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Article

A System Size Analysis of the Fireball Produced in Heavy-Ion Collisions †

1
Joint Institute for Nuclear Research, Dubna 141980, Russia
2
Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia
3
Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering, 077125 Magurele, Romania
4
Dalat Nuclear Research Institute, Vietnam Atomic Energy Institute, Dalat 670000, Vietnam
*
Author to whom correspondence should be addressed.
This paper is based on the talk at the 13th International Conference on New Frontiers in Physics (ICNFP 2024), Crete, Greece, 26 August–4 September 2024.
Particles 2025, 8(1), 34; https://doi.org/10.3390/particles8010034
Submission received: 29 December 2024 / Revised: 5 March 2025 / Accepted: 14 March 2025 / Published: 19 March 2025

Abstract

:
One of the main interests of high-energy physics is the study of the phase diagram and the localization of phase transitions from hadronic to quark–gluonic matter. There are different techniques to study the hot matter. One of them is femtoscopy, which uses two-particle correlations to extract spatiotemporal characteristics of the emission source. Another approach involves obtaining thermodynamic parameters from the momentum distributions of produced particles based on various theoretical models. In this research, we perform a comparative analysis of femtoscopic volumes and volumes obtained using the Tsallis statistical fit. This analysis allows us to estimate system size at the time of kinetic freeze-out and its dependence on collision centrality and energy. We observe that at high energies, the volume values estimated taking the two approaches diverge significantly, while at low energies, they are more consistent. In the future, these results can help to combine these two different methods and provide a more comprehensive picture of the fireball produced in heavy-ion collisions.

1. Introduction

Recently, a large amount of experimental data have been collected in high-energy physics for studying the properties of nuclear matter. One of the main interests is the study of phase transitions from regular hadronic matter to a deconfined state called quark–gluon plasma (QGP) [1,2,3,4]. This transition can occur at high temperatures and densities and can be shown on the Quantum ChromoDynamics (QCD) phase diagram in terms of temperature and baryonic chemical potential.
At low baryonic chemical potential, lattice calculations indicate the presence of a smooth transition—a crossover [5,6]. However, at higher chemical potential, lattice calculations are not possible, and effective models have to be used instead. These models suggest the presence of a first-order phase transition and the critical point separating it from the crossover.
QGP can be formed in ultrarelativistic heavy-ion collisions. After its formation, the hot matter expands and cools, which leads to the transition into the hadronic phase and reaches a state of statistical equilibrium until chemical freeze-out, when inelastic interactions between particles cease. Kinetic freeze-out occurs when hadrons stop interacting elastically, and their kinematic properties remain unchanged.

2. Materials and Methods

In our research, we use the published data from the STAR experiment obtained during the first stage of the Beam Energy Scan program at RHIC for Au+Au collisions at s NN = 7.7 , 11.5 , 19.6 , 27 and 39 GeV, aimed at localizing the phase transition boundary [7,8].
The main goal of this research is to estimate the size of the system formed in heavy-ion collisions using different methods and to compare the results obtained through different observables.
The first method is based on the application of a statistical model using Tsallis statistics [9,10,11]. The fireball is considered a thermodynamic system in the grand canonical ensemble (GCE), and thermodynamic characteristics, such as temperature and system volume, are extracted from the analysis of transverse momentum spectra.
The second method relies on femtoscopy. The interferometry method was initially proposed by Hanbury Brown and Twiss [12] in astrophysics and was later adapted for use in particle physics [13,14,15]. It has been applied in studies across a wide range of energies [16,17,18,19] to extract spatiotemporal properties of the homogeneity region at kinetic freeze-out by analyzing two-particle correlations of final particles.

2.1. Tsallis-3 Statistics

Tsallis entropy is defined as [9]
S = i p i q p i 1 q , i p i = 1 ,
where p i is the probability of i-th microscopic state of the system and q [ 0 , ] is the entropic parameter. In the Gibbs limit ( q 1 ), the Tsallis entropy (1) recovers the Boltzmann–Gibbs entropy [9]:
S = i p i ln p i .
We can see that the Tsallis entropy generalizes the Boltzmann–Gibbs entropy, with the parameter q indicating the degree of deviation of a given generalized probability distribution from the Boltzmann–Gibbs one. There are a few variants of the Tsallis statistics [10].
In the Tsallis-3 statistics (Tsallis statistics with escort probabilities) [10,11] in the grand canonical ensemble, the thermodynamic potential Ω of the system can be written as
Ω = H T S μ N ,
where H = 1 θ i p i q E i is the mean energy of the system, N = 1 θ i p i q N i is the mean number of particles, and θ = i p i q .
Let us consider a relativistic ideal gas in the Tsallis-3 statistics in the grand canonical ensemble with the Maxwell–Boltzmann statistics of particles. According to the principle of thermodynamic equilibrium for an open system, similar to the principle of maximum entropy for an isolated system, normalization expressions for the parameters Λ = θ T S + H μ N and θ can be expressed as [11]
1 = n = 0 n 0 ω n n ! Γ ( 1 q 1 ) 0 t 2 q q 1 n e t + β ( Λ + μ n ) ( K 2 ( β m ) ) n d t , θ = n = 0 n 0 ω n n ! Γ ( q q 1 ) 0 t 1 q 1 n e t + β ( Λ + μ n ) ( K 2 ( β m ) ) n d t ,
where ω = g V 2 π 2 m 2 T θ 2 q 1 , β = t ( 1 q ) T θ 2 and n 0 is the number of terms of the series to be taken into account, starting from zero. The values of Λ and θ are found by numerically solving the system of Equation (4). The expression for transverse momentum distribution in rapidity range y [ y m i n , y m a x ] takes the following form [11]:
d 2 N p T d p T d y y m i n y m a x = g V ( 2 π ) 2 m T y m i n y m a x d y cosh y 1 θ n = 0 n 0 ω n n ! Γ ( q q 1 ) 0 t 1 q 1 n e t + β ( Λ m T cosh y + μ ( n + 1 ) ) ( K 2 ( β m ) ) n d t .
The fitting of the final particle spectra was performed using the ROOT analysis framework with MINUIT2 numerical minimization software library [20]. The technique of simultaneous fitting of all particle types by χ 2 -minimization method was applied, where parameters of the medium, such as temperature T and the parameter q, are common for all types of particles. The fitting procedure involves substituting some initial guess fit parameters (the state variables of GCE) into the system of Equation (4), which is then solved numerically to determine the values of the thermodynamic normalization functions Λ and θ . Calculated values of the normalization functions, along with the initial parameters, are then used in the final Formula (5)) for the transverse momentum distribution for χ 2 minimization. The procedure can then be applied iteratively to reach the global minimum of fit parameters. For simultaneous fitting, we minimized the sum of χ 2 for all particle species. Figure 1 shows the Tsallis-3 model fits for K ± , π ± , at s NN = 7.7, 19.6, 39 GeV and centrality classes: 0–5%, 30–40%, 70–80%. The parameter values for all energies and centralities are presented in Table A1. The Tsallis-3 statistics are sensitive to the particle type; therefore, during the fitting process, protons were excluded, and only mesons were included in the fit. Since the next method uses combined data from π + and π , their spectra are also combined here. The low p T part of the pion spectra up to 0.5 GeV/c is affected by resonance decays and was excluded from the calculations. The fit ranges were [0.25, 2.00] GeV/c for kaons and [0.50, 2.00] GeV/c for pions. The volume values were obtained as a free fit parameter.

2.2. Femtoscopic Approach

The experimental correlation function is constructed using distributions of the relative pair momentum q = p 1 p 2 :
C ( q ) = N ( q ) D ( q ) ,
where the numerator N ( q ) is constructed with pairs of particles from the same event and, thus, contains information on quantum statistics and effects of particle final state interactions [21], while the denominator D ( q ) uses uncorrelated pairs obtained from mix-events. The relative pair momentum q , expressed in the longitudinal co-moving system (LCMS), is projected onto the Bertsch–Pratt coordinate system [22,23]. The q o u t component is directed along the pair transverse momentum k T = ( p T , 1 + p T , 2 ) / 2 , the q l o n g component is directed along the beam axis, and the q s i d e component is perpendicular to the other two directions. Further, the Bowler–Sinyukov Gaussian parametrization [24,25] is applied:
C ( q ) ( 1 λ ) + K Coul ( q inv ) λ exp ( q o 2 R o 2 q s 2 R s 2 q l 2 R l 2 2 q o q s R os 2 2 q o q l R ol 2 ) ,
where K Coul ( q inv ) is the Coulomb correction factor, λ is the correlation strength related to secondary particles from resonance decays, R out , side , long are the correlation radii, and R out - side , out - long are the cross terms. The particle emission source correlation radii are extracted by fitting. The radius R side depends on the geometrical size of the source, R out is influenced by both the geometrical size and the emission duration, and R long depends on the system’s lifetime.
We use the data measured by the STAR collaboration during the BES-I program [8]. The dependence of the correlation radii on transverse mass, m T , is considered below. The homogeneity region of the system is smaller for pairs with larger m T [26]. Therefore, to assess the entire volume of the system, we extrapolate the m T -dependence of femtoscopic radii to the minimum possible transverse mass, m T = m . We employed several parameterizations for this extrapolation: the first is grounded in basic hydrodynamic assumptions (known as m T -scaling [27])
R i 1 m T ,
the second uses a simple power-law model
R i m T α ,
and the third applies a model based on the Blast-Wave expansion scenario [28]
R side = R 0 1 + ρ 0 2 ( m / T + 1 / 2 ) , R long = τ T m T K 2 ( m T / T ) K 2 ( m T / T ) ,
where i = out , side , long , ρ 0 is the maximum flow rapidity, and T is the kinetic freeze-out temperature. The values of these parameters are obtained by fitting π ± , K ± , p , p ¯ transverse momentum spectra using the Blast-Wave model [29] and are fixed for further analysis. These parameter values are listed in Table A2. We use linear transverse flow rapidity profile ρ = ρ 0 ( r / R ) to remain consistent with previous HBT studies [18,30]. R 0 and τ are free normalization parameters that are not of interest in this study. Figure 2 shows the extrapolation of the femtoscopic π ± radii R side and R long to zero transverse momentum. The extrapolated values for all energies and centralities are presented in Table A3.
The radii obtained from femtoscopy follow Gaussian distributions and thus characterize only the average size of the system, whereas the Tsallis statistics provide the total volume of the system. To compare the femtoscopic radii with the statistical model, it is necessary to recalculate them by equating the variances of the Gaussian and hard sphere distributions:
σ hard 2 = x 2 = V x 2 d V V d V = R hard 2 5 , σ Gauss 2 = R Gauss 2 ,
where i = out , side , long , and d V = r 2 sin θ d r d φ d θ is the volume element in spherical coordinates. Then, the relation of the radii is
R hard = 5 R Gauss .
Subsequently, the system volume is calculated as
V = 4 3 π ( 5 ) 3 2 R side 2 R long ,
where we use R s i d e instead of R o u t because R o u t depends on the particle emission time [31] and does not reflect the exact geometric size of the system.

3. Results and Discussion

Figure 3 shows the volume values obtained by femtoscopy and by fitting the particle momenta distributions using the Tsallis-3 statistics. The volume values for all energies and centralities are presented in Table A4. The temperature values decrease from central to peripheral events, which is consistent with a previous study [32] using the same data and a similar model based on q-dual statistics [33]. However, the behavior of the parameter q and the values of temperature and q differ significantly. This is due to the fact that in this study, we used the exact Tsallis-3 statistics, taking into account the higher-order terms, while the previous study used a zeroth term approximation. As demonstrated in [11], including higher-order terms in the expansion significantly impacts the parameter values, increasing the temperature and decreasing the q parameter.
It was established that the m T -scaling (8) provides an unsatisfactory description of the femtoscopic radii, significantly overestimating R side and underestimating R long . The power-law dependence (9) demonstrates that the parameter α deviates substantially from the value equal to 0.5, which may indicate contributions from flow and collective effects. The Blast-Wave model (10) accounts for these factors and provides a satisfactory description of the correlation radii.
At low energies, the volumes agree with each other; however, with increasing energy, the volumes begin to diverge, with femtoscopic volumes appearing smaller. This can be explained by several factors:
  • Femtoscopic correlations measure only the region of homogeneity [34], while in the statistical approach using the Tsallis distribution, we obtain the volume of the whole system. If the collective expansion is strong, the region of homogeneity can be smaller than the entire source volume [21].
  • The correlation function may deviate from a Gaussian shape due to exponential tails caused by resonance decay contributions [35]. This can lead to an underestimation of the femtoscopic radii.
  • In the Tsallis statistics, the particle interactions and collective effects, including rescattering, are modeled phenomenologically by deformation of entropy using the nonextensive entropic parameter q.

4. Conclusions

In this study, we compared two different methods for the determination of the size of the system formed in ultrarelativistic heavy-ion collisions at kinetic freeze-out using femtoscopy and a statistical approach based on the Tsallis distribution. A procedure for recalculating correlation radii was presented, ensuring proper comparison between the methods. The results show that collective effects, such as flow, strongly influence the femtoscopic radii. Simple models like m T scaling fail to describe these radii accurately, while the Blast-Wave model provides a better description.
One key finding is that femtoscopic volumes tend to be smaller than the volumes obtained in the Tsallis model, particularly at higher collision energies. This difference points to the need for further investigation, as it suggests potential underlying physics not fully captured by either method in their current forms.
Future work could refine this approach by calculating the system volume as the volume of a cylinder at kinetic freeze-out instead of relying on a simple product of radii. This might give more precise estimates and make the comparison between the two methods more reliable. Adding Tsallis statistics to the Blast-Wave framework could also help to improve the analysis.
Combining these methods can provide a more complete understanding of the system’s size and behavior. This will help to study possible phase transitions and better understand the properties of hot and dense matter in heavy-ion collisions.

Author Contributions

Conceptualization, A.A.; Methodology, A.P. and V.B.L.; Formal analysis, E.N.; Writing—original draft, E.N.; Writing—review and editing, A.A., A.P. and V.B.L.; Visualization, E.N.; Supervision, A.A.; Funding acquisition, A.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Russian Science Foundation grant number 22-72-10028.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Acknowledgments

The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” under the project 24-2-1-62-1. The authors thank Richard Lednicky from the Joint Institute for Nuclear Research, Dubna, Russia, and the Institute of Physics AS CR, Praha, Czech Republic for his insightful feedback and valuable comments during the preparation of this work. A.P. acknowledges also the support of the Romanian Ministry of Research, Innovation and Digitalization, through Project PN 23 21 01 01/2023.

Conflicts of Interest

The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Appendix A

Table A1. Tsallis-3 model fit of π ± ,   K ±   p T -spectra at s NN = 7.7, 11.5, 19.6, 27, 39 GeV and centralities from 0–5%, to 70–80%. p T ranges: [0.5, 2], [0.25, 2] (GeV/c) are taken for π ± , K ± , correspondingly.
Table A1. Tsallis-3 model fit of π ± ,   K ±   p T -spectra at s NN = 7.7, 11.5, 19.6, 27, 39 GeV and centralities from 0–5%, to 70–80%. p T ranges: [0.5, 2], [0.25, 2] (GeV/c) are taken for π ± , K ± , correspondingly.
CentralityT (MeV)q χ 2 /NDF
7.7 GeV
0–5% 187.5 ± 1.5 1.0221 ± 0.0022 131 / 61
5–10% 184.9 ± 1.7 1.0227 ± 0.0031 93 / 63
10–20% 186.9 ± 1.1 1.0346 ± 0.0034 67 / 63
20–30% 178.2 ± 1.0 1.0331 ± 0.0029 73 / 63
30–40% 170.5 ± 0.8 1.0277 ± 0.0030 70 / 62
40–50% 161.7 ± 0.9 1.0326 ± 0.0029 65 / 57
50–60% 151.6 ± 0.9 1.0206 ± 0.0031 81 / 57
60–70% 143.8 ± 1.0 1.0157 ± 0.0032 73 / 53
70–80% 136.8 ± 0.9 1.0082 ± 0.0041 66 / 44
11.5 GeV
0–5% 201.3 ± 1.7 1.0330 ± 0.0029 107 / 63
5–10% 197.6 ± 0.8 1.0316 ± 0.0007 75 / 65
10–20% 195.6 ± 1.1 1.0384 ± 0.0031 56 / 65
20–30% 190.8 ± 0.8 1.0412 ± 0.0028 41 / 65
30–40% 182.8 ± 1.0 1.0447 ± 0.0024 55 / 64
40–50% 173.0 ± 0.8 1.0363 ± 0.0020 62 / 64
50–60% 164.5 ± 0.9 1.0323 ± 0.0019 60 / 63
60–70% 154.9 ± 0.8 1.0228 ± 0.0023 82 / 56
70–80% 147.1 ± 0.9 1.0163 ± 0.0025 100 / 51
19.6 GeV
0–5% 218.9 ± 3.5 1.0479 ± 0.0010 75 / 67
5–10% 216.3 ± 2.3 1.0520 ± 0.0005 70 / 67
10–20% 210.1 ± 0.9 1.0502 ± 0.0025 57 / 67
20–30% 201.9 ± 1.0 1.0527 ± 0.0017 27 / 67
30–40% 191.6 ± 0.9 1.0443 ± 0.0019 49 / 67
40–50% 182.1 ± 0.8 1.0377 ± 0.0017 92 / 65
50–60% 174.0 ± 0.8 1.0343 ± 0.0016 159 / 65
60–70% 165.2 ± 0.8 1.0279 ± 0.0017 229 / 61
70–80% 154.6 ± 1.1 1.0219 ± 0.0020 177 / 58
27 GeV
0–5% 221.8 ± 1.9 1.0421 ± 0.0031 103 / 66
5–10% 221.2 ± 1.4 1.0465 ± 0.0004 76 / 67
10–20% 211.4 ± 0.1 1.0490 ± 0.0001 117 / 67
20–30% 209.5 ± 1.2 1.0570 ± 0.0022 32 / 67
30–40% 201.3 ± 0.0 1.0528 ± 0.0009 28 / 67
40–50% 191.3 ± 1.1 1.0486 ± 0.0016 49 / 67
50–60% 182.1 ± 0.9 1.0424 ± 0.0014 94 / 67
60–70% 172.5 ± 0.9 1.0351 ± 0.0014 180 / 67
70–80% 163.8 ± 0.9 1.0284 ± 0.0013 255 / 67
39 GeV
0–5% 232.6 ± 1.2 1.0454 ± 0.0007 86 / 67
5–10% 231.6 ± 1.3 1.0499 ± 0.0007 64 / 67
10–20% 229.1 ± 1.0 1.0568 ± 0.0007 42 / 67
20–30% 218.6 ± 0.3 1.0614 ± 0.0008 19 / 67
30–40% 205.0 ± 0.2 1.0612 ± 0.0012 28 / 67
40–50% 199.9 ± 1.2 1.0521 ± 0.0016 56 / 67
50–60% 185.0 ± 0.1 1.0493 ± 0.0010 134 / 67
60–70% 182.2 ± 1.1 1.0422 ± 0.0014 143 / 67
70–80% 174.9 ± 1.1 1.0354 ± 0.0015 216 / 67
Table A2. Blast-Wave model fit results for π ± ,   K ± ,   p ,   p ¯   p T -spectra at s NN = 7.7, 11.5, 19.6, 27, 39 GeV and centralities from 0–5%, to 70–80%. p T ranges: [0.5, 1.50], [0.25, 1.5], [0.4, 1.5] (GeV/c) are taken for π ± , K ± , p( p ¯ ), correspondingly.
Table A2. Blast-Wave model fit results for π ± ,   K ± ,   p ,   p ¯   p T -spectra at s NN = 7.7, 11.5, 19.6, 27, 39 GeV and centralities from 0–5%, to 70–80%. p T ranges: [0.5, 1.50], [0.25, 1.5], [0.4, 1.5] (GeV/c) are taken for π ± , K ± , p( p ¯ ), correspondingly.
Centrality ρ 0 T (MeV) χ 2 /NDF
7.7 GeV
0–5% 0.743 ± 0.016 112 ± 3 76 / 100
5–10% 0.710 ± 0.017 115 ± 3 64 / 102
10–20% 0.649 ± 0.018 122 ± 3 48 / 106
20–30% 0.605 ± 0.021 123 ± 3 79 / 104
30–40% 0.544 ± 0.023 129 ± 3 81 / 104
40–50% 0.470 ± 0.028 133 ± 4 88 / 99
50–60% 0.391 ± 0.035 136 ± 5 101 / 97
60–70% 0.340 ± 0.034 136 ± 4 80 / 95
70–80% 0.182 ± 0.058 143 ± 4 68 / 81
11.5 GeV
0–5% 0.748 ± 0.014 115 ± 2 37 / 103
5–10% 0.721 ± 0.016 118 ± 3 33 / 106
10–20% 0.684 ± 0.016 121 ± 3 29 / 106
20–30% 0.616 ± 0.018 129 ± 3 29 / 107
30–40% 0.566 ± 0.022 134 ± 3 30 / 107
40–50% 0.421 ± 0.026 147 ± 4 58 / 104
50–60% 0.359 ± 0.029 151 ± 4 52 / 103
60–70% 0.290 ± 0.035 151 ± 4 66 / 99
70–80% 0.248 ± 0.036 149 ± 4 96 / 96
19.6 GeV
0–5% 0.765 ± 0.015 117 ± 3 25 / 105
5–10% 0.740 ± 0.015 120 ± 3 24 / 101
10–20% 0.710 ± 0.017 123 ± 3 24 / 101
20–30% 0.649 ± 0.019 129 ± 3 28 / 101
30–40% 0.573 ± 0.020 136 ± 3 35 / 102
40–50% 0.507 ± 0.022 141 ± 4 46 / 100
50–60% 0.414 ± 0.026 149 ± 4 59 / 100
60–70% 0.350 ± 0.026 152 ± 3 89 / 99
70–80% 0.250 ± 0.033 157 ± 3 133 / 99
27 GeV
0–5% 0.796 ± 0.015 115 ± 3 42 / 98
5–10% 0.771 ± 0.016 118 ± 3 31 / 99
10–20% 0.738 ± 0.017 122 ± 3 33 / 99
20–30% 0.683 ± 0.018 128 ± 3 27 / 99
30–40% 0.618 ± 0.021 136 ± 3 23 / 99
40–50% 0.534 ± 0.024 144 ± 4 30 / 99
50–60% 0.451 ± 0.026 151 ± 4 37 / 99
60–70% 0.352 ± 0.030 158 ± 4 59 / 99
70–80% 0.251 ± 0.035 163 ± 4 114 / 99
39 GeV
0–5% 0.824 ± 0.015 116 ± 3 23 / 99
5–10% 0.785 ± 0.015 121 ± 3 22 / 99
10–20% 0.758 ± 0.017 125 ± 3 20 / 99
20–30% 0.709 ± 0.018 130 ± 3 22 / 99
30–40% 0.632 ± 0.015 140 ± 2 17 / 99
40–50% 0.567 ± 0.024 145 ± 4 29 / 99
50–60% 0.476 ± 0.024 153 ± 4 43 / 99
60–70% 0.379 ± 0.031 164 ± 4 55 / 99
70–80% 0.280 ± 0.035 172 ± 4 97 / 99
Table A3. Femtoscopic π ± radii extrapolation to k T = 0 at s NN = 7.7, 11.5, 19.6, 27, 39 GeV and the centrality classes from 0–5% to 60–70% using Equations (8)–(10). The radii are given in fm.
Table A3. Femtoscopic π ± radii extrapolation to k T = 0 at s NN = 7.7, 11.5, 19.6, 27, 39 GeV and the centrality classes from 0–5% to 60–70% using Equations (8)–(10). The radii are given in fm.
Centrality R side ( m T α ) R long ( m T α ) R side ( 1 / m T ) R long ( 1 / m T ) R side ( BW ) R long ( BW )
7.7 GeV
0–5% 6.22 ± 0.16 7.54 ± 0.28 6.89 ± 0.20 6.51 ± 0.21 5.43 ± 0.13 8.06 ± 0.32
5–10% 5.58 ± 0.13 7.50 ± 0.28 6.54 ± 0.18 6.18 ± 0.19 5.11 ± 0.11 7.69 ± 0.29
10–20% 5.17 ± 0.11 7.28 ± 0.26 5.99 ± 0.15 5.80 ± 0.17 4.58 ± 0.09 7.27 ± 0.26
20–30% 4.61 ± 0.09 6.68 ± 0.22 5.38 ± 0.12 5.32 ± 0.14 4.08 ± 0.07 6.66 ± 0.22
30–40% 3.84 ± 0.06 5.69 ± 0.16 4.88 ± 0.10 4.85 ± 0.12 3.63 ± 0.06 6.11 ± 0.18
40–50% 3.31 ± 0.05 5.35 ± 0.14 4.37 ± 0.08 4.37 ± 0.09 3.17 ± 0.04 5.53 ± 0.15
50–60% 2.87 ± 0.04 5.85 ± 0.17 3.85 ± 0.06 3.79 ± 0.07 2.75 ± 0.03 4.79 ± 0.11
11.5 GeV
0–5% 5.83 ± 0.14 8.37 ± 0.34 6.89 ± 0.20 7.01 ± 0.24 5.42 ± 0.12 8.73 ± 0.37
5–10% 5.52 ± 0.13 8.13 ± 0.32 6.48 ± 0.18 6.73 ± 0.22 5.06 ± 0.11 8.41 ± 0.35
10–20% 5.03 ± 0.11 7.54 ± 0.28 6.03 ± 0.15 6.32 ± 0.20 4.65 ± 0.09 7.92 ± 0.31
20–30% 4.59 ± 0.09 6.86 ± 0.23 5.43 ± 0.12 5.72 ± 0.16 4.09 ± 0.07 7.22 ± 0.26
30–40% 4.01 ± 0.07 6.26 ± 0.19 4.90 ± 0.10 5.17 ± 0.13 3.63 ± 0.06 6.55 ± 0.21
40–50% 3.51 ± 0.05 5.81 ± 0.17 4.32 ± 0.08 4.65 ± 0.11 3.06 ± 0.04 5.97 ± 0.17
50–60% 2.98 ± 0.04 5.17 ± 0.13 3.77 ± 0.06 4.18 ± 0.09 2.62 ± 0.03 5.38 ± 0.14
19.6 GeV
0–5% 5.88 ± 0.15 8.94 ± 0.39 6.96 ± 0.21 7.53 ± 0.28 5.50 ± 0.13 9.38 ± 0.43
5–10% 5.52 ± 0.13 8.51 ± 0.36 6.57 ± 0.18 7.19 ± 0.25 5.16 ± 0.11 8.97 ± 0.39
10–20% 5.13 ± 0.11 7.99 ± 0.31 6.12 ± 0.16 6.72 ± 0.22 4.75 ± 0.10 8.44 ± 0.35
20–30% 4.57 ± 0.09 7.33 ± 0.26 5.52 ± 0.13 6.05 ± 0.18 4.20 ± 0.08 7.65 ± 0.29
30–40% 4.00 ± 0.07 6.47 ± 0.21 4.95 ± 0.10 5.47 ± 0.15 3.70 ± 0.06 6.93 ± 0.24
40–50% 3.47 ± 0.05 5.96 ± 0.17 4.36 ± 0.08 4.90 ± 0.12 3.20 ± 0.04 6.23 ± 0.19
50–60% 2.87 ± 0.04 5.00 ± 0.12 3.87 ± 0.06 4.34 ± 0.09 2.76 ± 0.03 5.54 ± 0.15
60–70% 2.62 ± 0.03 4.52 ± 0.10 3.34 ± 0.05 3.81 ± 0.07 2.33 ± 0.02 4.87 ± 0.12
27 GeV
0–5% 5.94 ± 0.15 9.17 ± 0.41 7.04 ± 0.21 7.77 ± 0.30 5.61 ± 0.13 9.67 ± 0.46
5–10% 5.65 ± 0.14 8.79 ± 0.38 6.66 ± 0.19 7.42 ± 0.27 5.27 ± 0.12 9.26 ± 0.42
10–20% 5.14 ± 0.11 8.14 ± 0.33 6.23 ± 0.16 6.90 ± 0.23 4.86 ± 0.10 8.65 ± 0.37
20–30% 4.63 ± 0.09 7.46 ± 0.27 5.60 ± 0.13 6.26 ± 0.19 4.31 ± 0.08 7.88 ± 0.30
30–40% 4.07 ± 0.07 6.79 ± 0.23 5.02 ± 0.11 5.63 ± 0.16 3.79 ± 0.06 7.14 ± 0.25
40–50% 3.55 ± 0.05 6.10 ± 0.18 4.47 ± 0.08 5.03 ± 0.12 3.29 ± 0.05 6.41 ± 0.20
50–60% 2.99 ± 0.04 5.27 ± 0.14 3.91 ± 0.07 4.46 ± 0.10 2.81 ± 0.03 5.72 ± 0.16
60–70% 2.61 ± 0.03 4.63 ± 0.11 3.40 ± 0.05 3.88 ± 0.07 2.37 ± 0.02 4.99 ± 0.12
39 GeV
0–5% 6.11 ± 0.16 9.32 ± 0.43 7.18 ± 0.22 7.96 ± 0.31 5.71 ± 0.14 9.96 ± 0.49
5–10% 5.74 ± 0.14 8.76 ± 0.38 6.81 ± 0.20 7.56 ± 0.28 5.36 ± 0.12 9.50 ± 0.44
10–20% 5.18 ± 0.11 8.50 ± 0.35 6.28 ± 0.17 7.04 ± 0.24 4.91 ± 0.10 8.88 ± 0.39
20–30% 4.68 ± 0.09 7.75 ± 0.30 5.69 ± 0.14 6.38 ± 0.20 4.39 ± 0.08 8.07 ± 0.32
30–40% 4.14 ± 0.07 6.97 ± 0.24 5.10 ± 0.11 5.76 ± 0.16 3.84 ± 0.06 7.35 ± 0.26
40–50% 3.61 ± 0.06 6.03 ± 0.18 4.53 ± 0.09 5.10 ± 0.13 3.35 ± 0.05 6.52 ± 0.21
50–60% 3.11 ± 0.04 5.39 ± 0.14 3.97 ± 0.07 4.52 ± 0.10 2.86 ± 0.04 5.80 ± 0.17
60–70% 2.73 ± 0.03 4.78 ± 0.11 3.45 ± 0.05 4.03 ± 0.08 2.42 ± 0.03 5.21 ± 0.13
Table A4. Comparison of volumes at s NN = 7.7, 11.5, 19.6, 27, 39 GeV and the centrality classes from 0–5% to 60–70% obtained using Tsallis-3 statistics and calculated using Equations (8)–(10). The volumes are given in fm3.
Table A4. Comparison of volumes at s NN = 7.7, 11.5, 19.6, 27, 39 GeV and the centrality classes from 0–5% to 60–70% obtained using Tsallis-3 statistics and calculated using Equations (8)–(10). The volumes are given in fm3.
CentralityTsallis-3 m T α 1 / m T BW
7.7 GeV
0–5%13,020 ± 1454 13,674 ± 2022 14,470 ± 2137 11,145 ± 1646
5–10%11,184 ± 869 10,931 ± 1618 12,370 ± 1826 9394 ± 1387
10–20% 12 , 171 ± 1279 9125 ± 1349 9750 ± 1440 7150 ± 1056
20–30% 9350 ± 338 6657 ± 985 7213 ± 1065 5187 ± 766
30–40% 5956 ± 537 3934 ± 582 5418 ± 800 3766 ± 556
40–50% 5161 ± 600 2734 ± 405 3911 ± 578 2605 ± 385
50–60% 2782 ± 189 2254 ± 336 2632 ± 389 1691 ± 251
11.5 GeV
0–5% 19 , 335 ± 1589 13 , 329 ± 1969 15 , 570 ± 2299 12 , 001 ± 1772
5–10% 15 , 199 ± 505 11 , 611 ± 1716 13 , 229 ± 1953 10 , 079 ± 1488
10–20% 14 , 691 ± 1562 8951 ± 1322 10 , 766 ± 1590 8022 ± 1185
20–30% 11 , 556 ± 1170 6775 ± 1001 7883 ± 1164 5654 ± 835
30–40% 9780 ± 1035 4717 ± 697 5800 ± 856 4051 ± 598
40–50% 5656 ± 450 3346 ± 494 4072 ± 601 2616 ± 386
50–60% 3410 ± 267 2152 ± 319 2781 ± 411 1730 ± 256
19.6 GeV
0–5% 32 , 249 ± 2714 14 , 479 ± 2138 17 , 107 ± 2526 13 , 294 ± 1963
5–10% 30 , 633 ± 1803 12 , 157 ± 1796 14 , 506 ± 2142 11 , 171 ± 1649
10–20% 23 , 912 ± 2270 9844 ± 1454 11 , 794 ± 1741 8920 ± 1317
20–30% 19 , 404 ± 1545 7155 ± 1057 8628 ± 1274 6323 ± 934
30–40% 11 , 305 ± 894 4844 ± 715 6277 ± 927 4434 ± 655
40–50% 6582 ± 470 3355 ± 496 4369 ± 645 2981 ± 440
50–60% 4010 ± 273 1929 ± 285 3037 ± 448 1977 ± 292
60–70% 2127 ± 144 1451 ± 215 1987 ± 293 1242 ± 184
27 GeV
0–5% 27 , 576 ± 2472 15 , 125 ± 2233 18 , 024 ± 2661 14 , 229 ± 2101
5–10% 26 , 184 ± 551 13 , 118 ± 1937 15 , 403 ± 2274 12 , 016 ± 1774
10–20% 26 , 783 ± 401 10 , 061 ± 1486 12 , 529 ± 1850 9586 ± 1415
20–30% 22 , 091 ± 2252 7482 ± 1105 9195 ± 1358 6870 ± 1014
30–40% 13 , 673 ± 486 5258 ± 776 6642 ± 981 4797 ± 708
40–50% 8441 ± 636 3591 ± 530 4695 ± 693 3251 ± 480
50–60% 4618 ± 302 2201 ± 325 3193 ± 472 2112 ± 312
60–70% 2381 ± 155 1476 ± 218 2095 ± 309 1314 ± 194
39 GeV
0–5% 27 , 434 ± 696 16 , 276 ± 2418 19 , 200 ± 2835 15 , 215 ± 2247
5–10% 26 , 424 ± 556 13 , 523 ± 1998 16 , 435 ± 2427 12 , 771 ± 1886
10–20% 25 , 733 ± 225 10 , 695 ± 1580 13 , 003 ± 1920 10 , 024 ± 1480
20–30% 23 , 522 ± 585 7944 ± 1173 9659 ± 1426 7281 ± 1075
30–40% 18 , 861 ± 781 5581 ± 825 7023 ± 1037 5076 ± 750
40–50% 8612 ± 660 3666 ± 542 4896 ± 723 3424 ± 506
50–60% 6334 ± 213 2445 ± 362 3335 ± 493 2221 ± 328
60–70% 2537 ± 173 1668 ± 247 2248 ± 332 1425 ± 211

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Figure 1. Tsallis-3 model fits of K ± , π ± at s NN = 7.7, 19.6, 39 GeV and the centrality classes 0–5%, 30–40%, 70–80%. Data points are taken from [7].
Figure 1. Tsallis-3 model fits of K ± , π ± at s NN = 7.7, 19.6, 39 GeV and the centrality classes 0–5%, 30–40%, 70–80%. Data points are taken from [7].
Particles 08 00034 g001
Figure 2. Femtoscopic π ± radii fits at s NN = 19.6 GeV and the centrality class 30–40% using Equations (8)–(10). Data points are taken from [8].
Figure 2. Femtoscopic π ± radii fits at s NN = 19.6 GeV and the centrality class 30–40% using Equations (8)–(10). Data points are taken from [8].
Particles 08 00034 g002
Figure 3. Comparison of volumes at s NN = 7.7, 19.6, 39 GeV and the centrality classes from 0–5% to 60–70% obtained using the Tsallis-3 statistics and calculated using Equations (8)–(10).
Figure 3. Comparison of volumes at s NN = 7.7, 19.6, 39 GeV and the centrality classes from 0–5% to 60–70% obtained using the Tsallis-3 statistics and calculated using Equations (8)–(10).
Particles 08 00034 g003
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Nedorezov, E.; Aparin, A.; Parvan, A.; Luong, V.B. A System Size Analysis of the Fireball Produced in Heavy-Ion Collisions. Particles 2025, 8, 34. https://doi.org/10.3390/particles8010034

AMA Style

Nedorezov E, Aparin A, Parvan A, Luong VB. A System Size Analysis of the Fireball Produced in Heavy-Ion Collisions. Particles. 2025; 8(1):34. https://doi.org/10.3390/particles8010034

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Nedorezov, Egor, Alexey Aparin, Alexandru Parvan, and Vinh Ba Luong. 2025. "A System Size Analysis of the Fireball Produced in Heavy-Ion Collisions" Particles 8, no. 1: 34. https://doi.org/10.3390/particles8010034

APA Style

Nedorezov, E., Aparin, A., Parvan, A., & Luong, V. B. (2025). A System Size Analysis of the Fireball Produced in Heavy-Ion Collisions. Particles, 8(1), 34. https://doi.org/10.3390/particles8010034

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