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Z. J. Jiang, Dongfang Xu, Yan Huang, "A Description of Pseudorapidity Distributions of Charged Particles Produced in Au+Au Collisions at RHIC Energies", Advances in High Energy Physics, vol. 2018, Article ID 1369098, 8 pages, 2018. https://doi.org/10.1155/2018/1369098
A Description of Pseudorapidity Distributions of Charged Particles Produced in Au+Au Collisions at RHIC Energies
In heavy ion collisions, charged particles come from two parts: the hot and dense matter and the leading particles. In this paper, the hot and dense matter is assumed to expand according to the hydrodynamic model including phase transition and decouples into particles via the prescription of Cooper-Frye. The leading particles are as usual supposed to have Gaussian rapidity distributions with the number equaling that of participants. The investigations of this paper show that, unlike low energy situations, the leading particles are essential in describing the pseudorapidity distributions of charged particles produced in high energy heavy ion collisions. This might be due to the different transparencies of nuclei at different energies.
The BNL Relativistic Heavy Ion Collider (RHIC) accelerates nuclei up to the center-of-mass energies from a dozen GeV to 200 GeV per nucleon. In the past decade, the measurements from such collisions have triggered an extensive research for the properties of matter at extreme conditions of very high temperature and energy densities [1–33]. One of the most important achievements from such research is the discovery that the matter created in nucleus-nucleus collisions at RHIC energies is in the state of strongly coupled quark-gluon plasma (sQGP) exhibiting a clear collective behavior nearly like a perfect fluid with very low viscosity [10–33].
The best approach for describing the space-time evolution of fluid-like sQGP is the relativistic hydrodynamics. However, owing to the formidable complexities of hydrodynamic equations, the most analytical work so far is mainly limited to 1+1 expansion for a perfect fluid with simple equation of state, which can be found as an important application in the analysis of the pseudorapidity distributions of charged particles in high energy physics. In this paper, combining the effects of leading particles, we will discuss such distributions in the framework of 1+1 hydrodynamic model including phase transition .
2. A Brief Introduction to the Model
Here, for the purpose of completeness and applications, we will list the key ingredients of the hydrodynamic model .
The expansion of fluid is subject to the conservation of energy and momentum. This is reflected in continuity equationwhere is the energy-momentum tensor. For a perfect fluidwhereis the 4-velocity of fluid and is its rapidity. and in (2) are the energy density and pressure of fluid, which meet the thermodynamical relationswhere and are the temperature and entropy density of fluid, respectively. To close (1), another relation, namely, the equation of stateis needed, where is the sound speed of fluid, which takes different values in sQGP and in hadronic phase.
Project (1) to the direction of and the direction perpendicular to , respectively. This leads to equationsEquation (6) is the continuity equation for entropy conservation. Equation (7) means the existence of a scalar function satisfying relationsFrom and Legendre transformation, Khalatnikov potential is introduced via relationIn terms of , the variables and can be expressed aswhere is the initial temperature of fluid and . Through the above equations, the coordinate base of is transformed to that of , and (6) is translated into the so-called telegraphy equation
Along with the expansions of matter created in collisions, it becomes cooler and cooler. As its temperature drops from the initial to the critical , phase transition occurs. The matter transforms from sQGP state to hadronic state. The produced hadrons are initially in the violent and frequent collisions. The major part of these collisions is inelastic. Hence, the abundances of identified hadrons are changing. Furthermore, the mean free paths of these primary hadrons are very short. The movement of them is still like that of a fluid meeting (11) with only difference being the value of . In sQGP, , which is the sound speed of a massless perfect fluid, being the maximum of . In the hadronic state, . At the point of phase transition, that is, as , is discontinuous.
The solution of (11) for the sector of sQGP is where is a constant determined by tuning the theoretical results to experimental data and is the 0th-order modified Bessel function of the first kind, and
3. The Pseudorapidity Distributions of Charged Particles
(1) The Invariant Multiplicity Distributions of Charged Particles Frozen Out from sQGP. From Khalatnikov potential , the rapidity distributions of charged particles frozen out from fluid-like sQGP read where is the area of overlap region of collisions, being the function of impact parameter or centrality cuts. Inserting (10) into the above equation, the part in the round brackets becomes
With the expansions of hadronic matter, it continues becoming cooler. According to the prescription of Cooper-Frye , as the temperature drops to the freeze-out temperature , the inelastic collisions among hadrons cease. The yields of identified hadrons remain unchanged becoming the measured results in experiments. The invariant multiplicity distributions of charged particles equal [10, 15, 34]where is the transverse mass of produced charged particle with rest mass . in (18) is the baryochemical potential. For Fermi charged particles, in the denominator of (18), and for Bosons, . The meaning of (18) is evident. It is the convolution of with the energy of the charged particles in the state with temperature .
The integral interval of in (18) is . The integrand is evaluated with . At this moment, the fluid freezes out into the charged particles. Replacing in (17) by of (14), it becomeswherewhere is the 1st-order modified Bessel function of the first kind.
(2) The Invariant Multiplicity Distributions of Leading Particles. Investigations have shown that the leading particles are formed outside the overlap region of collisions [35, 36]. The generation and movement of them are therefore beyond the scope of hydrodynamic description and should be treated separately.
In our previous work [24–26], we once argued that the rapidity distributions of leading particles take the Gaussian formwhere and are, respectively, the central position and width of distributions. It can be expected that should increase with increasing energies and centrality cuts, while should not, at least not apparently, depend on the energies, centrality cuts, and even colliding systems. The specific values of them can be determined by comparing the theoretical results with experimental data. in (21) represents the number of leading particles, which, for an identical nucleus-nucleus collision, equals half of the number of participants.
The investigations have shown that , for certain rapidity, the invariant multiplicity distributions of leading particles possess the formwhere is a constant. Then, as a function of rapidity, the invariant multiplicity distributions of leading particles can be written aswhich is normalized to .
(3) The Pseudorapidity Distributions of Charged Particles. Writing invariant multiplicity distributions in terms of pseudorapidity, we havewhereTo fulfill the transformation of (24), another relationis in order.
Substituting (18) and (23) into (24) and carrying out the integration of , we can get the pseudorapidity distributions of charged particles produced in high energy heavy ion collisions. Figures 1 and 2 show such distributions in Au+Au collisions at and 62.4 GeV, respectively. The solid dots in the figures are the experimental measurements . The dashed, dashed-dotted, and dotted curves are, respectively, the contribution from pions, kaons, and protons got from the hydrodynamic result of (18). The dotted-star curves are the components of leading particles obtained from (23). The solid curves are the sums of the four types of curves. for each curve is listed in Table 1. It can be seen that the combined contribution from both hydrodynamics and leading particles matches up well with experimental data.
Experiments have shown that the overwhelming majority of charged particles produced in Au+Au collisions at GeV consists of pions, kaons, and protons with proportions of about 84%, 12%, and 4%, respectively , which are roughly independent of energies, centrality cuts, and colliding systems. In calculations, the ratios of these three kinds of particles take about the same as these values. in (15) takes the values of and 0.42 for and 62.4 GeV from the investigations of [15, 39–41]. in (15) takes the well-recognized value of MeV. The freeze-out temperature takes the values of MeV from the studies of , which also shows that the baryochemical potential in (18) is about equal to 20 and 50 MeV for and 62.4 GeV, respectively. For the most central collisions at these two different energies, in (15) takes the values of and 0.68 GeV referring to those given in . This allows us to determine the constant in (16) to be , , and for pions, kaons, and protons, respectively. Keeping unchanged, is fixed for the rest centrality cuts by making theoretical results fit in with experimental data. The results are listed in Table 1. It can be seen that decreases slowly with increasing centralities especially in the first four cuts. Table 1 also lists the central position in (21). As addressed above, it increases with increasing energies and centralities. The width parameter in (21) takes the value of a constant of , being independent of energies and centrality cuts. The parameter in (23) takes the value of for the two different energies.
Figure 3 shows the pseudorapidity distributions of charged particles produced in Au+Au collisions at GeV. for each curve is listed in Table 1. The meanings of different types of curves are the same as those in Figures 1 and 2. It can be seen that, in the absence of leading particles, the hydrodynamics alone can give a good description to the experimental observations. This is different from Figures 1 and 2, where leading particles are essential in fitting experimental data. This difference might be caused by the different transparencies of nuclei in different energies. As the analyses given in , in central Au+Au collisions at GeV, the leading particles are located at about . This position is far away from the mid-rapidity region, where, relative to the low yields of charged particles frozen out from sQGP, the effect of leading particles is evident which should be considered separately. On the contrary, in case of Au+Au collisions at GeV, . This position is so close to the mid-rapidity region that the effect of leading particles is hidden by the large yields of charged particles generated from the freeze-out of sQGP. Therefore, there is no need to consider the contribution of leading particles separately.
By taking into consideration the effect of leading particles, the hydrodynamic model incorporating the phase transition is used to analyze the pseudorapidity distributions of charged particles produced in Au+Au collisions at RHIC energies.
The hydrodynamic model contains rich information about transport coefficients of sQGP, such as the sound speed in sQGP, the sound speed in hadronic phase, the phase transition temperature , the chemical freeze-out temperature , the baryochemical potential , and the initial temperature . With the exception of , the other five coefficients take the values either from the well-known theoretical results or from experimental measurements. As for , there are no widely accepted results so far. In our calculations, in the most central Au+Au collisions at and 62.4 GeV takes the value referring to those given by other investigations, which enables us to ascertain the constant in (16). In the rest centrality cuts and in Au+Au collisions at GeV, is determined by maintaining unchanged and comparing the theoretical results with experimental data.
The leading particles, by conventional definition, are the particles carrying on the quantum numbers of colliding nucleons and taking away the most part of incident energy. They are separately in projectile and target fragmentation region. The present investigations show that the importance of leading particles in describing the pseudorapidity distributions of charged particles produced in heavy ion collisions is related to the incident energy. At high energy, owing to the high transparency of nuclei, the contribution of leading particles is evident and indispensable, while, at low energy, as a result of poor transparency of nuclei the effect of leading particles is integrated with the results of freeze-out of sQCD. It does not need to be dealt with separately.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is supported by the Shanghai Key Lab of Modern Optical System.
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Copyright © 2018 Z. J. Jiang et al. 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.