Jun-Qi Tao · Hong-Bin He · Hua Zheng · Wen-Chao Zhang · Xing-Quan Liu · Li-Lin Zhu · Aldo Bonasera

Received: 11 July 2023 / Revised: 21 September 2023 / Accepted: 30 September 2023 / Published online: 18 November 2023? The Author(s) 2023

Abstract Thepseudo-rapiditydistributionsof the chargedparticlesproduced in theasymmetriccollisionsystemsp+Al,p+Auand 3 He+Au atGeVareevaluated intheframeworkofafireballmodelwithTsallis thermodynamics.Thefireball model assumes that the experimentally measured particles are produced by fireballs following the Tsallis distribution and it can effectively describe the experimental data.Our results as well as previous results for d+Au collisions atGeV and p+Pb collisions atTeV validate that the fireball model based on Tsallis thermodynamics can provide a universal framework for pseudo-rapidity distribution of the charged particles produced in asymmetric collision systems.We predict the centrality dependence of the total charged particle multiplicity in the p+Al, p+Au and 3He+Au collisions.Additionally, the dependences of the fireball model parameters ( y0a , y0A , σa and σA ) on the centrality and system size are studied.

Keywords Tsallis thermodynamics · Fireball model · Pseudo-rapidity distribution · Heavy-ion collisions · Charged particles

1 Introduction

High-energy heavy-ion collisions provide a unique way to understand the origin of the universe.However, their processes cannot be directly observed in experiments.We can only study the collision process indirectly by analyzing the properties of the final particles produced in the collisions.The pseudo-rapidity distribution of charged particles is one of the important experimental observables.The study of this observable could lead to a better understanding of the properties of the particles produced in the collisions, the particle production mechanism and so on.There have been numerous works in previous studies using different models, such as HIJING [1], AMPT [2—4], EPOS-LHC [5], a multi-source thermal model [6, 7], a new revised Landau hydrodynamics model [8], a 1 + 1-dimensional hydrodynamics model[9, 10], a dynamical initial state model coupled to (3 + 1)D viscous relativistic hydrodynamics [11] and so on, to analyze the existing experimental data of pseudo-rapidity distributions of the charged particles [12—30].Although these models are based on different physical ideas, valuable physical information on the collision process has been extracted and learned.

Recently, a fireball model based on Tsallis thermodynamics was utilized to analyze the pseudo-rapidity distribution of charged particles measured in high-energy heavy-ion

The paper is organized as follows.In Sect.2, the fireball model with Tsallis thermodynamics is briefly introduced.In Sect.3, the fitting results of the fireball model and the total charged particle multiplicities extracted from the fireball model are shown.The dependences of the model parameters on the centrality and size of the collision systems are also presented.A brief conclusion is drawn in Sect.4.

2 Theoretical descriptions

In the self-consistent Tsallis thermodynamics, the Tsallis distribution is proposed as a generalization of the Boltzmann-Gibbs distribution [34].To describe the transverse momentum spectrum of particles, the Tsallis distribution is written as [31—33]

wheregis the particle state degeneracy,Vis the volume,is the transverse mass andm0is the particle rest mass,yis the rapidity,qis the entropic factor, which measures the non-additivity of the entropy [34, 35],μis the chemical potential andTis the temperature.The Boltzmann distribution is recovered whenq=1.We takeμ=0 because the multiplicities ofπ+andπ-are equal and they are the majority of particles produced in the collision systems considered.For the middle rapidityy≈0 , Eq.(1) can be rewritten as

The parametersqandTare extracted from the experimental transverse momentum spectrum of the particles.

In the fireball model with Tsallis thermodynamics [31—33],the particles measured in the experiment were produced by fireballs following Tsallis distribution Eq.(1).The density distribution of these fireballs in the rapidity space isν(yf) ,whereyfis the rapidity of the fireball.Therefore the transverse momentum spectrum of particles can be written as

whereNis the total particle multiplicity andAis the normalization constant such that

Sometimes, the experimental data are measured in the pseudo-rapidityηspace.To describe the experimental data, we substitute the relation between rapidity and pseudorapidity [36]

into Eq.(3) and integrate the transverse momentum in the equation to obtain [32, 33]

where

In this paper the asymmetric collision systems are studied, and the distributionν(yf) is assumed to be the sum of two asymmetricq-Gaussian functions [33],

wherey0a(A)andσa(A)are the centroid position and width of the fireball distribution in the direction of the light (heavy)nucleus beam, respectively.The normalization of Eq.(8) is handled by the normalization constantAin Eq.(3).xis the parameter to characterize the extent of asymmetry, which was first proposed in our previous work [33].In this work,we takeq′=qas in [31—33].A representative figure of Eq.(8) is shown in Appendix 1 with the parameters obtained for the p+Al collisions at 0-5% centrality and

3 Results and discussion

Fig.1 (Color online) The pseudo-rapidity distributions of the charged particles produced in p+Al collisions at =200 GeV for different centralities.The symbols are experimental data taken from [30].The curves are the results from Eqs.(6) and (8)

Fig.2 (Color online) Same as Fig.1, but for p+Au collisions at =200 GeV

property of the collision system.We take the parametersTandqfrom the closest centrality when the centrality of the particle transverse momentum spectrum and centrality of the charged particle pseudo-rapidity distribution are not the same.These two parameters and the fireball model with Tsallis thermodynamics, Eqs.(6) and (8), are then utilized to study the pseudo-rapidity distribution of the charged particles produced in the collisions.The corresponding values ofxin Eq.(8) are also listed in Table 3 in Appendix 2.

Fig.3 (Color online) Same as Fig.1, but for 3He+Au collisions at =200 GeV

In Figs.1, 2 and 3, the results of the pseudo-rapidity distributions of the charged particles from the fireball model with Tsallis thermodynamics for different centrality bins in p+Al, p+Au and3He+Au collisions atGeV are shown.The fireball model effectively describes the experimental data within the errors.Notably, the data quality of the pseudo-rapidity distributions of the charged particles is not as good as that for the d+Au collisions atGeV shown in [33], i.e., in terms of larger errors, a fewer number of data points as well as a lower pseudo-rapidity coverage, which leads to larger uncertainties to the fireball model parameters and affects our analyses of the fireball model parameters versus collision centrality and the collision system size to some extent later in the following.The pseudo-rapidity distribution of the charged particles for centrality 5—10% is lower than the case for centrality 10—20% in some pseudo-rapidity regions for the3He+Au collisions, which is observed in Fig.3.A largerxat centrality 5—10% compared with the others for the3He+Au collisions is also observed in Table 3 in Appendix 2.We emphasize that the same fitting protocol is applied for all the pseudo-rapidity distribution data of the charged particles.Because the collision system is asymmetric, the pseudo-rapidity distribution of the charged particles has significant forward/backward asymmetry.Fewer particles are produced in the direction of the light nucleus (p,3He) beam compared to the heavy nucleus (Al, Au) beam.As the d+Au collision system atGeV and the p+Pb collision system atTeV we studied in [33], the pseudo-rapidity distributions of the charged particles produced by these collision systems also become more symmetric from the central to peripheral collisions.This is because the peripheral collisions for asymmetric collision systems are more similar to the symmetric p+p collisions according to collision geometry.

We then evaluate the centrality dependence of the total multiplicities of the charged particles produced in these collision systems.Integrating Eq.(6) over theη∈[-10,10] we obtain the total multiplicity of the charged particles for each centrality from the fireball model.Because the corresponding experimental data are not yet available, we only analyze the results extracted from the fireball model and treat them as predictions.Figure 4 shows the total multiplicities of the charged particles calculated from the fireball model versus the collision centralityc.c=0 represents the most central collisions, andc=1 represents the most peripheral collisions.It can be observed that the fitting function taken from[14] can effectively describe the centrality dependence of the total multiplicities of the charged particles.As the centrality changes from the central to peripheral collisions, fewer charged particles are produced.increasing centrality in the direction of the heavy nucleus beam.In the p+Al collision system,σaandσAhave opposite trends with increasing centrality to their counterparts in the other collision systems.These different patterns indicate the complex dynamics in the asymmetric collisions relevant to the combinations of the projectile and target as well as the collision energy, which needs more investigations.

Fig.4 Total charged particle multiplicities produced in the p+Al, p+Au and 3He+Au collisions at =200 GeV versus the collision centrality c.The squares are the fireball model results.The lines are the fitting results.The fitting function is from [14] and specified in the legend

Fig.5 (Color online) Centrality dependence of model parameters y0a , y0A , σa and σA in p+Al,p+Au and 3He+Au collision at =200 GeV.The lines are the linear fit results to guide the eyes

Fig.6 (Color online) Collision system size dependence of model parameters y0a , y0A ,σa , σA for p+p, p+Al (0-5%),p+Au (0-5%), d+Au (0-20%),3He+Au (0-5%) and Au+Au(0-6%) collisions at =200 GeV.The parameters for the p+p, d+Au and Au+Au collisions are taken from [33]

Table 1 Results of the linear fits are shown in Fig.5.The c represents the centrality

Figure 6 shows the collision system size dependence of the fireball model parameters atGeV.For collision systems other than p+p, the parameters of the most central collisions are considered.In the p+p and Au+Au collisions, the parametersy0a=y0A=y0,σa=σA=σ, wherey0andσare the rapidity centroid and width of fireball distribution in the symmetric collision system, as detailed in [33].It can be deduced that when the light nucleus is p,y0adecreases as the size of the heavy nucleus increases, whereasy0Ashows the opposite trend.This indicates that a larger heavy nucleus has stronger stopping power for p.When the heavy nucleus is Au,y0aincreases as the size of the light nucleus increases,whereasy0Ashows the opposite trend.This means that a larger light nucleus is more difficult to stop by Au but has a stronger stopping power for Au.For parametersσaandσA, no conclusive patterns are observed.We expect that more discussions can be added when the data quality of the pseudo-rapidity distributions of the charged particles is improved by experimentalists.These phenomena manifest the complex dynamics in the asymmetric collisions.

4 Summary

Appendix 1: Fireball distribution of Eq.(8)

A representative figure of the fireball distribution of Eq.(8) using the parameters obtained for the p+Al collisions at 0—5% centrality andGeV is shown in Fig.7.

Fig.7 (Color online) Fireball distribution with the parameters obtained from p+Al collisions for 0—5% centrality at=200 GeV is shown.a The value of x is varied; b the value of q′ is varied

Appendix 2: Parameters q, T and x

The parameters ofqandTextracted by fitting the transverse momentum spectrum of particles [37] with Tsallis distribution Eq.(2) as well as parameterxin Eq.(8) are listed in Tables 2 and 3.

Table 2 Parameters q and T for the p+Al, p+Au and 3He+Au collisions at =200 GeV for different centralities

Table 2 Parameters q and T for the p+Al, p+Au and 3He+Au collisions at =200 GeV for different centralities

System Centrality q T (GeV) χ2∕NDF p+Al 0—5% 1.092 ± 0.004 0.127 ± 0.012 0.129 0—20% 1.098 ± 0.004 0.115 ± 0.012 0.151 20—40% 1.101 ± 0.004 0.107 ± 0.010 0.288 40—60% 1.101 ± 0.004 0.108 ± 0.012 0.223 60—72% 1.103 ± 0.005 0.098 ± 0.013 0.232 p+Au 0—5% 1.089 ± 0.004 0.138 ± 0.012 0.261 0—20% 1.090 ± 0.004 0.138 ± 0.012 0.117 20—40% 1.095 ± 0.004 0.127 ± 0.012 0.122 40—60% 1.096 ± 0.004 0.123 ± 0.012 0.204 60—84% 1.105 ± 0.004 0.103 ± 0.010 0.164 3He+Au 0—5% 1.092 ± 0.004 0.133 ± 0.010 0.082 0—20% 1.095 ± 0.004 0.127 ± 0.011 0.066 20—40% 1.098 ± 0.004 0.118 ± 0.011 0.046 40—60% 1.101 ± 0.004 0.112 ± 0.011 0.164 60—84% 1.103 ± 0.004 0.104 ± 0.011 0.157

Table 3 Parameter x for the p+Al, p+Au and 3He+Au collisions at =200 GeV for different centralities

Table 3 Parameter x for the p+Al, p+Au and 3He+Au collisions at =200 GeV for different centralities

System Centrality (%) x p+Al 0—5 1.760 ± 0.796 5—10 1.313 ± 0.620 10—20 1.247 ± 0.596 20—40 1.430 ± 0.840 40—74 4.767 ± 4.522 p+Au 0—5 4.098 ± 1.540 5—10 2.318 ± 0.190 10—20 4.690 ± 1.637 20—40 3.173 ± 1.196 40—60 1.470 ± 0.194 60—84 1.092 ± 5.980 3He+Au 0—5 5.912 ± 1.926 5—10 13.055 ± 7.709 10—20 5.470 ± 1.900 20—40 5.414 ± 1.919 40—60 2.210 ± 0.957 60—88 1.184 ± 0.550

Appendix 3: The particle spectra fit

Fig.8 (Color online) Transverse momentum spectra of π0 produced in p+Al collisions at =200 GeV [37].The curves are the corresponding fittings using Eq.(2)

Author contributions All authors contributed to the study conception and design.Material preparation, data collection and analysis were performed by Jun-Qi Tao, Hong-Bin He, Hua Zheng, Wen-Chao Zhang,Xing-Quan Liu, Li-Lin Zhu and Aldo Bonasera.The first draft of the manuscript was written by Jun-Qi Tao and Hua Zheng, and all authors commented on previous versions of the manuscript.All authors read and approved the final manuscript.

Data availability The data that support the findings of this study are openly available in Science Data Bank at https:// doi.org/ 10.57760/ scien cedb.j00186.00293 and https:// cstr.cn/ 31253.11.scien cedb.j00186.00293.

Declarations

Conflict of interest The authors declare that they have no competing interests.

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