# Study of Proton, Deuteron, and Triton at 54.4 GeV

Authors: M. Waqas; G. X. Peng
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

Transverse momentum spectra of proton, deuteron, and triton in gold-gold (Au-Au) collisions at 54.4 GeV are analyzed in different centrality bins by the blast wave model with Tsallis statistics. The model results are approximately in agreement with the experimental data measured by STAR Collaboration in special transverse momentum ranges. We extracted the kinetic freeze-out temperature, transverse flow velocity, and freeze-out volume from the transverse momentum spectra of the particles. It is observed that the kinetic freeze-out temperature is increasing from the central to peripheral collisions. However, the transverse flow velocity and freeze-out volume decrease from the central to peripheral collisions. The present work reveals the mass dependent kinetic freeze-out scenario and volume differential freeze-out scenario in collisions at STAR Collaboration. In addition, parameterq characterizes the degree of nonequilibrium of the produced system, and it increases from the central to peripheral collisions and increases with mass .


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## Body

## 1. Introduction

The two important stages in the evolution system are chemical and kinetic freeze-out. The degree of excitation of the interacting system at the two stages is different from each other. The chemical and kinetic freeze-out temperatures are used to describe different excitation degrees of interacting system of the two stages. In general, the ratios of different kinds of particles are no longer changed at the stage of chemical freeze-out. The chemical freeze-out temperature can be obtained from different particle ratios in the framework of the thermal model [1–3]. The transverse momentum spectra of different particles are no longer changed at the stage of kinetic freeze-out and thermal/kinetic freeze-out temperature can be obtained from the transverse momentum spectra according to the hydrodynamical model [4].It is important to point out that the transverse momentum spectra even though in a narrowpT range, but it contains both the contribution of thermal motion and transverse flow velocity. The random thermal motion reflects the excitation, and the transverse flow velocity reflects the degree of expansion of the interacting system. In order to extract the kinetic freeze-out temperature (T0), we have excluded the contribution of transverse flow velocity (βT), that is, to disengage the random thermal motion and transverse flow velocity. There are various methods to disengage the two issues. The methods include but are not limited to blast wave fit with Boltzmann Gibbs statistics [5–7], blast wave model with Tsallis statistics [8–10], and alternative methods [11–17].The dependence ofT0 and βT on centrality is a very complex situation. There are two schools of thought. (1) T0 increase decrease from the central to peripheral collisions [18–21] and (2) T0 increase from the central to peripheral collisions [22, 23]. Both have their own explanations. Larger T0 in the central collisions explain higher degree of excitation of the system due to more violent collisions, while smaller T0 in the central collisions indicates longer lived fireball in the central collisions. It is very important to find out which collision system contains larger T0. Furthermore, there are several opinions about the freeze-out of particles which include single, double, and multiple kinetic freeze-out. It is also very important to dig out the correct freeze-out scenario.In the present work, we will analyze thepT spectra of proton, deuteron, and triton and will extract T0 and βT. Deuteron and triton are light nuclei. The fundamental mechanism for light nuclei production in relativistic heavy ion collision is not well understood [24–26]. Coalescence of anti(nucleons) is a possible approach [27–31]. Because of small binding energies (d with 2.2 MeV and t with 8.8 MeV), the light nuclei cannot persists when the temperature is much higher than their binding energy. The typical kinetic freeze-out temperature is around 100 MeV for light hadrons, so they might disintegrate and be formed again by final-state coalescence after nucleons are decoupled from the hot and dense system. Hence, the study of the light nuclei can be useful in the extraction of information of nucleon distribution at the freeze-out [27, 30, 32].Before going to the formalism, we would point out that the concept of the volume is also important in high-energy collisions. The volume occupied by the ejectiles, when the mutual interactions become negligible and the only force they feel is the columbic repulsive force, is called the kinetic freeze-out volume (V). Various freeze-out volumes occur at various freeze-out stages, but we are only focusing on the kinetic freeze-out volume V in the present work. The freezeout volume gives information about the coexistence of phase transition and is important in the extraction of multiplicity, microcanonical heat capacity, and its negative branch or shape of the caloric curves under the thermal constraints can be obtained V.The remainder of the paper consists of method and formalism in Section2, followed by the results and discussion in Section 3. In Section 4, we summarized our main observations and conclusions.


## 2. The Method and Formalism

In high-energy collisions, there are two types of particle production process: (1) soft process ad (2) hard process. For soft process, there are various methods which include but are not limited to blast wave model with boltzmann Gibbs statistics [5–7], blast wave model with Tsallis statistics [8–10], Hagedorn thermal model [20], and standard distribution [33, 34]. We are interested in the blast wave model with Tsallis statistics. According to [8], the blast wave fit with Tsallis statistics results in the probability density function be
(1)f1pT=1NdNdpT=CgV2π2pTmT∫−ππdϕ∫0Rrdr×1+q−1T0mTcoshρ−pTsinhρ×cosϕ−1/q−1,where C denotes the normalization constant that leads the integral in Equation (1) to be normalized to 1, g is the degeneracy factor which is different for different particles based on gn=2Sn+1, mT=pT2+m02 is the transverse mass, m0 denotes rest mass of the particle, ϕ shows the azimuthal angle, r is the radial coordinate, R is the maximum r, q represents the measure of degree of deviation of the system from an equilibrium state, ρ=tanh−1βr is the boost angle, βr=βSr/Rn0 is a self-similar flow profile, βS represents the flow velocity on the surface, as a mean of βr,βr=2/R2∫0Rrβrdr=2βs/n0+2=2βs/3, and n0=1. Furthermore, the index −1/q−1 in Equation (1) can be substituted by −q/q−1 due to the reason that q is being close to 1. This substitution results in a small and negligible divergence in the Tsallis distribution.In case of a not too widepT range, the above equation can be used to describe the pT spectra and we can extract T0 and βT. But, if we use the wide pT spectra, then the contribution of the hard scattering process can be considered. According to quantum chromodynamics (QCD) calculus [35–37], the contribution of hard process is parameterized to be an inverse power law.
(2)fHpT=1NdNdpT=ApT1+pTp0−n,which is the Hagedorn function [38, 39], A is the normalization constant while p0 and n are the free parameters.The superposition of the soft and hard scattering processes can be used if thepT spectra are distributed in a wide range. If Equation (1) describes the contribution of the soft process, then the contribution of the hard process can be described by Equation (2). To describe the spectrum in a wide pT range, one can superpose the two-component superposition like this
(3)f0pT=kfSpT+1−kfHpT,where k denotes the contribution fraction of soft excitation and 1−k shows the hard scattering process, fS denotes the soft process which contributes in the low pT region and fH is the hard process which contributes in a whole pT region. The two contributions overlap each other in the low pT region.We may also use the usual step function to superpose the two functions. According to the Hagedorn model [38],
(4)f0=pT=A1θp1−pTfSpT+A2θpT−p1fHpT,where A1 and A2 are the normalization constants that synthesize A1fSp1=A1fHp1 and θx is the usual step function.


## 3. Results and Discussion

Figure1 presents the transverse momentum (pT) spectra 1/2πpTd2N/dydpT of proton, deuteron, and triton in Au-Au collisions at sNN=54.4 GeV. The pT spectra is distributed in different centrality bins of 0–10%, 10−20%, 20−40%, 40−60%, and 60−80% for p and d, and 0–10%, 10−20%, 20−40%, and 40−60% for triton at ∣y∣<0.5, where ∣y∣ denotes the rapidity. The symbols represent the experimental data measured by the STAR Collaborations [40], and the curves are our fitting results by using the blast wave model with Tsallis statistics. Each panel is followed by its corresponding data/fit. The related parameters, χ2 and degree of freedom (dof) are listed in Table 1. One can see that Equation (1) fits well the data in Au-Au collisions at 54.4 GeV at the RHIC.Figure 1
Transverse mass spectra of (a–c)p, d, and t produced in different centrality bins in Au-Au collisions at sNN=54.4 GeV. The symbols represent the experimental data measured by the STAR Collaboration at ∣y∣<0.5 [40]. The curves are our fitted results by Equation (1). Each panel is followed by its corresponding ratios of data/fit.
(a)(b)(c)Table 1
Values of free parameters (T0 and βT), entropy index (q), normalization constant (N0), χ2, and degree of freedom (dof) corresponding to the curves in Figures 1.
CollisionsCentralityParticleT0βTV (fm3)qN0χ2/dofFigure1Au-Au 54.4 GeV0–10%p0.082±0.0060.454±0.0094000±2201.012±0.0040.019±0.0055/1310–20%—0.086±0.0050.434±0.0083800±3001.015±0.0050.0072±0.00046/1320–40%—0.090±0.0070.420±0.0103650±2341.018±0.0070.00194±0.000511/1340–60%—0.094±0.0060.370±0.0083400±1901.022±0.0066×10−4±6×10−57/1360–80%—0.099±0.0020.340±0.0113100±2001.024±0.0071.2×10−4±4×10−529/130–10%d0.127±0.0070.360±0.0103000±1771.004±0.0064.15×10−4±4×10−510/1210–20%—0.130±0.0050.340±0.0092700±2201.007±0.0051.5×10−5±5×10−511/1220–40%—0.134±0.0050.280±0.0102500±1801.011±0.0045.1×10−5±6×10−639/1240–60%—0.138±0.0060.224±0.0082200±2001.013±0.0051.2×10−6±6×10−63/1260–80%—0.142±0.0080.130±0.0092000±2001.016±0.0062.7×10−7±4×10−716/100–10%t0.136±0.0050.340±0.0072200±1651.0002±0.0049.17×10−7±3×10−84/310–20%—0.139±0.0060.320±0.0092000±1501.0006±0.0064×10−7±4×10−83/320–40%—0.148±0.0070.250±0.0081823±1351.0009±0.0071.4×10−8±6×10−812/840–60%—0.154±0.0060.170±0.0101500±1801.0012±0.0063.7×10−6×10−96/3To show the trend of the extracted parameters, Figure2 shows the dependence of kinetic freeze-out temperature on centrality. One can see that T0 in central collisions is smaller, and it is increasing with the decrease of centrality which indicates a decrease of lifetime of fireball from central collisions to peripheral collisions. Furthermore, T0 is observed to be mass dependent, as it is larger from triton, followed by deuteron and then proton which means heavy particles freeze-out early than lighter particles.Figure 2
The dependence ofT0 on centrality.Figure3 shows the centrality dependence of transverse flow velocity. βT is observed to decrease with the decrease of centrality due to the reason that in central collisions the system undergoes more violent collisions and the system expands very rapidly. In addition, βT is observed for smaller for heavy particles.Figure 3
The dependence ofβT on centrality.Figure4 is same as Figure 3 but shows the centrality-dependent freeze-out volume. The freeze-out volume decreases with the decrease of centrality due to decreasing the number of participant nucleons. There are large number of binary collisions due to the rescattering of partons in central collisions, and therefore, the system with more participants reaches quickly to the equilibrium state. Furthermore, the volume differential scenario is observed and heavy particles are observed to have less freeze-out volume and this shows the early freeze-out heavier particles. The different freeze-out of different particles exhibits different freeze-out surfaces of different particles. Figures 5 and 6 are the same. Figure 5 shows the correlation of T0 and βT and Figure 6 shows the correlation of T0 and V. One can see that both T0 and βT, and T0 and V exhibit a two-dimensional anticorrelation band. The larger the T0, the smaller the β and V. Figure 7 shows the dependence of q on centrality. The parameter q is smaller in central collisions, but as we go from the central to peripheral collisions, it is going to increase. The parameter is varying with mass of the particle, larger q is observed for light particles. The interacting system stays at equilibrium because q is being very close to 1. In Figure 8, the parameter N0 decreases with the mass of the particle. N0 basically shows the multiplicity, and it is larger in the central collision which decreases towards periphery.Figure 4
The dependence ofV on centrality.Figure 5
Variation ofT0 with βT.Figure 6
Variation ofT0 with V.Figure 7
Dependence ofq on centrality.Figure 8
Dependence ofN0 on mass and centrality.


### 3.1. Further Discussion

The study ofpT spectra of the particles may give some fruitful information about effective temperature (Teff), initial temperature (Ti), thermal/kinetic freeze-out temperature (T0), thermal freeze-out volume (V) of the interacting system, and transverse flow velocity (βT) of the final-state particles. We use the fitting method to extract this information by using different models and distributions. In the present work, the blast wave model with Tsallis statistics is used.The structure of transverse momentum (pT) spectra of charged particles generated in high-energy heavy ion collisions is very complex. It is not enough to use only one probability density function to describe the pT spectra, though this function can be of various forms, particularly, in the case when the maximum pT reaches to 100 GeV at LHC collisions [41]. Several pT regions are observed by the model analysis [42], including the first region pT<4-6 GeV/c, 4−6GeV/c<pT<17−20GeV/c, and the third region with pT>17−20 GeV/c. The boundaries of different pT regions at the RHIC are slightly lower. It is expected that different pT regions correspond to different interacting mechanisms. Even for the same pT region, there are different explanations due to different model methods and microcosmic pictures.According to [42], different whole features of fragmentation and hadronization of partons through the string dynamics corresponds to different pT regions. The effects and changes by the medium take part in the main role in the first pT region, while it has weak appearance in second pT region. At the same time, in the third region, the nuclear transparency results in a negligible impact of the medium. From number of strings’ point of view, the maximum number of strings is in the second pT region that results in fusion and creation of strings and collective behavior of partons. The second pT region is proposed as a possible area of quark-gluon plasma (QGP) through the string fusion. Due to direct hadronization of the low-energy strings into mesons [42], the first pT region has the minimum number of strings and maximum number of hadrons. In some cases, there may be the contribution region (pT<0.2-0.3 GeV/c) of the very soft process which is due to resonant production of charged particles, e.g., pions, and this region is considered as the fourth pT region. Different components in a unified superposition can describe the four pT regions. We have two methods in order to structure the unified superposition. The first method is the common method of overlapping of the contribution regions of various components; however, the second method is the Hagedorn model [38] which excludes this overlapping. If the contribution of the hard component in the first method is neglected in the low pT region due to its small value, the first method can be changed into the second method. Indeed the contribution to T0 and βT is less for the hard component. If the spectra in the low pT region is analyzed to extract only T0 and βT, then we can give up the second part of Equations (3) and (4). That is, fSpT can be used directly from Equation (1) which also includes the contribution of the very soft component that comes from resonance decays if available in the data. In the present work, the contribution of the hard component in the low pT region if available is included in the extraction of T0 and βT which may cause a slight increase in T0 and/or βT but the relative increase can be neglected due to small values [43]. In the present work, we only use Equation (1), which means that the fraction of the hard component is zero in the low pT region. But, we also show Equations (3) and (4) to show a method for further analysis if necessary.


## 3.1. Further Discussion

The study ofpT spectra of the particles may give some fruitful information about effective temperature (Teff), initial temperature (Ti), thermal/kinetic freeze-out temperature (T0), thermal freeze-out volume (V) of the interacting system, and transverse flow velocity (βT) of the final-state particles. We use the fitting method to extract this information by using different models and distributions. In the present work, the blast wave model with Tsallis statistics is used.The structure of transverse momentum (pT) spectra of charged particles generated in high-energy heavy ion collisions is very complex. It is not enough to use only one probability density function to describe the pT spectra, though this function can be of various forms, particularly, in the case when the maximum pT reaches to 100 GeV at LHC collisions [41]. Several pT regions are observed by the model analysis [42], including the first region pT<4-6 GeV/c, 4−6GeV/c<pT<17−20GeV/c, and the third region with pT>17−20 GeV/c. The boundaries of different pT regions at the RHIC are slightly lower. It is expected that different pT regions correspond to different interacting mechanisms. Even for the same pT region, there are different explanations due to different model methods and microcosmic pictures.According to [42], different whole features of fragmentation and hadronization of partons through the string dynamics corresponds to different pT regions. The effects and changes by the medium take part in the main role in the first pT region, while it has weak appearance in second pT region. At the same time, in the third region, the nuclear transparency results in a negligible impact of the medium. From number of strings’ point of view, the maximum number of strings is in the second pT region that results in fusion and creation of strings and collective behavior of partons. The second pT region is proposed as a possible area of quark-gluon plasma (QGP) through the string fusion. Due to direct hadronization of the low-energy strings into mesons [42], the first pT region has the minimum number of strings and maximum number of hadrons. In some cases, there may be the contribution region (pT<0.2-0.3 GeV/c) of the very soft process which is due to resonant production of charged particles, e.g., pions, and this region is considered as the fourth pT region. Different components in a unified superposition can describe the four pT regions. We have two methods in order to structure the unified superposition. The first method is the common method of overlapping of the contribution regions of various components; however, the second method is the Hagedorn model [38] which excludes this overlapping. If the contribution of the hard component in the first method is neglected in the low pT region due to its small value, the first method can be changed into the second method. Indeed the contribution to T0 and βT is less for the hard component. If the spectra in the low pT region is analyzed to extract only T0 and βT, then we can give up the second part of Equations (3) and (4). That is, fSpT can be used directly from Equation (1) which also includes the contribution of the very soft component that comes from resonance decays if available in the data. In the present work, the contribution of the hard component in the low pT region if available is included in the extraction of T0 and βT which may cause a slight increase in T0 and/or βT but the relative increase can be neglected due to small values [43]. In the present work, we only use Equation (1), which means that the fraction of the hard component is zero in the low pT region. But, we also show Equations (3) and (4) to show a method for further analysis if necessary.


## 4. Conclusions

The main observations and conclusions are summarized here.(a)
The transverse momentum spectra of proton (p), deuteron (d), and triton (t) are analyzed by the blast wave model with Tsallis statistics and the bulk properties in terms of the kinetic freeze-out temperature, transverse flow velocity, and kinetic freeze-out volume are extracted(b)
The kinetic freeze out temperature (T0) is observed to increase from the central to peripheral collisions. However, the transverse flow velocity and freeze-out volume is decreasing from the central to peripheral collisions(c)
The entropy index (q) increasing with from centrality while parameter N0 is decreasing with centrality(d)
The kinetic freeze-out temperature, transverse flow velocity, and freeze-out volume decrease with the increasing mass of the particle. Therefore, the mass differential kinetic freeze-out scenario and volume differential freeze out scenario are observed(e)
Both the entropy index (q) and the parameter N0 decrease with the mass of particle



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*Source: 6674470-2021-04-01.xml*