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M. Waqas, G. X. Peng, "Study of Proton, Deuteron, and Triton at 54.4 GeV", Advances in High Energy Physics, vol. 2021, Article ID 6674470, 9 pages, 2021. https://doi.org/10.1155/2021/6674470
Study of Proton, Deuteron, and Triton at 54.4 GeV
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, parameter characterizes the degree of nonequilibrium of the produced system, and it increases from the central to peripheral collisions and increases with mass .
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 .
It is important to point out that the transverse momentum spectra even though in a narrow 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 (), we have excluded the contribution of transverse flow velocity (), 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 of and on centrality is a very complex situation. There are two schools of thought. (1) increase decrease from the central to peripheral collisions [18–21] and (2) increase from the central to peripheral collisions [22, 23]. Both have their own explanations. Larger in the central collisions explain higher degree of excitation of the system due to more violent collisions, while smaller in the central collisions indicates longer lived fireball in the central collisions. It is very important to find out which collision system contains larger . 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 the spectra of proton, deuteron, and triton and will extract and . 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 (). Various freeze-out volumes occur at various freeze-out stages, but we are only focusing on the kinetic freeze-out volume 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 .
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 , and standard distribution [33, 34]. We are interested in the blast wave model with Tsallis statistics. According to , the blast wave fit with Tsallis statistics results in the probability density function be where denotes the normalization constant that leads the integral in Equation (1) to be normalized to 1, is the degeneracy factor which is different for different particles based on , is the transverse mass, denotes rest mass of the particle, shows the azimuthal angle, is the radial coordinate, is the maximum , represents the measure of degree of deviation of the system from an equilibrium state, is the boost angle, is a self-similar flow profile, βS represents the flow velocity on the surface, as a mean of , and . Furthermore, the index in Equation (1) can be substituted by due to the reason that is being close to 1. This substitution results in a small and negligible divergence in the Tsallis distribution.
In case of a not too wide range, the above equation can be used to describe the spectra and we can extract and . But, if we use the wide 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. which is the Hagedorn function [38, 39], is the normalization constant while and are the free parameters.
The superposition of the soft and hard scattering processes can be used if the 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 range, one can superpose the two-component superposition like this where k denotes the contribution fraction of soft excitation and shows the hard scattering process, denotes the soft process which contributes in the low region and is the hard process which contributes in a whole region. The two contributions overlap each other in the low region.
We may also use the usual step function to superpose the two functions. According to the Hagedorn model , where and are the normalization constants that synthesize and is the usual step function.
3. Results and Discussion
Figure 1 presents the transverse momentum () spectra of proton, deuteron, and triton in Au-Au collisions at GeV. The spectra is distributed in different centrality bins of 0–10%, 10−20%, 20−40%, 40−60%, and 60−80% for and , and 0–10%, 10−20%, 20−40%, and 40−60% for triton at , where denotes the rapidity. The symbols represent the experimental data measured by the STAR Collaborations , 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, 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.
To show the trend of the extracted parameters, Figure 2 shows the dependence of kinetic freeze-out temperature on centrality. One can see that 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, 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 3 shows the centrality dependence of transverse flow velocity. 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, is observed for smaller for heavy particles.
Figure 4 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 and and Figure 6 shows the correlation of and . One can see that both and , and and exhibit a two-dimensional anticorrelation band. The larger the , the smaller the and . Figure 7 shows the dependence of on centrality. The parameter 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 is observed for light particles. The interacting system stays at equilibrium because is being very close to 1. In Figure 8, the parameter decreases with the mass of the particle. basically shows the multiplicity, and it is larger in the central collision which decreases towards periphery.
3.1. Further Discussion
The study of spectra of the particles may give some fruitful information about effective temperature (), initial temperature (), thermal/kinetic freeze-out temperature (), thermal freeze-out volume () of the interacting system, and transverse flow velocity () 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 () 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 spectra, though this function can be of various forms, particularly, in the case when the maximum reaches to 100 GeV at LHC collisions . Several regions are observed by the model analysis , including the first region -6 GeV/c, , and the third region with GeV/c. The boundaries of different regions at the RHIC are slightly lower. It is expected that different regions correspond to different interacting mechanisms. Even for the same region, there are different explanations due to different model methods and microcosmic pictures.
According to , different whole features of fragmentation and hadronization of partons through the string dynamics corresponds to different regions. The effects and changes by the medium take part in the main role in the first region, while it has weak appearance in second 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 region that results in fusion and creation of strings and collective behavior of partons. The second 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 , the first region has the minimum number of strings and maximum number of hadrons. In some cases, there may be the contribution region (-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 region. Different components in a unified superposition can describe the four 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  which excludes this overlapping. If the contribution of the hard component in the first method is neglected in the low region due to its small value, the first method can be changed into the second method. Indeed the contribution to and is less for the hard component. If the spectra in the low region is analyzed to extract only and , then we can give up the second part of Equations (3) and (4). That is, 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 region if available is included in the extraction of and which may cause a slight increase in and/or but the relative increase can be neglected due to small values . In the present work, we only use Equation (1), which means that the fraction of the hard component is zero in the low region. But, we also show Equations (3) and (4) to show a method for further analysis if necessary.
The main observations and conclusions are summarized here. (a)The transverse momentum spectra of proton (), deuteron (), and triton () 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 () 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 () increasing with from centrality while parameter 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 () and the parameter decrease with the mass of particle
The data used to support the findings of this study are included within the article and are cited at relevant places within the article with in the text as references.
The authors declare that they are in compliance with ethical standards regarding the content of this paper.
The funding agencies have no role in the design of this study; in the collection, analysis, or interpretation of the data in writing 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.
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Copyright © 2021 M. Waqas and G. X. Peng. 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.