AHEP Advances in High Energy Physics 1687-7365 1687-7357 Hindawi 10.1155/2020/1787183 1787183 arXiv:1909.11339v1 Research Article Kinetic Freeze-Out Temperature and Transverse Flow Velocity in Au-Au Collisions at RHIC-BES Energies https://orcid.org/0000-0003-2799-0262 Waqas Muhammad https://orcid.org/0000-0003-0291-7042 Li Bao-Chun Sahoo Raghunath Institute of Theoretical Physics & Department of Physics & State Key Laboratory of Quantum Optics and Quantum Optics Devices Shanxi University Taiyuan Shanxi 030006 Chinasxu.edu.cn 2020 812020 2020 25 09 2019 21 11 2019 13 12 2019 812020 2020 Copyright © 2020 Muhammad Waqas and Bao-Chun Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

By using the method of data-driven reanalysis, the midrapidity transverse momentum pT spectra of charged hadrons (π+, K+, and p) produced in central and peripheral gold-gold (Au-Au) collisions from the Beam Energy Scan (BES) program at the Relativistic Heavy Ion Collider (RHIC) are fitted by using the blast-wave model with the Boltzmann-Gibbs statistics. The model results are in agreement with the experimental data measured by the STAR Collaboration at the RHIC-BES energies. We observe that the kinetic freeze-out temperature T0, transverse flow velocity βT, mean transverse momentum pT, and initial temperature Ti increase with collision energy as well as with event centrality.

Fund for Shanxi “1331 Project” Key Subjects ConstructionNatural Science Foundation of Shanxi Province201701D121005Scientific and Technological Innovation Programs (STIP) of Higher Education Institutions in Shanxi201802017China Scholarship CouncilNational Natural Science Foundation of China11575103 1. Introduction

One of the most fundamental questions in high energy and nuclear physics is to determine the phase structure of the strongly interacting quantum chromodynamics (QCD) matter [1–3]. The yield ratios, transverse momentum pT spectra, and other data for various identified particles produced in proton-proton pp, proton-nucleus (pA), and nucleus-nucleus (AA) collisions at high energies are important observable quantities for determining the phase structure. The experimental facilities such as the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) provide excellent tools to study the properties of Quark-Gluon Plasma (QGP) [4–6], which are expected to create collision events with high multiplicities.

The phase diagram of the QCD matter is usually expressed in terms of the chemical freeze-out temperature (Tch) and the baryon chemical potential (μB) [7, 8]. Besides, other quantities such as the kinetic freeze-out temperature (Tkin or T0) and transverse flow velocity (βT) are useful to understand the phase diagram [9]. To search for the possible critical energy in the phase transition from hadronic matter to QGP in high-energy collisions, the STAR Collaboration has been performing the Beam Energy Scan (BES) program [10–13] at the RHIC. Besides, other experiments at similar or lower energies at other accelerators are scheduled [14, 15].

Generally, the processes of high-energy collisions result possibly in three main stages [16–18]:

The initial stage: at this stage, the collisions are in the beginning. The temperature at this stage is called the initial temperature which is one of the main factors to affect the particle spectra, which is less studied in the community comparatively. After the initial state, the “fireball” leads to a decrease in the temperature and finally to the hadronization

The chemical freeze-out stage: at this stage, the inner collisions among various particles are elastic and the yield ratios of differential types of particles remain invariant. The chemical freeze-out temperature Tch can be obtained from the particle ratios, which is much studied in the community comparatively

The kinetic freeze-out stage: at this stage, the scattering processes stop, the hadrons decouple from the rest of the system, and the hadron’s energy/momentum spectra freeze in time. The temperature at this stage is known as the kinetic freeze-out temperature T0 which can be obtained from the pT spectra

When one studies T0 from the pT spectra, the effect of βT should be eliminated. If the effect of βT is not eliminated in the temperature, this temperature is called the effective temperature (Teff or T). At the stage of kinetic freeze-out, T0 and βT are two important parameters which describe the thermal motion of the produced particles and the collective expansion of the emission source, respectively. The spectra in a low-pT region (pT=2–3 GeV/c) which is mainly contributed by the soft excitation process essentially separate the contribution of the thermal motion and the collective expansion, if one only extracts T0 and βT. The spectra in a high-pT region are contributed by the hard scattering process which is not needed in extracting T0 and βT.

We are very interested in the extraction of T0 and βT in collisions at the RHIC-BES energies which are very suitable to study the spectra in a low-pT region, where the spectra in a high-pT region are not produced due to not too high energies. In this work, the double-differential pT spectra of charged particle dependences on collision energy and event centrality in gold-gold (Au-Au) collisions are analyzed with the blast-wave model with the Boltzmann-Gibbs statistics by means of data-driven analysis. The model results are compared with the data measured by the STAR Collaboration at the RHIC-BES energies [19, 20].

The remainder of this work consists of The Method and Formalism, Results and Discussion, and Conclusions. We shall describe the remanent parts orderly.

 2. The Method and Formalism

Various methods can be used for the extraction of T0 and βT, e.g., the blast-wave model with the Boltzmann-Gibbs statistics [21–23], the blast-wave model with the Tsallis statistics [24–26], an alternative method by using the Boltzmann-Gibbs statistics [22, 27–33], and the alternative method by using the Tsallis distribution [33–39]. In this work, we choose the blast-wave model with the Boltzmann-Gibbs statistics due to its similarity with the ideal gas model in thermodynamics and few parameters. However, these methods only describe the spectra in the low-pT region. For the spectra in the high-pT region if available, the Hagedorn function which is known as the inverse power law [40, 41] can be used. We shall discuss these issues in detail as follows.

In general, there are two main processes responsible for the contribution of pT spectra. They are (i) the soft excitation process which contributes the soft component in the low-pT region and (ii) the hard scattering process which contributes the hard component in the whole pT region if one uses the general superposition function or in the high-pT region if one uses the usual step function.

For the soft component, according to Refs. [21–23], the probability density function of the pT spectra in the blast-wave model with the Boltzmann-Gibbs statistics results in (1)f1pT=1NdNdpT=CpTmT∫0Rrdr×I0pTsinhρT0K1mTcoshρT0,where N is the number of particles, C is the normalization constant, mT=pT2+m02 is the transverse mass, m0 is the rest mass of the considered particle, r and R are the radial position and the maximum radial position, respectively, I0 and K1 are the modified Bessel functions of the first and second kinds, respectively, ρ=tanh−1βr is the boost angle, βr=βSr/Rn0 is a self-similar flow profile, βS is the flow velocity on the surface, and n0=2 is used in the original form [21]. Particularly, βT=2/R2∫0Rrβrdr=2βS/n0+2=0.5βS. The parameter n0 is used in different works, e.g., n0=1 or noninteger in Refs. [22, 24, 42], which corresponds to the centrality from the center to the periphery.

Equation (1) and similar or related functions are not enough to describe the whole pT spectra. In particular, the maximum pT reaches up to 100 GeV/c in collisions at the LHC [43]. Then, one needs other functions such as the Tsallis–Lévy- [44, 45] or Tsallis–Pareto-type function [44, 46] and the Hagedorn function [40, 41] or inverse power law [47–49] to the spectra in high- and very high-pT regions. In this work, the hard component is simply represented by the inverse power law. That is, (2)f2pT=1NdNdpT=ApT1+pTp0−n,where p0 and n are free parameters and A is the normalization constant which is related to the free parameters.

However, the structure of pT spectra is very complex. In fact, several regions have been observed and analyzed in Ref. [50]. These regions include the first one with pT<4–6 GeV/c, the second one with 4–6 GeV/c<pT<17–20 GeV/c, and the third one with pT>17–20 GeV/c. Different regions may correspond to different mechanisms. The first pT region in our discussion is regarded as the region of the soft excitation process, while the second and third pT regions are regarded as the regions of the hard and the very hard excitation process, respectively. In particular, a special region with pT<0.2–0.3 GeV/c is considered due to the resonant production in some cases, and it is regarded as the region of the very soft excitation process.

Generally, the whole pT region discussed above can be uniformly superposed by two methods: (i) the general superposition in which the contribution regions of different components overlap each other and (ii) the Hagedorn model (the usual step function) [40] in which there is no overlapping of different regions of different components.

Considering f1pT, f2pT, fVSpT, and fVHpT which denote the probability density functions by the soft, hard, very soft, and very hard components, respectively, where fVSpT and fVHpT are assumed to be in the form of f1pT and f2pT, respectively, the unified superposition according to the first method is (3)f0pT=1NdNdpT=kVSfVSpT+kf1pT+1−k−kVS−kVHf2pT+kVHfVHpT,where kVS is the contribution fraction of the very soft component, while k and kVH denote the contributions of the soft and very hard components, respectively.

The step function can be used to structure the superposition according to the Hagedorn model [40]; i.e., (4)f0pT=1NdNdpT=AVSθpVS−pTfVSpT+A1θpT−pVSθp1−pTf1pT+A2θpT−p1θpVH−pTf2pT+AVHθpT−pVHfVHpT,where AVS, A1, A2, and AVH are the constants which make the interfacing components link to each other perfectly.

Particularly, if the contributions of the very soft and very hard components can be neglected, Equations (3) and (4) are, respectively, simplified to be (5)f0pT=1NdNdpT=kf1pT+1−kf2pT,(6)f0pT=1NdNdpT=A1θp1−pTf1pT+A2θpT−p1f2pT.Further, if the contribution of the hard component at the RHIC-BES energies can be neglected, Equations (5) and (6) are simplified to be the same form: (7)f0pT=1NdNdpT=f1pT.

This work deals with Au-Au collisions at the RHIC-BES energies, for which Equation (7), i.e., Equation (1), is suitable. In the following section, we shall use Equation (1) to fit the experimental data measured by the STAR Collaboration at the RHIC-BES energies [19, 20].

In particular, the mean pTpT and the root-mean-square pTpT2 can be expressed, respectively, as (8)pT=∫0pTmaxpTf0pTdpT,(9)pT2=∫0pTmaxpT2f0pTdpT,due to (10)∫0pTmaxf0pTdpT=1,where pTmax denotes the maximum pT considered by us. In this work, we take pTmax=2.5 GeV/c.

It should be noted that although only Equation (1) is used in the analysis, we would like to continue to have the statement and formalism for other functions or distributions such as the inverse power law and its superposition with thermal distribution and the discussions on the very soft, hard, and very hard components. In fact, due to the existence of other functions or distributions, the mentioned method of data-driven reanalysis can be used in the spectra in wide pT coverage, which is not the case in this work. In addition, it is possible to use simultaneously the (very) soft and (very) hard components in other cases which are more universal.

 3. Results and Discussion

Figure 1 presents the event centrality-dependent double-differential pT spectra, 1/2πpTd2N/dpTdy, of π+, K+, and p produced in the midrapidity interval ∣y∣<0.1 in Au-Au collisions at the center-of-mass energy per nucleon pair sNN=7.7 GeV at the RHIC-BES, where y denotes the rapidity. The symbols represent the experimental data measured by the STAR Collaboration [19], and the curves are our fitting results by using the blast-wave model with the Boltzmann-Gibbs statistics, Equation (1) [21–23]. The spectra in centrality classes 0–5%, 5–10%, 10–20%, 20–30%, 30–40%, 40–50%, 50–60%, 60–70%, and 70–80% are scaled by 1, 1/2, 1/4, 1/6, 1/8, 1/10, 1/12, 1/14, and 1/16, respectively. In the fit, the least-square method is used to determine the best values of parameters. The related parameters along with χ2 and degree of freedom (dof) are listed in Table 1, where the centrality classes are listed together. One can see that Equation (1) fits well the data in Au-Au collisions at 7.7 GeV at the RHIC.

Transverse momentum spectra of (a–c) π+, K+, and p produced in different centrality bins in Au-Au collisions at sNN=7.7 GeV. The symbols represent the experimental data measured by the STAR Collaboration in the midrapidity interval ∣y∣<0.1 [19]. The curves are our fitted results by Equation (1).

Values of free parameters (T0 and βT), normalization constant (N0), χ2, and dof corresponding to the curves in Figures 1–6.

Figure Particle Centrality T0 βT N0 χ2 dof
Figure 1Au-Au7.7 GeV π+ 0–5% 0.130±0.004 0.306±0.006 15.00±1.00 17 26
5–10% 0.129±0.005 0.305±0.005 6.02±0.9 14 26
10–20% 0.128±0.004 0.303±0.007 2.35±1.05 20 26
20–30% 0.126±0.005 0.302±0.006 1.09±0.60 18 26
30–40% 0.124±0.004 0.300±0.006 0.54±0.43 19 23
40–50% 0.122±0.003 0.297±0.007 0.28±0.02 13 23
50–60% 0.120±0.004 0.292±0.004 0.14±0.02 20 22
60–70% 0.118±0.004 0.280±0.005 0.067±0.001 16 21
70–80% 0.115±0.005 0.269±0.007 0.030±0.008 11 18
K+ 0–5% 0.133±0.005 0.305±0.005 3.65±0.20 99 20
5–10% 0.131±0.004 0.304±0.005 1.50±0.15 76 22
10–20% 0.130±0.004 0.302±0.007 0.53±0.05 55 22
20–30% 0.128±0.005 0.301±0.007 0.24±0.17 27 22
30–40% 0.127±0.004 0.299±0.009 0.11±0.01 27 21
40–50% 0.125±0.006 0.296±0.008 0.050±0.002 16 20
50–60% 0.123±0.005 0.278±0.007 0.23±0.02 34 19
60–70% 0.118±0.004 0.265±0.008 0.0093±0.0007 34 18
70–80% 0.116±0.004 0.250±0.008 0.0034±0.0003 29 15
p 0–5% 0.134±0.004 0.340±0.005 9.31±0.60 54 29
5–10% 0.133±0.005 0.328±0.006 3.90±0.15 47 29
10–20% 0.132±0.006 0.318±0.008 1.50±0.20 46 29
20–30% 0.130±0.005 0.310±0.008 0.65±0.05 28 29
30–40% 0.128±0.004 0.301±0.006 0.32±0.04 14 25
40–50% 0.126±0.005 0.280±0.007 0.15±0.03 13 25
50–60% 0.124±0.004 0.271±0.006 0.070±0.008 7 24
60–70% 0.122±0.003 0.250±0.005 0.030±0.004 8 25
70–80% 0.120±0.006 0.204±0.009 0.013±0.001 14 18
Figure 2Au-Au11.5 GeV π+ 0–5% 0.132±0.004 0.315±0.005 19.11±1.60 5 26
5–10% 0.130±0.005 0.313±0.005 7.60±1.40 16 26
10–20% 0.129±0.004 0.312±0.006 2.90±0.50 14 26
20–30% 0.128±0.003 0.311±0.007 1.36±0.10 34 26
30–40% 0.127±0.003 0.310±0.008 0.38±0.05 15 26
40–50% 0.126±0.006 0.307±0.007 0.34±0.03 16 26
50–60% 0.124±0.004 0.305±0.005 0.16±0.01 7 23
60–70% 0.121±0.004 0.296±0.006 0.080±0.007 6 21
70–80% 0.119±0.005 0.288±0.008 0.040±0.005 10 21
K+ 0–5% 0.135±0.005 0.314±0.009 4.23±0.30 64 22
5–10% 0.133±0.004 0.312±0.008 1.72±0.10 62 23
10–20% 0.132±0.006 0.310±0.010 0.60±0.05 46 23
20–30% 0.130±0.003 0.308±0.004 0.27±0.02 45 23
30–40% 0.129±0.004 0.307±0.006 0.13±0.01 59 23
40–50% 0.128±0.005 0.306±0.007 0.060±0.006 11 23
50–60% 0.126±0.004 0.300±0.006 0.026±0.002 15 22
60–70% 0.124±0.003 0.288±0.005 0.011±0.001 6 20
70–80% 0.122±0.004 0.264±0.011 0.0050±0.0003 26 19
p 0–5% 0.136±0.005 0.323±0.007 7.74±1.00 55 25
5–10% 0.135±0.005 0.321±0.007 2.90±0.25 56 26
10–20% 0.134±0.005 0.318±0.006 1.12±0.15 40 26
20–30% 0.132±0.004 0.311±0.007 0.50±0.03 24 26
30–40% 0.130±0.005 0.308±0.007 0.24±0.01 13 26
40–50% 0.128±0.003 0.285±0.006 0.12±0.01 10 25
50–60% 0.125±0.004 0.274±0.008 0.55±0.01 13 25
60–70% 0.123±0.00414 0.251±0.006 0.024±0.004 7 25
70–80% 0.121±0.004 0.231±0.007 0.010±0.003 23 26
Figure 3Au-Au14.5 GeV π+ 0–5% 0.135±0.003 0.320±0.005 22.14±2.00 4 25
5–10% 0.133±0.003 0.318±0.006 8.50±2.00 10 25
10–20% 0.132±0.004 0.317±0.006 3.50±0.25 12 25
20–30% 0.131±0.004 0.315±0.005 1.60±0.13 15 25
30–40% 0.130±0.004 0.314±0.007 0.80±0.06 10 25
40–50% 0.128±0.004 0.311±0.006 0.40±0.03 16 25
50–60% 0.126±0.004 0.307±0.007 0.19±0.02 7 25
60–70% 0.124±0.004 0.299±0.007 0.093±0.014 8 25
70–80% 0.120±0.005 0.291±0.005 0.040±0.006 14 25
K+ 0–5% 0.137±0.005 0.318±0.007 4.36±0.40 16 23
5–10% 0.136±0.004 0.314±0.008 1.86±0.20 8 23
10–20% 0.135±0.005 0.313±0.008 0.70±0.08 21 23
20–30% 0.134±0.004 0.312±0.006 0.31±0.03 15 23
30–40% 0.132±0.004 0.310±0.009 0.14±0.01 9 23
40–50% 0.130±0.003 0.308±0.008 0.067±0.006 7 21
50–60% 0.128±0.006 0.305±0.010 0.025±0.003 9 21
60–70% 0.127±0.004 0.294±0.007 0.012±0.001 3 19
70–80% 0.125±0.005 0.267±0.008 0.0060±0.0006 4 17
p 0–5% 0.139±0.005 0.335±0.009 6.47±0.70 22 23
5–10% 0.137±0.004 0.328±0.008 2.90±0.30 20 23
10–20% 0.135±0.003 0.326±0.008 1.03±0.12 18 23
20–30% 0.134±0.004 0.323±0.007 0.46±0.07 15 23
30–40% 0.132±0.004 0.315±0.008 0.21±0.03 12 23
40–50% 0.130±0.005 0.311±0.007 0.096±0.012 11 23
50–60% 0.128±0.005 0.294±0.005 0.042±0.007 13 23
60–70% 0.126±0.005 0.270±0.008 0.018±0.004 18 23
70–80% 0.123±0.004 0.236±0.007 0.0080±0.0019 26 23
Figure 4Au-Au19.6 GeV π+ 0–5% 0.138±0.004 0.322±0.004 24.14±2.00 9 23
5–10% 0.137±0.004 0.321±0.005 9.50±0.80 6 23
10–20% 0.135±0.005 0.319±0.008 3.75±0.25 7 23
20–30% 0.134±0.004 0.317±0.005 1.97±0.16 12 23
30–40% 0.132±0.003 0.315±0.005 0.87±0.06 18 23
40–50% 0.130±0.004 0.312±0.006 0.43±0.04 20 23
50–60% 0.129±0.005 0.311±0.008 0.22±0.02 16 23
60–70% 0.128±0.004 0.308±0.008 0.10±0.01 16 23
70–80% 0.125±0.005 0.303±0.009 0.048±0.004 14 23
K+ 0–5% 0.140±0.003 0.320±0.007 4.98±0.30 24 23
5–10% 0.138±0.005 0.319±0.009 2.00±0.20 33 23
10–20% 0.136±0.005 0.318±0.007 0.75±0.05 33 23
20–30% 0.135±0.005 0.314±0.007 0.34±0.02 21 23
30–40% 0.133±0.004 0.311±0.007 0.16±0.03 12 23
40–50% 0.130±0.005 0.309±0.009 0.075±0.005 15 22
50–60% 0.129±0.004 0.307±0.008 0.036±0.005 8 22
60–70% 0.128±0.006 0.300±0.010 0.016±0.001 23 20
70–80% 0.126±0.004 0.294±0.009 0.0070±0.0003 25 19
p 0–5% 0.142±0.005 0.338±0.006 5.84±0.70 41 26
5–10% 0.140±0.006 0.336±0.008 2.40±0.30 28 22
10–20% 0.138±0.005 0.334±0.005 0.91±0.12 19 20
20–30% 0.136±0.005 0.324±0.007 0.37±0.06 41 20
30–40% 0.133±0.004 0.316±0.005 0.18±0.03 20 20
40–50% 0.131±0.006 0.312±0.008 0.090±0.016 8 20
50–60% 0.129±0.005 0.295±0.009 0.042±0.007 14 20
60–70% 0.128±0.004 0.275±0.008 0.019±0.003 2 20
70–80% 0.125±0.004 0.237±0.007 0.0080±0.0013 11 20
Figure 5Au-Au27 GeV π+ 0–5% 0.139±0.004 0.326±0.006 26.14±1.80 5 23
5–10% 0.138±0.004 0.324±0.008 11.07±2.00 8 23
10–20% 0.136±0.005 0.323±0.004 4.25±0.25 10 23
20–30% 0.135±0.004 0.322±0.004 1.90±0.15 14 23
30–40% 0.133±0.003 0.321±0.004 0.98±0.10 23 23
40–50% 0.131±0.004 0.320±0.006 0.50±0.03 26 23
50–60% 0.130±0.005 0.318±0.005 0.23±0.03 19 23
60–70% 0.129±0.005 0.317±0.009 0.11±0.01 21 23
70–80% 0.128±0.004 0.315±0.005 0.048±0.005 19 23
K+ 0–5% 0.142±0.005 0.324±0.008 5.11±0.60 53 23
5–10% 0.140±0.005 0.322±0.007 2.15±0.20 52 23
10–20% 0.139±0.003 0.321±0.008 0.82±0.10 55 23
20–30% 0.137±0.006 0.321±0.005 0.37±0.03 43 23
30–40% 0.136±0.004 0.320±0.007 0.18±0.01 25 23
40–50% 0.134±0.004 0.318±0.008 0.86±0.01 9 23
50–60% 0.132±0.005 0.316±0.005 0.040±0.005 8 23
60–70% 0.130±0.004 0.311±0.006 0.014±0.003 12 23
70–80% 0.128±0.004 0.304±0.007 0.0070±0.0004 27 23
p 0–5% 0.144±0.004 0.343±0.007 5.31±0.40 34 20
5–10% 0.143±0.004 0.341±0.007 2.20±0.22 27 20
10–20% 0.141±0.005 0.336±0.007 0.84±0.09 21 20
20–30% 0.139±0.005 0.330±0.006 0.37±0.05 15 20
30–40% 0.137±0.005 0.326±0.008 0.18±0.03 9 20
40–50% 0.134±0.005 0.318±0.005 0.090±0.016 8 20
50–60% 0.131±0.004 0.300±0.008 0.042±0.005 3 20
60–70% 0.129±0.004 0.280±0.005 0.019±0.002 6 20
70–80% 0.126±0.004 0.257±0.007 0.0070±0.0003 9 20
Figure 6Au-Au39 GeV π+ 0–5% 0.141±0.004 0.330±0.007 27.84±2.30 7 23
5–10% 0.139±0.005 0.328±0.007 11.60±0.70 14 23
10–20% 0.138±0.004 0.326±0.006 4.50±0.30 23 23
20–30% 0.136±0.004 0.325±0.005 2.12±0.10 38 23
30–40% 0.135±0.003 0.324±0.008 1.05±0.08 42 23
40–50% 0.135±0.005 0.322±0.005 0.52±0.02 36 23
50–60% 0.134±0.004 0.321±0.008 0.27±0.02 39 23
60–70% 0.132±0.004 0.320±0.007 0.12±0.01 36 23
70–80% 0.130±0.005 0.319±0.008 0.062±0.005 51 23
K+ 0–5% 0.148±0.004 0.328±0.005 5.29±0.40 35 23
5–10% 0.147±0.004 0.327±0.006 2.30±0.15 15 23
10–20% 0.146±0.005 0.328±0.005 0.90±0.08 29 23
20–30% 0.145±0.006 0.324±0.009 0.40±0.03 19 23
30–40% 0.144±0.005 0.323±0.008 0.19±0.01 12 23
40–50% 0.143±0.005 0.321±0.006 0.090±0.010 10 23
50–60% 0.142±0.003 0.317±0.006 0.0040±0.0004 12 23
60–70% 0.140±0.004 0.316±0.005 0.019±0.001 15 23
70–80% 0.138±0.005 0.313±0.008 0.0083±0.0003 18 23
p 0–5% 0.149±0.005 0.359±0.008 4.38±0.50 34 19
5–10% 0.148±0.004 0.348±0.006 1.94±0.30 36 19
10–20% 0.146±0.005 0.346±0.006 0.80±0.12 22 19
20–30% 0.145±0.004 0.340±0.007 0.33±0.05 16 19
30–40% 0.144±0.004 0.335±0.005 0.16±0.03 8 19
40–50% 0.144±0.004 0.330±0.006 0.078±0.014 13 19
50–60% 0.143±0.004 0.300±0.006 0.040±0.005 1 19
60–70% 0.139±0.004 0.281±0.005 0.017±0.002 4 19
70–80% 0.127±0.004 0.274±0.007 0.0080±0.0006 10 19

Figure 2 is the same as Figure 1, but it shows the pT spectra at sNN=11.5 GeV. One can see that Equation (1) fits well the data in Au-Au collisions at 11.5 GeV at the RHIC-BES.

The same as Figure 1, but showing the results at sNN=11.5 GeV.

Figure 3 is also the same as Figure 1, but it shows the pT spectra at sNN=14.5 GeV, where the data are cited from Ref. [20]. Once again, Equation (1) fits well the data in Au-Au collisions at 14.5 GeV at the RHIC-BES.

The same as Figure 1, but showing the results at sNN=14.5 GeV, where the data are cited from Ref. [20].

Figures 4–6 are also the same as Figure 1, but they show the pT spectra at sNN=19.6, 27, and 39 GeV, respectively. Once more, Equation (1) fits well the data in Au-Au collisions at other RHIC-BES energies.

The same as Figure 1, but showing the results at sNN=19.6 GeV.

The same as Figure 1, but showing the results at sNN=27 GeV.

The same as Figure 1, but showing the results at sNN=39 GeV.

It is noteworthy to point out that Equation (1) for the blast-wave model in the system is assumed to be in local thermodynamic equilibrium, and therefore, a single T0 and βT should be obtained by the weight average of different particle species. To see clearly the trends of weight average parameters, Figures 7(a) and 7(b) show the dependences of weight averages T0 and βT on sNN for different event centralities. The symbols represent the parameter values averaged by weighting the yields of different particles which are listed in Table 1. One can see that T0 and βT increase with the increase in sNN from 7.7 to 39 GeV. Meanwhile, T0 and βT increase with the increase in the event centrality from the periphery to the center.

Dependences of weighted averages (a) T0 and (b) βT on sNN for different event centralities as well as (c) T0 on βT for different collision energies and event centralities. The different symbols display different centrality classes in (a) and (b) or different collision energies in (c), which are averaged by weighting the yields of different particles which are listed in Table 1.

In addition, the variation of weight averages T0 on βT for different collision energies and event centralities is displayed in Figure 7(c), where the symbols represent the parameter values averaged by weighting the yields of different particles. One can see that T0 increases with the increase in βT. At higher energy and in central collisions, one sees larger T0 and βT. There is a positive correlation between T0 and βT.

The dependences of mean transverse momentum pT and initial temperature (Ti) on sNN for different event centralities obtained by weighting the yields of different particles are shown in Figures 8(a) and 8(b), respectively, where Ti=pT2/2 according to the color string percolation model [51–53]. One can see that pT and Ti increase with the increase in sNN from 7.7 to 39 GeV. Meanwhile, pT and Ti increase with the increase in event centrality from the periphery to the center.

Dependences of weighted averages (a) pT and (b) Ti on sNN for different event centralities. The different symbols display different centrality classes, which are averaged by weighting the yields of different particles which are listed in Table 1.

We notice that Ti=0.28–0.38 GeV which is quite high for the considered collision energies. Because we obtain Ti from the spectra of particles with nonzero masses, it is possible to have a high value. If we obtain Ti from the spectra of photons, the value will be small. High Ti renders that the excitation degree of the emission source at the stage of the initial state is high. Meanwhile, one of the key issues is whether the transverse flow should be also considered at the initial stage. Naturally, after considering the transverse flow at the initial stage, Ti will be small. It is regretful that we have no clear idea on the extraction of transverse flow at the initial stage. As an alternative method, if we redefine Ti=k0pT2/2 as the initial temperature and βTi=1−k0pT2/2 as the initial transverse flow velocity, where 0<k0<1 is to be determined, we may obtain a small Ti and a nonzero βTi.

The reason for the increase in T0 and βT with the increase in collision energy is due to the fact that more energies are deposited in collisions at higher energy in the considered RHIC-BES energy range. Meanwhile, the system size at higher energy decreases due to a relativistic constriction effect, which results in a smaller volume and then a larger energy density and larger T0. Meanwhile, at higher energy, the squeeze is more violent, which results in a rapider expansion and larger βT.

The reason for the increase in T0 and βT with the increase in event centrality is due to the fact that the central collisions contain more nucleons than the peripheral collisions; then, more energies are deposited in central collisions. Meanwhile, a rapider expansion appears due to more violent squeeze in central collisions, compared to peripheral collisions. As a result, both T0 and βT in central collisions are larger than those in peripheral collisions.

Because of pT and Ti being positive correlation with T0 and βT, with increasing collision energy and event centrality, the increasing trend of T0 and βT results naturally in the increasing trend of pT and Ti. This work shows that the two free parameters T0 and βT and the two derived parameters pT and Ti appear to be similar law on the dependences of collision energy and event centrality. In particular, pT and Ti are model-independent, though we obtain them from model-dependent free parameters T0 and βT in this work. In fact, pT and Ti can be obtained by the pT data themselves if the data are across the possible pT range.

It seems that there are nonmonotonous changes at 11.5 GeV in the excitation functions of βT, pT, and Ti in the most central Au-Au collisions. These nonmonotonous changes reflect the minimum or maximum point of equation of state (EoS) of the matter formed in collisions. At a few GeV to about 10 GeV, the matter formed in collisions is baryon-dominant. At above 10 GeV, the matter formed in collisions is meson-dominant. At around 10 GeV, the baryon number density is the largest [54] due to the competition between projectile/target penetrating/stopping and longitudinal contraction.

It is hard to say whether the minimum or maximum point of EoS of the matter formed in the most central Au-Au collisions at 11.5 GeV is related to the search for the QCD critical end point (CEP) which is the main objective of the BES program performed by the STAR Collaboration. Generally, large nonmonotonous changes or saturations or a slight increase should appear in the excitation functions of some quantities at the critical energy which is the energy corresponding to the CEP. The excitation functions considered in this paper change slightly. Although there is no value in the energy range of less than 7.7 GeV, it is expected that the excitation function increases quickly in the energy range of a few GeV while the onset stage of a slight increase appears at around 10 GeV in the excitation functions of pT and Ti.

It should be noted that there is entanglement in the extraction of T0 and βT. In fact, if one uses smaller T0 and larger βT for central collisions, a decreasing trend for T0 from peripheral to central collisions can be obtained. Meanwhile, a negative correlation between T0 and βT can also be obtained. Thus, this situation is in agreement with some current references [19, 55, 56]. If one even uses almost invariant or slightly larger T0 and properly larger βT for central collisions, an almost invariant or slightly increased trend for T0 from peripheral to central collisions can be obtained [57]. To show the flexibility in the extraction of T0 and βT, this work has reported an increasing trend for T0 from peripheral to central collisions and a positive correlation between T0 and βT.

In addition, we have taken n0=2 in this work, which closely resembles the hydrodynamic profile as mentioned in Ref. [21]. Although Ref. [58] shows that n0=1 is the closest approximation to hydrodynamics at freeze-out, Ref. [36] shows that n0=2 or 1 does not affect obviously the fit curve and free parameters T0 and βT. If we consider that βr decays quickly from the surface to the center of the emission source, we are inclined to use n0=2. Anyhow, we are not inclined to regard n0 as a free parameter which is too mutable and debatable in our opinion. In current analysis with the blast-wave model [22], not only is n0 mutable (from 0.0±10.1 to 4.3±1.7) but also the pT coverage is narrow and particle-dependent (pT≈0.20–0.70 GeV/c for π+, 0.25–0.75 GeV/c for K+, and 0.35–1.15 GeV/c for p), which uses a single kinetic freeze-out scenario and results in different trends of T0 versus βT from this work. If we also regard n0 as a free parameter and use narrow and particle-dependent pT coverage, consistent result with current analysis [22] can be naturally obtained by us.

Indeed, there are too much uncertainties arising from the choice of fit function and flow profile and from the well-known ambiguity in the fit results—in a single pT spectrum, it is always possible to trade βT against T0. That is, T0 and βT is negatively correlative for a given pT spectrum. It is possible that we may use suitable T0 and βT for a set of pT spectra and obtain a positive or negative correlation. In a positive correlation, decreasing T0 and increasing βT will result in a negative correlation. Contrarily, in a negative correlation, increasing T0 and decreasing βT will result in a positive correlation. Indeed, there is an influence if we use a changeable pT coverage and/or n0 choice on the extraction of the two free parameters. In our opinion, to reduce the uncertainties, one should use a fixed flow profile (n0) and wide and fixed pT coverage for different particles. In fact, we have used n0=2 and pT<2.5 GeV/c for different particles in this work and used a multiple freeze-out scenario such as that used in Ref. [59].

Our result (Table 1) shows that the heavier the particle is, the higher T0 and the smaller βT correspond. This result is in agreement with the hydrodynamic-type behavior [4]. The final T0 and βT are averaged by weighting different particle yields, which shows a positive correlation between T0 and βT (Figure 7). Our result is in agreement with the alternative method [36, 37] in which T0 is regarded as the intercept in the linear relation of T versus m0 and βT is regarded as the slope in the linear relation of pT versus m0γ¯, where γ¯ is the mean Lorentz factor in the source rest frame. Our result is also in agreement with a very recent work [60] which uses the same method as ours. If the negative correlation can be explained as the result of a longer lifetime (lower excitation degree) which corresponds to lower T0 and a quicker expansion (stronger squeeze) which corresponds to larger βT, the positive correlation can be explained as the result of a high excitation degree which corresponds to high T0 and quick expansion (strong squeeze) which corresponds to large βT.

This whole phenomenal analysis results in degrees of thermal motion and collective expansion that are reflected by T0 and βT, respectively. With the increasing collision energy, the system may undergo different evolution processes. In the considered RHIC-BES energy range, the violent degree of collisions increases with increasing collision energy. The trends of T0 and βT show approximately monotonous increase in which large fluctuation does not appear, though there are nonmonotonous changes at 11.5 GeV in some cases. The evolution processes at the considered six energies show similar behaviors to each other.

 4. Conclusions

The main observations and conclusions are summarized as follows.

By using the method of data-driven reanalysis, the blast-wave model with the Boltzmann-Gibbs statistics is used to analyze the collision energy-dependent and event centrality-dependent double-differential transverse momentum spectra of charged particles (π+, K+, and p) produced in the midrapidity interval in Au-Au collisions at the RHIC-BES energies. The contribution of soft excitation is considered in this work, but the contribution of the hard process is not excluded if available

As the free parameters, the kinetic freeze-out temperature T0 and transverse flow velocity βT are extracted with the blast-wave model. Both T0 and βT increase with the increase in collision energy due to more violent collisions at higher energy. The two parameters also increase with the increase in event centrality, as the central collisions contain more nucleons which means more energy deposited and more violent collisions and squeeze, compared with peripheral collisions

As the derived parameters, the mean transverse momentum pT and initial temperature Ti appear to be similar law to the free parameters T0 and βT when we study the dependences of parameters on collision energy and event centrality. Although T0 and βT are model-dependent, pT and Ti are generally model-independent. There is no large fluctuation in the excitation function of the considered parameters at the RHIC-BES, which means a similar collision mechanism

 Data Availability

The data used to support the findings of this study are included within the article and are cited at relevant places within the text as references.

 Ethical Approval

The authors declare that they are in compliance with ethical standards regarding the content of this paper.

 Disclosure

The funding agencies have no role in the design of the study; in the collection, analysis, or interpretation of the data; in the writing of the manuscript; or in the decision to publish the results.

 Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

 Acknowledgments

We thank Dr. Muhammad Usman Ashraf for his kind help. This work was supported by the National Natural Science Foundation of China under Grant No. 11575103, the Chinese Government Scholarship (China Scholarship Council), the Scientific and Technological Innovation Programs (STIP) of Higher Education Institutions in Shanxi under Grant No. 201802017, the Shanxi Provincial Natural Science Foundation under Grant No. 201701D121005, and the Fund for Shanxi “1331 Project” Key Subjects Construction.

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