. The abundance of “s” quark is a useful tool to analyze the heavy-ion and proton-proton collisions. Additionally, abundance of strangeness is considered as an abundance in the degree of freedom.

The strangeness enhancement is combined with gluon existence in QGP, in which the gluon dissolves to a pair of strange quarks rapidly [2]. An early study explored the phase transition from hadronic matter to QGP [3] and postulated the idea of chemical and thermal equilibrium which, in turn, developed the explanation of thermodynamics at high temperature and various kinds of chemical potentials.

It is noted that QGP comprises an equal number of strange and antistrange quarks [4]. Therefore, the density of strange quarks raises, and more multistrange hadrons are produced. This occurs during the hadronization process [5]. Recently, a good review [6] handled the strangeness production as a signature for QGP formation. The theoretical and experimental procedures are discussed besides tackling the nonstrange signature of QGP such as suppression.

In the present work, the hadron resonance gas (HRG) model is utilized as a powerful tool to analyze the production of hadrons resulting from various heavy-ion experiments such as AGS, SPS [7, 8], RHIC [9, 10], and LHC [11]. This is in addition to the previous work that investigated RHIC, LHC, and NICA energies [12, 13].

An earlier work [1] studied the strange and nonstrange production in the framework of excluded volume model which fits good with the experimental data. The strange to nonstrange ratios are analyzed, in particular the kaon to pion and to pion ratios in a canonical ensemble [14]. The results showed an effect of the system size, and as a consequence, the peak (horn) of such ratios is noticed at different energies. Another interesting work of the system-size dependence of hadrochemistry is applied [15]. There is an enhancement of multistrange hadrons in high-multiplicity pp collisions [16].

In addition, the particles have a vital role in heavy-ion collisions [17] throughout the evolution process. This could be attributed to their short life times (, ), respectively, that facilitate analyzing the system at various times. The enhanced contribution of these particles is essential due to the strangeness enhancement.

The particle production is studied [18] in lead-lead (Pb-Pb) and proton-proton (p-p) collisions at nucleon center of mass energy (= TeV). For a good knowledge, a pedagogic review in strangeness enhancement and papers therein is provided [19].

The main target of the present work is to investigate the strangeness enhancement in terms of various particle ratios such as , , , , , , , , and and strange and multistrange particles such as , , , , , and from low to high energies using different models, namely, HRG, EPOS , and EPOSlhc. Both event generators, EPOS and EPOSlhc, are executed through the CRMC interface model to produce the abovementioned particles for an ensemble of 100,000 events where the fusion option is turned on. The production of kaons and lambda particles is studied in terms of the mean multiplicity in p-p collisions at energies ranging from 4 to 26 GeV. First, we have used both event generators to produce well-identified particles and comparing the obtained results with the available experimental data. This encourages us to use it for a production of particles which have no experimental data. Also, the freeze-out parameters, i.e., the temperature () and baryon chemical potential (), are estimated as a result of fitting the obtained results from the HRG model of a combination of the used particle ratios with both LHC and EPOSlhc results. The obtained values of and are compared to those presented in Ref. [20].

The present study is organized as follows: in Section 2, the main equations of the hadron gas model are discussed. A general introduction about the event generator is presented in Section 3. Section 4 presents the obtained results. Finally, the conclusion is represented in Section 5.

#### 2. Formalism

In the present work, the grand canonical ensemble (GCE) is used in the framework of the HRG model. In GCE ensembles, the energy exchanges freely with the surrounding medium, so that the number of particles is no longer fixed. Such a system possess thermodynamic properties which can be obtained from the GCE partition function. The GCE has rigorous conserved quantum numbers such as the charge, strangeness, and baryon quantum numbers. Thus, the GCE partition function is defined as follows [21, 22]: where is the Hamiltonian, is the different conserved charges, are the corresponding chemical potentials, and is natural units (). The Hamiltonian in HRG includes all the degree of freedom. Then, the partition function in the hadron resonance gas can be written as a sum of partition functions of hadrons and resonances as follows [21, 22]: where the signs refer to fermions and bosons, respectively; with as the mass of “” particle; and is the fugacity factor and is given by [21, 22] where , and are the strange, quark, and baryon chemical potentials, respectively, and , and are the corresponding quantum numbers for particle species “.” These quantities should fulfill the conservation laws such as strangeness, , and charge and baryon number, , where and are the atomic number and mass number of the colliding nuclei, respectively. The integration in equation (2) has been performed over “” resulting in Bessel function [22].

Therefore, the thermodynamic quantities can be obtained from equation(4). Then, the number density of particles is given by [22] where is the average number of the particles. In order to include all hadrons with their resonance decay, the average number can be rewritten as where the first term represents the average number of thermal particles of species and the second term represents all resonance contributions to the particle multiplicity of species where “Br” stands for the branching ratio for the decay from particle (). All particle ratios are calculated using equation (5).

#### 3. Cosmic Ray Monte Carlo (CRMC)

CRMC is an interface which gives access to various Monte Carlo event generators such as EPOS 1.99, EPOSlhc, SIBYLL 2.1/2.3, and QGSJet 01/II.03/II.04 [23–25]. CRMC provides a full background description taking into account the produced diffraction. It is built on various types of interactions which are depending on the Gribov-Regee model such as EPOS 1.99 and EPOSlhc.

EPOS 1.99 and EPOSlhc are designed to explain both cosmic and noncosmic air showers and could be used to describe data produced from various collision systems such as proton-proton “p-p” or proton-nucleus “p-A” or deuteron-nucleus “d-Au” gold. Others in [23] presented a phenomenological approach based on the parton model trying to understand different experiments by a unified approach. They introduced EPOS, which stands for energy-conserving quantum mechanical multiple scattering approach, based on partons (parton ladders), off-shell remnants, and splitting of parton ladders [23]. EPOS is a sort of Monte-Carlo (MC) generator valid for heavy ion interactions and cosmic ray air shower simulations [24]. EPOS is confronted to Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) data [23, 24].

Such (MC) models are essential to analyze the acceptance of the detector, the hadrons in the universe, and other impacted effect in astrophysics; all of them are confronted with high energy experiments [24]. In order to reproduce the LHC data [26–28] for p-p, p-Pb and Pb-Pb interactions, Pierog et al. [24] made the necessary modification in the model. There is another promising work [29] for the future analyzing the data from proton-oxygen (p-O) reaction at LHC energies. However, in the latter, they simulated the pseudorapidity spectra of charged pions, charged kaons, and protons at 13 TeV in p-p and p-O collisions at 10 TeV with CRMC.

In the present work, we utilize two different event generator EPOS 1.99 and EPOSlhc [30] at energies ranging from 0.001 up to 13 TeV for 100,000 events per energy to calculate the particle ratios , , , , , , , , , and strange and multistrange particles such as , , , , , and . EPOS 1.99 is performed at 7.7, 11.5, 19.6, 27, 39, 62.4, 130, and 200 GeV while EPOSlhc is executed at 0.9, 2.76, 5.02, 7, and 13 TeV for Pb-Pb collision. The resulting particle ratios are used to explain the strangeness enhancement signature.

#### 4. Results and Discussion

In this section, the obtained results of different particle ratios using the HRG model are presented from up to TeV. All results are compared with the available experimental data. For some suggested strange and multistrange particles, there is a lack of experimental data; thus, we used two different generators, i.e., EPOS 1.99 and EPOSlhc, to predict their results. Also, the freeze-out parameters, i.e., and , are estimated as a result of fitting the obtained results from the HRG model of a combination of the calculated particle ratios with both LHC data and EPOSlhc event generator results for two different collisions systems, i.e., p-p and Pb-Pb, at , 13 TeV, respectively. The obtained values of and are compared to the values presented in Ref. [20]. The calculated particle ratios as a function of various centers of mass energies are then used to explain the strangeness enhancement signature.

The first experimental data of strangeness enhancement in high-multiplicity pp collision is presented in [31] for strange and multistrange particles. These kick-off results motivated the authors of the current work to study the strange and multistrange particle enhancement. Additionally, they observed that there is a similarity in the strangeness production between and collisions for high-multiplicity events where the deconfined phase of matter (i.e., QGP) is formed. This conclusion is impacted again in different interesting works [20, 32]. The results are divided into three groups: (i)Particle multiplicity versus the center of mass energy

Strangeness enhancement is considered as a signal of deconfinement in the ultrarelativistic heavy-ion collisions where there is an enhancement of the yields of hyperons relative to that of p-p nucleus collisions [33]. In this section, the EPOS 1.99 event generator is used to predict the mean multiplicities of the strange particles, , , and , from p-p collisions at energies ranging from up to GeV in a rapidity range of as shown in Figure 1. The obtained results are confronted to those measured in NA61/SHINE experiment [33]. EPOS 1.99 event generator is succeeded very well to describe the multiplicity of as seen in Figure 1(a). In the case of , there is a small deviation at and 19 GeV as shown in Figure 1(b). The multiplicity of particle predicted by EPOS 1.99 event generator is shown in Figure 1(c) and has a good agreement with the experimental data taken from [33]. (ii)Particle ratios versus the center of mass energy

**(a)**

**(b)**

**(c)**

The particle ratios with the including some heavier and strange particles are calculated by the HRG model and both event generators, i.e., EPOS 1.99 and EPOSlhc, at various energies spanning from up to TeV. The dependence of the baryon chemical potential and the temperature on the center of mass energy is taken from [34], which has an agreement with the parameterization in [35].
where GeV and GeV^{−1}. The temperature can also be defined in terms of the center of mass energy [34].
where is taken in GeV and MeV. The quark structure of the strange and multistrange particles suggested here is listed in Table 1.

The dependence of different particle ratios on at LHC energies from GeV to TeV is studied utlizing the HRG model. Figure 2 illustrates the ratios of strange to nonstrange particles (upper panel) such as and pure nonstrange and strange ratios (lower panel) for versus the center of mass energy. These ratios are confronted to the experimental data [36–38], and EPOS 1.99 (used at low energies) and EPOSlhc (used at high energies) event generators from TeV. Figure 2(a) shows the important particle ratio of which is used as characterising tool to describe the strangeness enhancement in the quantum chromodynamic (QCD) matter. This ratio shows a peak at GeV which is known as the horn puzzle and might be considered as an indication of the QCD phase transition. The EPOS 1.99 event generator can describe the lowest NA61/SHINE data produced from p-p collisions at the center of mass energy and 12.5 GeV and the highest STAR Pb-Pb collisions at GeV while the EPOSlhc event generator can describe the ALICE data at TeV. The wide range of energy shows the expected results in which there is a rapid enhancement in the strange particles only in the ratios () as in Figures 2(b) and 2(d). However, Figure 2(a) shows a monotonic increasing (horn) up to GeV, then begin to decrease with increasing the energy, and a clear deceasing in the pure nonstrange particles with increasing the energy such as Figure 2(c).

**(a)**

**(b)**

**(c)**

**(d)**

Figure 3 presents the energy dependence of the particle ratios , , and in comparison with the experimental data taken from [36–38] and the estimated results from both EPOS 1.99 and EPOSlhc event generators. We notice that the horn puzzle appears again in the ratio still at the range of GeV energy.

**(a)**

**(b)**

**(c)**

Figures 4 and 5 show a series of strange and multistrange particles such as , , , , , , , and which are calculated in the framework of the HRG model and compared with the results obtained from both EPOS 1.99 and EPOSlhc event generators. It is clear that most of the strange and multistrange particles show strangeness enhancement as the energy increases up to 13 TeV. The [18] ratio shows a rapid enhancement at energies in GeV and smoothly increases at TeV. This ensures that the strangeness enhancement is a strong signature for the quark gluon plasma (QGP) creation at very high energy. (iii)Fitting -tuning

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

Recently, fitting with particle ratios for both and collisions has been made in Ref. [32, 47]. They found that the HRG model fits very well and the grand canonical description is valid for the highest multiplicities. Figure 6 shows a fitting between the calculated particle ratios, i.e., particles shown in Table 2, from the HRG model using equation (5), and both LHC data [20] and EPOSlhc event generator results, using equation (9) in and collision systems at energies 5.02 and 13 TeV, respectively.

**(a)**

**(b)**

It is noticed that at 5.02 TeV for collision, the theoretical results are rather matched with the experimental data compared to the fitted one in a previous study [20]. In the future work, we would focus on p-p collision [48] at LHC 7 TeV and high multiplicity and for collisions at 2.76 TeV energies [49–52] due to its importance in the studying of the hadronic matter under extreme conditions.

The extracted chemical freeze-out temperature and baryon chemical potential are shown in Table 3. It seems that, hadronization occurs at 135 MeV and small 0.6 MeV as compared to a previous study [20] in which hadronization occurred at 140-160 MeV and (dof: degree of freedom). However, in Ref. [20], the fitted particle ratios are 3 only; the present work is closer to the minimum fitted temperature MeV. On the other hand, the expected critical temperature in [53] is MeV. (iv)Analysis of particle ratios

The statistical fit between the obtained results from the HRG model and both LHC data and EPOSlhc event generator are done as follows: where and represent the experimental and computed values of the particle ratios, respectively. is the error in the experimental results.

In the present work, all hadrons and resonances are implemented from the particle data group (PDG) up to 11 GeV [17]. The main point in Table 3 is that the data agree fairly well with Pb-Pb collision but shows a great discrepancy for p-p collision. This note is marked in an earlier work [32], between the different system sizes and using the canonical and grand canonical ensemble, in which the latter was used in the present work.

#### 5. Conclusion

In the present work, various well-identified, strange, and multistrange particle ratios as a function of different centers of mass energies are studied using various models, namely, HRG model, EPOS 1.99, and EPOSlhc event generators. The aim of using both event generators is to predict the particle ratios that are not measured yet. The obtained results from both event generators and the HRG model are compared with the available experimental data and show a good agreement along the whole range of the considered . The strangeness enhancement is studied in terms of the strange particle multiplicities and the dependence of the center of mass energies of the mentioned particle ratios. The production of particles that contain one strange quark or multistrange quarks is enhanced. The ratio of strange particles is doubled from 0.5 to 1 at energies to 13 TeV; this is clearly shown in Figures 4 and 5. Such particles have no quarks in the colliding nuclei (Pb-Pb) or colliding nucleons (p-p). Therefore, the enhancement of these particles is considered a strong probe for the QGP formation. Particularly, the strange quarks may be produced from the deconfinement of matter phase. For more investigation of strangeness enhancement, EPOSlhc event generator is used alongside with the HRG model for strange, nonstrange, and multistrange particles; it is elaborated for Pb-Pb collision.

Fitting the HRG results with the experimental data has an important impact of using canonical and grand canonical ensemble, in which the latter has a global conservation of the different quantum numbers. The comparison of different ensembles has been carried out [32], and it was concluded that there are discrepancies between the different ensembles and with the various collision system sizes such as , , and . This final conclusion motivates the authors to extend this work to study the different ensembles at LHC energies in TeV range.

#### Data Availability

Data supporting this systematic review or meta-analysis are from previously reported studies and datasets, which have been cited.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This paper has been assigned the permanent arXiv identifier 2105.14303.