On pseudorapidity distribution and speed of sound in high energy heavy ion collisions based on a new revised Landau hydrodynamic model

We propose a new revised Landau hydrodynamic model to study systematically the pseudorapidity distributions of charged particles produced in heavy ion collisions over an energy range from a few GeV to a few TeV per nucleon pair. The interacting system is divided into three sources namely the central, target, and projectile sources respectively. The large central source is described by the Landau hydrodynamic model and further revised by the contributions of the small target/projectile sources. In the calculation, to avoid the errors caused by an unapt conversion or non-division, the rapidity and pseudorapidity distributions are obtained respectively. The modeling results are in agreement with the available experimental data at relativistic heavy ion collider (RHIC), large hadron collider (LHC), and other energies for different centralities. The value of square speed of sound parameter in different collisions has been extracted by us from the widths of rapidity distributions. Our results show that, in heavy ion collisions at RHIC and LHC energies, the central source undergoes through a phase transition from hadronic gas to quark-gluon plasma (QGP) liquid phase; meanwhile, the target/projectile sources remain in the state of hadronic gas. The present work confirms that the QGP is of liquid type rather than that of a gas. The whole region of participants undergoes through a mixed phase consisting of a large quantity of (>90%) QGP liquid and a small quantity of (<10%) hadronic gas.


Introduction
In fields of particle physics and nuclear physics, heavy ion (nucleus-nucleus) collisions at high energies are a very important research subject. Many charged and neutral particles are produced in final state of the collisions and can be measured in experiments. To study the behavior of the particles could help us to understand the processes of interacting system in the collisions. The pseudorapidity distributions of charged particles are an important quantity which can be measured in the early stage of the measurements. According to ecumenical textbooks, the pseudorapidity  is simply defined as   ln tan 2 , where  is the emission angle of the considered particle.
The pseudorapidity distributions can be used to study stopping and penetrating powers of the target and projectile nuclei, positions and contribution ratios of different 1 E-mail: fuhuliu@163.com; fuhuliu@sxu.edu.cn emission sources, contribution ratios of leading nucleons, square speed of sound, and other related topics. Many models have been introduced in order to study the pseudorapidity distributions, transverse momentum distributions, azimuthal correlations, and other distributions and correlations in relativistic heavy ion collisions. These theoretical models can be classified mainly into two classes: i) thermal and statistical model, and ii) transport and dynamical model. Among them, the three-fireball model [1][2][3][4][5][6], the three-source relativistic diffusion model [7][8][9][10][11][12], and the Landau hydrodynamic model [13][14][15][16][17][18][19][20][21] are of great interested for us and will be used in the present work.
A lot of experimental data on nuclear collisions at high energies has been published in literature. The relativistic heavy ion collider (RHIC) has performed gold-gold (Au-Au), copper-copper (Cu-Cu), deuteron-gold (d-Au), and other collisions at various GeV energies [22][23][24][25]. Also, the large hadron collider (LHC) has performed lead-lead (Pb-Pb), proton-lead (p-Pb), and other collisions at TeV energies [26]. These collider experiments show rich and exciting results on the pseudorapidity distributions and other distributions. In fixed target experiments as well (such as in nuclear emulsion experiments at high energies), proton to gold induced emulsion (p-Em and Au-Em) collisions have presented pseudorapidity distributions with abundant structures [27][28][29]. More other ions such as helium, carbon, oxygen, neon, silicon, sulphur, krypton, and etc. have also been used [30,31].
In this paper, we propose a new revision on the Landau hydrodynamic model based on the three-source picture to describe systematically the pseudorapidity distributions of charged particles produced in Au-Au, Cu-Cu, Pb-Pb, d-Au, Au-Em, and p-Em collisions at high energies. The central source with a large enough contribution is described by the Landau hydrodynamic model. The small contributions of the target and projectile sources are considered as a revision on the Landau hydrodynamic model for the central source. Based on the descriptions of pseudorapidity distributions, the values of square speed of sound parameter are extracted from the widths of rapidity distributions.

The model and calculation method
Enlightening by the three-fireball model [1][2][3][4][5][6] and the three-source relativistic diffusion model [7][8][9][10][11][12], we classify the particle emission sources into three types: a central source (C), a target source (T), and a projectile source (P). Generally, the central source's center stays at the mid-rapidity, and its contribution covers a wide enough region in the whole rapidity distribution. The central source includes the contributions of all produced particles and most non-leading nucleons. The target source stays in the left side of the central source's center and covers an appropriate region. It includes the contributions of all leading and a few non-leading target nucleons. The projectile source stays in the right side of the central source's center and covers an appropriate region. It includes the contributions of all leading and a few non-leading projectile nucleons. As revisions of the central source, the distribution ranges of particles produced from the target and projectile sources are covered by that from the central source.
The central source can be described by the Landau Hydrodynamic model [13][14][15][16][17][18][19][20][21]. The target and projectile sources are assumed to emit isotropically particles in their respective rest fames. The three sources may have different contribution ratios which are regarded as free parameters in the model. The central source can produce pions, kaons, nucleons, and other particles. The target and projectile sources emit only nucleons due to the two sources consisting of leading and non-leading nucleons. In fact, the target/projectile sources are a complementarity and revision of the central source. Generally, spectator nucleons appear in the very backward or forward rapidity region in non-central collisions; their contributions should be subtracted in our analyses. Most experimental distributions don't contain the contributions of spectator nucleons due to relative central collisions and narrow region of measurement. Our treatment is in fact a new revision of the Landau Hydrodynamic model [13][14][15][16][17][18][19][20][21].
In center-of-mass reference frame and for symmetric collisions such as Au-Au and Pb-Pb collisions, the central source stays at the peak position at 0 C y  , the target source stays at a peak position in the range of 0 T y  , and the projectile source stays at a peak position in the range of 0 P y  . In an actual calculation for symmetric collisions, T y and P y are regarded as free parameters. The distributions contributed by the target/projectile sources revise (in fact increase) somewhere the probabilities underestimated by the central source. In the case of considering asymmetric collisions such as d-Au collisions, the central source stays at a peak position at 0 C y  , and the situations for the target/projectile sources are similar to those in symmetric collisions. In an actual calculation for asymmetric collisions, C y , T y , and P y are regarded as free parameters, where C y is close to the peak position. In the fixed target experiments in laboratory reference system, 0 C y  for both symmetric and asymmetric collisions. Obviously, we have T C P y y y   in any case. In center-of-mass or laboratory reference frame, according to the Landau hydrodynamic model [13][14][15][16][17][18][19][20][21], particles produced in the central source can be described by a Gaussian rapidity ( y ) distribution with a width of  [15][16][17][18][19][20][21]. We where N denotes the multiplicity, 0 N is the normalization constant, and  should be large enough to cover a wide enough rapidity region. At the same time, the transverse momentum ( T p ) distribution is assumed to obey the simplest form of Boltzmann distribution [32] where 0 C is the normalization, 0 m denotes the rest mass of the considered particle, k denotes the Boltzmann constant, and T denotes the source temperature. In the simplest form [Eq. (2)], the chemical potential and the distinction for fermions and bosons are not included due to small effects at high energy.
Based on Eqs. (1) and (2), we can get a series of values of pseudorapidity for the particles produced from the central source by using the Monte Carlo method. Then, we can get the pseudorapidity distribution contributed by the central source by the statistical method. Now we describe the process which calculates one of values of pseudorapidity. Let A given T p satisfies Repeating the above process [Eqs.
(3)- (7)] for many times, a series of values of pseudorapidity for the particles produced in the central source can be obtained. The situation for the target/projectile sources is somewhere different from that for the central source. We now describe the process which calculates one of values of pseudorapidity for the particles produced from the target or projectile source. Let 4 R denote random number in [0,1]. In the rest frame of the target or projectile source, particles are assumed to emit isotropically. The Monte Carlo method gives the emission angle /  to be where 0 0   (or  ) in the case of the first item in the above equation being greater than 0 (or less than 0). This isotropic emission results in a Gaussian pseudorapidity distribution with the width of 0.91-0.92 [27]. In this study, we have not studied further the width corresponding to the target and projectile sources. The transverse momentum T p has the same expression as Eqs. (2) and (4). The longitudinal momentum / The square speed of sound parameter can be obtained from the above equation. It should be noted that not only the central source but also the target/projectile sources discussed in the present work are formed in the participants, but not in the spectators which are mainly appeared in non-central or peripheral collisions. The relation between the square speed of sound and rapidity distribution width can be applied for the groups of particles produced from the three sources. We would like to point out that, as the previous version, it applies the hydrodynamic model for hadron production that was originally developed by Landau [13], and Belen'kji and Landau [14]. Subsequently other researchers have extended it [15][16][17][18][19][20], always based on an expanding central source for the produced particles in the rapidity/pseudorapidity space. The variance of this Gaussian source in Landau's original work is in the pseudorapidity space and later works in the rapidity space. Generally, the variance of this Gaussian source is related analytically to the logarithm of the center-of-mass energy, and the speed of sound. Hence the speed of sound can in principle be inferred from a comparison of calculated (pseudo-)rapidity distributions with data given this particular model. We think that the application of Gaussian source in the rapidity space is a revision of that in the pseudorapidity space. Therefore, the Gaussian rapidity distribution is used in the present work.
In addition, according to the Landau hydrodynamic model [13][14][15][16][17][18][19][20][21], in hadron-nucleus and nucleus-nucleus collisions, the number and energy of primary hadrons, which form an ideal liquid, fluctuate from event to event. Therefore, in already formed ideal liquid in laboratory system, the (pseudo)rapidity of its center-of-mass also fluctuates on the (pseudo)rapidity scale. The total inclusive distribution is then the sum of Gaussian distributions with different centers and widths, and the shape of the inclusive distributions differs from the form of a Gaussian distribution, in the laboratory system. These situations are changed in the center-of-mass system, where the mentioned centers are the same, and the mentioned widths fluctuate slightly from event to event within a given centrality range. Because our calculation is performed in the center-of-mass system and for the given centrality ranges in most cases, we will not take into account the fluctuations of the center and width of Gaussian distribution for the central source from event to event for the purpose of convenience. Even if for a few mini-bias samples and emulsion experiments in the laboratory system, our treatment gives an average for different events. For the target and projectile sources in their respective rest frames, we will not take into account the fluctuations of the center either, and the width is fixed due to isotropic emission and given temperature.
In the calculation, we don't need either to consider the Jacobian transformation between the rapidity and pseudorapidity spaces explicitly, but instead calculate pseudorapidity distribution for produced charged hadrons directly in the Monte Carlo approach, in which we assume Gaussian rapidity distribution and Boltzmann transverse momentum distribution for the central sources, and isotropic emission and Boltzmann transverse momentum distribution for the target and projectile sources in their respective rest frames. At the same time, we assume Landau's prescription for the speed of sound to be valid for the three sources. Based on the description of experimental pseudorapidity distribution, speed of sound parameters are then determined for the three sources from their respective widths of rapidity distributions.

Comparisons with experimental data
The pseudorapidity distributions,  Table 1 will be discussed later. To estimate the values of parameters, the least-squared fitting method is used. For the symmetric collisions, the peak position P y and contribution ratio P k of the projectile source, and the peak position C y and contribution ratio C k of the central source, can be given by P  Table 2 will be discussed later. Likewise, one can see that the modeling results describe the experimental data of the PHOBOS collaboration [22]. In the considered energy and centrality ranges, the dependence trends of parameters on  Table 3. The last two columns in Table 3 will be discussed later. One can see that the results calculated by us are in agreement with the experimental data. All of the three free parameters have no obvious dependence on NN s and C in the error range and in the centrality range of 0-30%. Because T k in Table 3 is very small (  0.3%), the contributions of target and projectile sources can be neglected.
The pseudorapidity distributions of charged particles produced in d-Au collisions for minimum-bias sample and different centralities at  Table 4. The last two columns in Table 4 will be discussed later. One can see that the results calculated by us are in agreement with the experimental data. Because the considered collisions are asymmetric, we have to use more parameters. From Table 4 one can see that T y and T k have almost no change, C y and P y have right shifts, P k increases, and  decreases with increase of C. These results are caused by the very asymmetric collisions. It should be noted that d-Au collisions have been analyzed by Wolschin et al. in their previous work [11] within their three-source model [7][8][9][10][11][12]. The present work is to some extent a repetition of the original work of Wolschin et al. [11], although different methods are used. Figure 10 shows the pseudorapidity distributions,   1 EV N dN dy , of shower (relativistic singly charged) particles produced in Au-Em collisions with different impacting types at beam energy beam E  10.6-10.7A GeV, where N and N EV denotes the numbers of shower particles and events respectively. The circles represent the experimental data quoted in reference [27], and the curves are the results calculated by us. The impacting types such as very central, semi-central, peripheral, and very peripheral are assumed by us to correspond to centrality percentages 10%, 40%, 60%, and 90%, respectively. The values of T y , C y , P y , T k , P k ,  , and 2 / dof  are listed in Table 5. The last two columns in Table 5 will be discussed later. One can see that the experimental data can be described by our modeling results. The parameters T y , T k , and P k don't show an obvious change, the parameters C y and P y show right shifts, and the parameter  shows an increase when the centrality percentage increases.
The pseudorapidity distributions of combined shower and grey (cascaded singly charged) particles produced in minimum-bias p-Em collisions at NN s  6.7, 11.2, 19.4, 23.7, and 38.7 GeV are presented in Figures 11(a)-11(e), respectively. The circles represent the experimental data quoted in references [28,29], and the curves are our modeling results. By fitting the experimental data of the fixed target emulsion experiments, we get the values of T y , C y , P y , T k , P k ,  , and 2 / dof  listed in Table 6. The last two columns in Table 6 will be discussed later. The modeling results describe the experimental data. It is shown that T y doesn't change obviously, C y , s NN c s  relations for the target and central sources are presented due to these relations for the projectile source to be trivial. All information on the energies and types are marked in the figure panels. The intercepts, slopes, and 2 / dof  corresponding to the lines in Figure 14 are listed in Table 9. One can see that, in most cases, 2 s c for the central source is greater than that for the target source, which renders that the central source has more density due to more energy deposit. The values of 2 ( ) s c C for central Au-Au and Cu-Cu collisions are less than those for peripheral collisions, which renders that the central collisions undergoes through longer transverse expanding time which results in lower density due to larger volume.
Although we studied the square speed of sound parameter in our previous work [36, 37], the physics picture and calculation method used there are different from the present work. In our previous work, we used a four-source picture, two (target and projectile) participants and two (target and projectile) spectators. The rapidity and pseudorapidity distributions were not distinguished for the purpose of convenience. In addition, our previous work seems not to correspond with the three-source picture [1][2][3][4][5][6][7][8][9][10][11][12]. The present work uses a large enough central source which is described by the Landau hydrodynamic model [13][14][15][16][17][18][19][20][21] and two small (target and projectile) sources which revise somewhere the contribution of the central source. At the same time, the rapidity and pseudorapidity distributions are strictly distinguished, which avoids the errors caused by non-division or inapposite conversion. Comparing with the results of the four-source picture in our previous work, larger 2

Discussions
Although we have used the four-source picture in our previous work [36, 37] which tries to combine the Landau hydrodynamic model [13][14][15][16][17][18][19][20][21] with the Glauber model [50] for participants and spectators and ends up with four Gaussian sources for particle production, it doesn't mean that a mid-rapidity source is completely missing. In fact, the separation for target and projectile participants give us only a convenience to consider the contributions of target and projectile nuclei in collisions. The sub-sources in the target participant are mainly contributed by the target nucleus, and the sub-sources in the projectile participant are mainly contributed by the projectile nucleus. The separation for target and projectile participants doesn't mean that there is no sub-source in the mid-rapidity region. Instead, we can regard together the target and projectile participants as whole of the three fireballs or sources in the three-fireball [1][2][3][4][5][6] or three-source model [7][8][9][10][11][12].
In our opinion, the central source discussed in the revised Landau hydrodynamic model in the present work contains almost all of the target and projectile participants, which results in a wide enough Gaussian rapidity distribution. The target and projectile sources contain only a few leading and non-leading nucleons, which revise the Gaussian rapidity distribution somewhere. The related picture in the revised Landau hydrodynamic model is set up at the level of nucleon, which is different from, but not inharmonious to, the three-fireball model [1][2][3][4][5][6] and the three-source model [7][8][9][10][11][12], which separate the whole participants into three parts: the central (mid-rapidity), target, and projectile sources.
Obviously, the central (mid-rapidity) fireball in the three-fireball model [1][2][3][4][5][6] or the central (mid-rapidity) source in the three-source model [7][8][9][10][11][12] is a part of the central source in the present work. Similarly, the target/projectile fireballs (sources) in the former two models are parts of the central source in the present work. Meanwhile, in the present work, the target/projectile sources are revisions of the central source. Particularly, the physics picture of three-source model [7][8][9][10][11][12] is set up at the level of quark and gluon. According to the three-source model [7][8][9][10][11][12], in the approximate energy range of NN s  20 GeV, the low-x gluon density rises rapidly with center-of-mass energy, and particles produced from the mid-rapidity source are most relevant at RHIC energies and dominant at LHC energies; while in the approximate energy range of NN s  20 GeV, no mid-rapidity source occurs [12]. These results contain more underlying and embedded physics such as interactions among quarks and gluons. We are interested in details of the three-source model [7][8][9][10][11][12], and hope to use this model in our future work.
From above calculations, comparisons, and discussions we see that, based on a new revised Landau hydrodynamic model, we present a new parameterization of pseudorapidity distributions of charged particles produced in heavy ion collisions for different energies and centralities. This parameterization is not baseless. From this parameterization, we have obtained a series of square speed of sound, particularly some of them being  in heavy ion collisions at RHIC and LHC energies that the central source undergoes through the phase transition from the hadronic gas to the QGP liquid. This confirms the QGP is of liquid type rather than of a gas.

Conclusions
From the above discussions, we obtain following conclusions: (a) The interacting system formed in high energy heavy ion collisions can be divided into three sources: a central source, a target source, and a projectile source. The central source is large enough covering the whole rapidity region and can be described by the Landau hydrodynamic model. The small target and projectile sources which isotropically emit particles in their respective rest frames are used revising the rapidity or pseudorapidity distributions described by the Landau hydrodynamic model. All the three sources are assumed to be formed in the participants. Our treatment on the sources is a new revision of the Landau hydrodynamic model, and this revision is different from previous revisions. The three-source picture used in the present work is also different from previous three-fireball or three-source model.
(b) The rapidity and pseudorapidity distributions are strictly dipartite in the modeling calculation. There is no conversion being used between the two distributions, where an unapt conversion may cause some errors. At the same time, a non-division for the two distributions may also cause some errors. The present work confirms that the QGP is of liquid type rather than that of a gas. The whole region of participants undergoes through a mixed phase consisting of a large quantity of (>90%) QGP liquid and a small quantity of (<10%) hadronic gas. These highlighted results are obtained by us from the extraction of square speed of sound from rapidity distribution width based on the parameterization of pseudorapidity distributions in the framework of new revised Landau Hydrodynamic model.     Table 3. As for Table 1 Table 4. Values of peak positions ( T y , P y , C y ), contribution ratios ( T k , P k ), wide  , and 2 / dof  for the curves in Figure 9 which shows d-Au collisions at 0.2 TeV for minimum-bias sample [ Figure 9 Table 5. As for Table 4, but showing the results for the curves in Figure 10 which shows Au-Em collisions at beam E  10.6A GeV with different impacting types.   Table 6. As for Table 4, but showing the results for the curves in Figure 11 which shows p-Em collisions at different NN s .       Figure 14(h).              Tables 1-6, and the lines are our fitted results.