Pseudorapidity Distribution of Charged Particles and Square Speed of Sound Parameter in or Collisions over an Energy Range from 0.053 to 7 TeV
Pseudorapidity distributions of charged particles produced in proton-proton () or proton-antiproton () collisions over an energy range from 0.053 to 7 TeV are studied by using the four-component Landau hydrodynamic model. The results calculated by the model are in agreement with the experimental data of the UA5, PHOBOS, UA1, P238, CDF, ALICE, and CMS Collaborations which present orderly from low to high energies. According to the distribution widths of different components, the values and some features of square speed of sound parameter for “participant” and “spectator” quark components are obtained. It is shown that the speed of sound for “participant” quark components agrees approximately with that for “spectator” quark components in the error ranges. The present work is useful for studying nucleus-nucleus collisions in the related energy range.
In the laboratory conditions, the only way for creating and studying the interacting systems with hadronic or partonic degrees of freedom at extremely high energy and density is to investigate heavy-ion (nucleus-nucleus) collisions at very high center-of-mass energies. The relativistic heavy ion collider (RHIC) in the United States and the large hadron collider (LHC) in Switzerland have been built, respectively [1, 2]. Particularly, the LHC was originally designed to accelerate two protons up to total energy of 14 TeV, and it will accelerate heavy ions to collide at center-of-mass energy () of 5.5 TeV. Such high energy collisions offer a new environment which leads to a new significant extension of the kinematic range in longitudinal rapidity and transverse momentum . It also provides a new chance for us to understand systematically the particle statistical behavior, production process, interaction mechanism, and correlation phenomena.
The multiplicity and (pseudo)rapidity distributions of final-state particles can often be used to test different theoretical models and wonderful ideas. The rapidity distribution and the pseudorapidity distribution of charged particles are very important quantities for the study of mainly particle production mechanism in high energy nucleon-nucleon and nucleus-nucleus collisions. The studies of and in proton-proton (-) or proton-antiproton (-) collisions not only can provide reference and baseline for heavy-ion collisions, but also can reflect the main particles contribution to heavy-ion collisions. In - or - collisions at present colliders, the center-of-energies reach a region of ultrahigh energies. Meanwhile, the highest energy nuclear experiments can be performed. Most of high-density experimental data can help us realize the creation of a new state of matter, namely, the quark gluon plasma (QGP) [4, 5], which is a thermalized system consisting of strong coupling quarks and gluons in a very small region. It is believed that the QGP was the main matter in our early universe shortly after the Big Bang.
Many models have been introduced in the field of high energy collisions, for example, the hydrodynamics model , the thermodynamics model , the hadron resonance gas model , the fireball model [9, 10], and so forth. Generally speaking, a given model treats the different collisions by different considerations, and different models approach the same collisions in different methods. In our previous work , based on the participant-spectator model  and Landau hydrodynamic model [13–17], the pseudorapidity distributions of charged particles produced in nucleus-nucleus collisions at the RHIC and LHC have been analyzed, and the values of square speed of sound () have been obtained. We notice that the related data for - or - collisions are absent in our previous work .
To make up the limitation of our previous work , in this work, we will use the same method to study the pseudorapidity distributions of charged particles produced in - or - collisions over an energy range from 0.053 to 7 TeV and to extract the values of speed of sound parameter for the “participant” and “spectator” quark components which correspond, respectively, to the participant and spectator nucleons in nucleus-nucleus collisions. The quoted experimental data are taken from the UA5 (- collisions at 0.053, 0.2, 0.546, and 0.9 TeV) , PHOBOS (- collisions at 0.2 and 0.41 TeV) , UA1 (- collisions at 0.54 TeV) , P238 (- collisions at 0.63 TeV) , CDF (- collisions at 0.63 and 1.8 TeV) , ALICE (- collisions at 0.9 and 2.36 TeV) , and CMS (- collisions at 0.9, 2.36, and 7 TeV) [24, 25] Collaborations.
2. The Model
The picture of participant-spectator model  is the base of many theoretical models such as the nuclear fireball model , the two-component model , the nuclear overlap model , and the multisource thermal model [29, 30]. In the framework of participant-spectator model , the projectile and target nuclei penetrating through each other in collisions. Then, a central participant region, a target spectator region, and a projectile spectator region are expected to form in the collisions. Further, we can divide the participant region into two parts: a target participant region and a projectile participant region. In the multisource thermal model [29, 30], these regions can extend to different cylinders in rapidity space. In addition, the contributions of leading nucleons can fall under the spectator regions. In high energy nucleus-nucleus collisions, both the participant and spectator regions consist of nucleons.
In high energy hadron-hadron collisions, the situation is different. If we use the picture of participant-spectator model , both the participant and spectator regions will be quarks. Similarly, we have four source components: the target spectator quark component, the target participant quark component, the projectile participant quark component, and the projectile spectator quark component, which are presented orderly from low to high in the rapidity space. According to [31, 32], if we use the Landau hydrodynamic model [13–17] to describe the hadron-hadron collisions, only the central pseudorapidity region can be fitted. This means that the Landau hydrodynamic model [13–17] gives a narrow pseudorapidity distribution, and the contributions of leading nucleons are not included in the model. To describe the pseudorapidity distribution in a wide region, we can apply the model for each quark component in the hadron-hadron collisions.
The pseudorapidity distribution of charged particles produced in each quark component is given by a Gaussian function [17, 33] with where is the normalization constant, is in the units of GeV, denotes the speed of sound, denotes the proton mass in the units of GeV/ with , and is the pseudorapidity distribution width.
Strictly speaking, there are some differences between the rapidity distribution and the pseudorapidity distribution. In fact, the differences between the two distributions at high energies can be neglected due to the small values . If we need to distinguish the two distributions, we should calculate them severally or make a complete transformation . In the present work, we do not distinguish the two distributions for the purpose of simpleness. Equation (1) can be rewritten as the function of due to (2). The final-state pseudorapidity distribution is a weighted sum of four equations (3) with different rapidity shifts and weights.
For - or - collisions, the rapidity shift () of target spectator is opposite to the rapidity shift () of projectile spectator, and the rapidity shift () of target participant is opposite to the rapidity shift () of projectile participant. The weight () of target participant is equal to that of projectile participant, and the weight of target spectator is equal to that of projectile spectator.
3. Comparisons with Experimental Data
Figures 1(a)–1(d) show the pseudorapidity distributions of charged particles produced in inelastic - collisions at = 0.053, 0.2, 0.546, and 0.9 TeV, respectively. The symbols in the range of represent the experimental data of the UA5 Collaboration , and those in the range of are symmetrical reflection at the midpseudorapidity. The curves are our calculated results with the four-component Landau hydrodynamic model. The values of parameters obtained by fitting the experimental data are given in Table 1 with the values of ( per degree of freedom), where and denote the square speeds of sound for the participants and spectators, respectively. One can see that the calculated results are in agreement with the experimental data at the four energies. Both values of for the participants and spectators are approximately equal to each other in the error ranges, and they do not depend obviously on in the considered energy range.
The pseudorapidity distributions of charged particles produced in inelastic - collisions at and 0.41 TeV measured by the PHOBOS Collaboration , - collisions at TeV measured by the UA1 Collaboration , - collisions at TeV measured by the P238 Collaboration with ranging from 1.5 to 5.5 , and - collisions at and 1.8 TeV measured by the CDF Collaboration with  are presented in Figures 2(a)–2(f), respectively. The symbols and curves represent the same meanings as those in Figure 1. By fitting the experimental data, the obtained values of and are given in Table 1. Once again the four-component Landau hydrodynamic model describes the experimental data well. Although the energy has a large change, the values of for the participants and spectators are approximately the same in the error ranges, and they have no obvious change with changing the energy.
Figure 3 gives the pseudorapidity distributions of charged particles produced in non-single-diffractive - collisions at = 0.9, 2.36, and 7 TeV. The symbols in the left panel represent the experimental data obtained by the CMS Collaboration [24, 25], and those in the right panel stand for the experimental data of the ALICE Collaboration . The curves are our calculated results. The obtained values of and are given in Table 1. We notice that the model describes the experimental data at the LHC energies. Both values of for the participants and spectators seem to be the same in the error ranges, and they do not depend obviously on the energy.
To see clearly the dependences of and on , the values of and in Table 1 are displayed in Figure 4. Different symbols represent the results extracted from different experiments as marked in the figure. The open and closed symbols represent the values of and , respectively. One can see that both the and fall mainly into a range consisting of two lines: and . These results are in agreement with the hadron resonance gas model which results in, respectively, and for the existing regions of hadron resonances including and excluding pions at the temperature of ~85 MeV and for the two existing regions at the temperature of ~190 MeV [36, 37]. The present results are also in agreement with the lattice quantum chromodynamics theory which gives and 0.31 at the temperatures of 125 and 400 MeV, respectively .
Both values of for the participants and spectators are approximately the same in the error ranges and they do not depend obviously on over an energy range from 0.053 to 7 TeV. From Figure 4, one can also see that seems to be less than at TeV energies. Because the fitted errors are large for both , it is hard to determine which one is less than the other.
The correlations between and as well as and are presented in Figures 5(a) and 5(b), respectively, where is in the units of GeV. The symbols represent the values of rapidity shifts and which are obtained in Figures 1–3 and listed in Table 1. The lines are our fitted results which are described by with and with , respectively. One can see the linear relation existing between the rapidity shifts and . With the increase of the logarithmic center-of-mass energy, both rapidity shifts increase slightly.
The correlation between and is shown in Figure 6, where is in the units of GeV. The symbols represent the values of contribution ratio of the projectile (or target) participant quark component which are obtained in Figures 1–3 and listed in Table 1. The line is our fitted result described by with . One can see the linear relation existing between the contribution ratio and . Particularly, the contribution ratio of participants decreases and that of spectators increases with the increase of the logarithmic center-of-mass energy.
To conclude, the pseudorapidity distributions of charged particles produced in - collisions at , 0.41, 0.9, 2.36, and 7 TeV and in - collisions at , 0.2, 0.54, 0.546, 0.63, 0.9, and 1.8 TeV have been studied in the present work. The calculated results of the four-component Landau hydrodynamical model satisfactorily describe the experimental data of the UA5, PHOBOS, UA1, P238, CDF, ALICE, and CMS Collaborations. In the model, we have considered the application of the participant-spectator model  in hadron-hadron collisions. Then, the interacting system of hadron-hadron collisions is divided into four components: the target and projectile “spectator” quark components and the target and projectile “participant” quark components. For each component, the Landau hydrodynamical model [13–17] is applied, which results in the pseudorapidity distribution being a Gaussian type.
Based on the above successful descriptions of the pseudorapidity distributions of charged particles produced in - or - collisions over an energy range from 0.053 to 7 TeV, the square speeds of sound, , for both the participants and spectators are obtained and found to be the same in the fitted error ranges. Both are approximately independent of in the considered energy range and fall mainly into the range of 0.12–0.30 which is in agreement with hadron resonance gas model which gives at the temperature of ~85 MeV and at ~190 MeV [36, 37]. The values of obtained in the present work are also in agreement with the lattice quantum chromodynamics theory which gives in the temperature range of 125–400 MeV .
In the process of extracting the speed of sound, the rapidity shifts and contribution ratio of the participant and spectator quark components are naturally obtained. There are linear relations existing between and , and , as well as and . Both rapidity shifts increase slightly with the increase of the logarithmic center-of-mass energy. The contribution ratio of participants decreases and that of the spectators increases with the increase of the logarithmic center-of-mass energy.
The basis for Landau’s equation (essentially a Gaussian pseudorapidity distribution) is the applicability of Landau hydrodynamics, namely, the creation of an initial state of dense Lorentz contracted matter that expands later according to the equations of 2 + 1 dimensional relativistic hydrodynamics. It seems that such an initial state is not created in hadron-hadron collision at intermediate or not too high energy, and the evolution of the interacting system would not be covered by hydrodynamics. However, hadron-hadron collision at high or ultrahigh energy can be regarded as a small dense system comparing with nucleus-nucleus collisions. This small dense system can produce a multihadron final-state state and can be approximately described by Landau’s equation.
The square speed of sound extracted from the present work is in agreement with that from nucleus-nucleus collisions at RHIC and LHC energies in our previous work . The similar values of speed of sound in both hadron-hadron and nucleus-nucleus collisions provides the insight into some common laws and universal mechanisms in multihadron productions [31, 32]. We believe that there are more similar quantities in multihadron productions in the “elementary” and “complex” collisions.
We would like to point out that the present work is the first one which extracts the square speed of sound in hadron-hadron collisions from the pseudorapidity distributions of charged particles over a very wide energy range. What we use is the four-component Landau hydrodynamic model which is based on the participant-spectator model  and the Landau hydrodynamic model [13–17]. Our work can provide references and baselines for nucleus-nucleus collisions at high energies. We hope to extract the square speed of sound in proton-proton and lead-lead collisions at the highest energy at the LHC in the near future.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was partly finished at the State University of New York at Stony Brook, USA. One of the authors (Fu-Hu Liu) thanks Professor Dr. Roy A. Lacey and the members of the Nuclear Chemistry Group of Stony Brook University for their hospitality. The authors acknowledge the supports of the National Natural Science Foundation of China (under Grant nos. 10975095, 11247250, and 11005071), the China National Fundamental Fund of Personnel Training (under Grant no. J1103210), the Open Research Subject of the Chinese Academy of Sciences Large-Scale Scientific Facility (under Grant no. 2060205), the Shanxi Scholarship Council of China, and the Overseas Training Project for Teachers at Shanxi University.
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