Table of Contents Author Guidelines Submit a Manuscript
Advances in High Energy Physics
Volume 2013 (2013), Article ID 450247, 8 pages
http://dx.doi.org/10.1155/2013/450247
Research Article

Speed of Sound Parameter from RHIC and LHC Heavy-Ion Data

Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China

Received 28 June 2013; Revised 25 August 2013; Accepted 27 August 2013

Academic Editor: George Siopsis

Copyright © 2013 Li-Na Gao 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.

Abstract

In framework of combing the participant-spectator model and the Landau hydrodynamic model, the pseudorapidity distributions of charged particles produced in heavy-ion (or nucleus-nucleus) collisions at RHIC and LHC energies are described by a modified Landau hydrodynamic model, where the Landau hydrodynamic model is applied to the target/projectile spectators and the target/projectile participants, respectively. The modeling results are in agreement with the PHOBOS and ALICE experimental data. Then, the values of square speed of sound () for the participants and spectators can be obtained from the widths of charged particle pseudorapidity distributions. Some features of for different centralities and center-of-mass energies are obtained too.

1. Introduction

High energy collisions are an important research topic in fields of particle and nuclear physics. Many charged and neutral particles are produced in the collisions. Particularly, the Relativistic heavy ion collider (RHIC) and the large hadron collider (LHC) have been opening a new epoch for high energy heavy-ion collider. Many experimental results are reported and many theoretical models are proposed to describe the data.

Among the theoretical models, the Landau hydrodynamic model [14] is one of the most useful treatment methods in high energy heavy-ion (or nucleus-nucleus) collisions [513]. We hope to use the Landau hydrodynamic model to analyze experimental data measured in nucleus-nucleus collisions at RHIC [6, 1418] and LHC energies [1921]. The Landau hydrodynamic model was proposed at lower energy. In [13], the model is revised for RHIC energies. In [22, 23], it is shown that the model in the frame of quark participants works well from the alternating gradient synchrotron (AGS) to the RHIC energies. We notice that these revision and application of the Landau hydrodynamic model are in a narrow or central rapidity region. Contributions of leading nucleons are not included in these studies [13, 22, 23].

To analyze a wider pseudorapidity distribution, we need to consider the contributions of leading nucleons. In the framework of participant-spectator model [24], the spectators are one of the main sources of leading nucleons. We may apply Landau hydrodynamic model to the target/projectile spectators and the target/projectile participants, respectively. Then, we can use a modified Landau hydrodynamic model which contains four components to describe wider pseudorapidity distributions of charged particles produced in Au-Au and Cu-Cu collisions at RHIC energy [2527] and Pb-Pb collisions at LHC energy [1921]. Because the square speed of sound parameter () is related to the width of pseudorapidity distribution, we can obtain naturally the values of .

In this paper, to extract the parameter , a modified Landau hydrodynamic model is introduced in Section 2. Comparisons of modeling results with experimental data of PHOBOS [27] and ALICE Collaborations [21] and the obtained parameter values are given in Section 3. Finally, we give conclusions of the present work in Section 4.

2. The Model

According to the participant-spectator model [24], the overlapping regions between the target and projectile nuclei in collisions are called the participants, and the other parts outside the overlapping regions are called the spectators. Then, we divide the interacting system of nucleus-nucleus collisions into four parts (sources). From low rapidity region to a high one, the four sources are a target spectator, a target participant, a projectile participant, and a projectile spectator. Because leading nucleons in the overlapping regions contribute large pseudorapidities, we may consider these leading nucleons not belonging to the participants. Instead, they belong to the spectators. The participants contain only the nonleading nucleons in the overlapping regions. The spectators contain in fact the other parts outside the overlapping regions and the leading nucleons in the overlapping regions.

For each source, we can use the Landau hydrodynamic model to describe rapidity () or pseudorapidity () distribution. Generally speaking, the rapidity distribution and the pseudorapidity distribution are not exactly equal to each other. From the rapidity/pseudorapidity distribution to pseudorapidity/rapidity distribution, we need a conversion. However, at very high energy such as RHIC and LHC energies, we use approximately which renders a small dispersion in analyzing the experimental data [28].

According to the Landau hydrodynamic model [14], the evolvement of interacting system in nucleus-nucleus collisions can be described by the hydrodynamic method. After a series of calculated treatments, the pseudorapidity distribution of charged particles produced in the collisions can be described by a Gaussian function [3, 12, 13, 2931]: where denotes the square of distribution width, is a normalization constant, is the center-of-mass energy per nucleon pair in the units of GeV, is the mass of a proton in the units of GeV, and is the square speed of sound.

It should be noticed that Landau considered originally pseudorapidity, and the “modified” form by Shuryak introduced rapidity instead [3, 29]. The rapidity was also introduced by Milekhin [32] in the form of , where is the beam energy in the units of GeV in fixed target experiments. Considering at very high energy, we do not need to distinguish between and at RHIC and LHC energies. The Gaussian function is not only an approximation of the original Landau distribution [2931], but also a close representation of the modified distribution discussed in [13].

Substituting (2) in (1), we have

Then, we can use (3) to describe the pseudorapidity distribution of charged particles produced in a given source. For a distribution in the full phase space, we need a modified Landau hydrodynamic model which contains four components corresponding to four sources in collisions [33]. The four sources are the target spectator/participant and projectile participant/spectator, respectively. The participants contain the nonleading nucleons in the overlapping regions, and the spectators contain the nucleons outside the overlapping regions and the leading nucleons in the overlapping regions.

As a main free parameter, the (square) speed of sound is related to the width of pseudorapidity distribution due to (2). If we can describe the particle pseudorapidity distribution corresponding to a given source, then the square speed of sound of the source can be obtained. In fact,

3. Comparison with Experimental Data

The pseudorapidity distributions of charged particles produced in Au-Au collisions at , 62.4, 130, and 200 GeV are presented in Figures 1, 2, 3, and 4, respectively. The circles represent the experimental data with different centralities measured by the PHOBOS collaboration [27], and the curves are our results calculated by the modified Landau hydrodynamic model. In the calculation, the values of are obtained by fitting the experimental data. The obtained values of and /dof ( per degree of freedom) are given in Table 1. One can see that the model describes the experimental data at RHIC energies. In the range of errors, the values of for the participants and spectators are the same. Both values of for the participants and spectators do not depend obviously on the centrality and in the considered energy range.

tab1
Table 1: Values of for the participants (P) and spectators (S) as well as /dof for the curves in Figures 14 which represent Au-Au collisions at RHIC energies.
450247.fig.001
Figure 1: versus for different centrality bins in Au-Au collisions at  GeV. The circles and curves represent the experimental data of the PHOBOS collaboration [27] and our calculated results, respectively.
450247.fig.002
Figure 2: As Figure 1 but showing the results in Au-Au collisions at  GeV.
450247.fig.003
Figure 3: As Figure 1 but showing the results in Au-Au collisions at  GeV.
450247.fig.004
Figure 4: As Figure 1 but showing the results in Au-Au collisions at  GeV.

The pseudorapidity distributions of charged particles produced in Cu-Cu collisions at , 62.4, and 200 GeV are presented in Figures 5, 6, and 7, respectively. The circles represent the experimental data with different centralities measured by the PHOBOS collaboration [27], and the curves are our results calculated by the modified Landau hydrodynamic model. The obtained values of and /dof are given in Table 2. One can see again that the model describes the experimental data. The values of for the participants and spectators are the same in the range of errors. Both values of for the participants and spectators do not depend obviously on the centrality and at RHIC energies.

tab2
Table 2: Values of for the participants and spectators as well as /dof for the curves in Figures 57 which represent Cu-Cu collisions at RHIC energies.
450247.fig.005
Figure 5: As Figure 1 but showing the results in Cu-Cu collisions at  GeV.
450247.fig.006
Figure 6: As Figure 1 but showing the results in Cu-Cu collisions at  GeV.
450247.fig.007
Figure 7: As Figure 1 but showing the results in Cu-Cu collisions at  GeV.

Figure 8 shows the pseudorapidity distributions of charged particles produced in Pb-Pb collisions at  TeV. The circles represent the experimental data of the ALICE collaboration [21], and the curves are our results calculated by the modified Landau hydrodynamic model. The obtained values of and /dof are given in Table 3. Once more the model describes the experimental data. The values of for the participants and spectators at LHC energy are the same, and they are the same as those at RHIC energies in the range of errors. Both the values of for the participants and spectators do not depend obviously on the centrality.

tab3
Table 3: Values of for the participants and spectators as well as /dof for the curves in Figure 8 which represent Pb-Pb collisions at 2.76 TeV (LHC energy).
450247.fig.008
Figure 8: versus for different centrality bins in Pb-Pb collisions at  TeV. The circles and curves represent the experimental data of the ALICE collaboration [21] and our calculated results, respectively.

From Figures 18, one can see that the modified Landau hydrodynamic model proposed by us is in agreement with the experimental data at RHIC and LHC energies. In the model, both the contributions of participants and spectators (or nonleading nucleons and leading nucleons) are included. We notice that in recent works [21, 34] both the Landau-Carruthers Gaussian [11, 12] and the Landau-Wong function [13] overestimate/underestimate the distributions in central/forward rapidity regions. If we consider the contributions of leading nucleons in the Landau-Carruthers Gaussian [11, 12] and in the Landau-Wong function [13], respectively, both the situations will change to be better.

The values of obtained by us are in agreement with the hadron resonance (HRG) gas model [35, 36]. According to the HRG model, the existing region of hadron resonances including the pions has a large (~0.23), and the existing region of hadron resonances excluding the pions has a small (~0.12) at low temperature (~85 MeV). The two regions trend approximately the same one (~0.14–0.15) at high temperature (~190 MeV). In [37], based on the lattice quantum chromodynamics (QCD) theory, the values of and other parameters are obtained in the temperature range from 125 MeV to 400 MeV. According to [37], the values of are 0.12–0.16 at 125 MeV and 0.31 at 400 MeV. One can see that these values are in agreement with our results in the range of errors.

We notice that the errors for in the present work are universally large. This renders that is not a sensitive quantity. As an intensive quantity, is a reflection of the ratios of pressure to energy density or entropy to specific heat in the participant and spectator regions and is related to the pseudorapidity distribution width which depends on the rapidity shifts of the participants and spectators.

The same for the participants and spectators and for different centrality bins at RHIC and LHC energies reflect the same matter density at the freeze-out stage of particle production. In collisions, the spectators have nearly a normal nuclear density, and the participants have a much larger density than the normal nuclear one. After collisions, the participants expand rapidly in volume and reach the normal nuclear density. Then, the participants produce many final-state particles. For the participants, to expand to the normal nuclear density and then to produce final-state particles are a fact that does not depend on centrality and energy at RHIC and LHC energies.

4. Conclusions

From the above discussions, we obtain following conclusions.(a)The pseudorapidity distributions of charged particles produced in nucleus-nucleus collisions at RHIC and LHC energies can be described by the modified Landau hydrodynamic model. The modeling results are in agreement with the PHOBOS experimental data of Au-Au collisions at , 62.4, 130, and 200 GeV and Cu-Cu collisions at , 62.4, and 200 GeV, as well as ALICE experimental data of Pb-Pb collisions at  TeV.(b)In the concerned energy range from 19.6 GeV to 2.76 TeV, the values of for the participants and spectators are the same in the range of errors and do not depend obviously on the centrality and .(c)The values of obtained in the present work are in agreement with the results of the HRG model and the lattice QCD theory. To extract of interacting system, our work provides a new method which bases on the width of particle pseudorapidity distribution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 10975095, 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, and the Shanxi Scholarship Council of China.

References

  1. L. D. Landau, “Izvestiya akademii nauk: series fizicheskikh 17 51 (1953),” in English-Translation: Collected Papers of L. D. Landau, D. Ter-Haarp, Ed., p. 569, Pergamon, Oxford, UK, 1965. View at Google Scholar
  2. S. Z. Belenkij and L. D. Landau, “Soviet Physics Uspekhi 56 309 (1955),” in English-Translation: Collected Papers of L.D. Landau, D. Ter-Haarp, Ed., p. 665, Pergamon, Oxford, UK, 1965. View at Google Scholar
  3. O. V. Zhirov and E. V. Shuryak, “Multiple production of hadrons and predictions of the Landau theory,” Yadernaya Fizika, vol. 21, no. 4, pp. 861–867, 1975. View at Google Scholar
  4. W. Florkowski, Phenomenology of Ultra-Relativistic Heavy-Ion CollisIons, World Scientific, Singapore, 2010.
  5. B. B. Back, M. D. Baker, D. S. Barton et al., “Charged-particle pseudorapidity density distributions from Au+Au collisions at SNN=130 GeV,” Physical Review Letters, vol. 87, no. 10, Article ID 102303, 2001. View at Publisher · View at Google Scholar
  6. B. B. Back, M. D. Baker, D. S. Barton et al., “Significance of the fragmentation region in ultrarelativistic heavy-ion collisions,” Physical Review Letters, vol. 91, no. 5, Article ID 052303, 2003. View at Publisher · View at Google Scholar
  7. I. G. Bearden, D. Beavisa, C. Besliuj et al., “Charged particle densities from Au+Au collisions at SNN=130 GeV,” Physics Letters B, vol. 523, no. 3-4, pp. 227–233, 2001. View at Publisher · View at Google Scholar
  8. I. G. Bearden et al., “Pseudorapidity distributions of charged particles from Au+Au collisions at the maximum RHIC energy, SNN=200 GeV,” Physical Review Letters, vol. 88, no. 20, Article ID 202301, 2002. View at Publisher · View at Google Scholar
  9. P. Steinberg, “Bulk dynamics in high energy collisions,” Journal of Physics: Conference Series, vol. 9, no. 1, pp. 280–285, 2005. View at Publisher · View at Google Scholar
  10. P. Steinberg, “Bulk dynamics in heavy ion collisions,” Nuclear Physics A, vol. 752, pp. 423–432, 2005. View at Publisher · View at Google Scholar
  11. P. Carruthers and M. Duong-Van, “New scaling law based on the hydrodynamical model of particle production,” Physics Letters B, vol. 41, no. 5, pp. 597–601, 1972. View at Google Scholar · View at Scopus
  12. P. Carruthers and M. Duong-Van, “Rapidity and angular distributions of charged secondaries according to the hydrodynamical model of particle production,” Physical Review D, vol. 8, no. 3, pp. 859–874, 1973. View at Publisher · View at Google Scholar · View at Scopus
  13. C.-Y. Wong, “Landau hydrodynamics reexamined,” Physical Review C, vol. 78, no. 5, Article ID 054902, 2008. View at Publisher · View at Google Scholar
  14. B. B. Back, M. D. Baker, D. S. Barton et al., “Charged-particle multiplicity near midrapidity in central Au+Au collisions at SNN=56 and 130 GeV,” Physical Review Letters, vol. 85, no. 15, pp. 3100–3104, 2000. View at Publisher · View at Google Scholar
  15. B. B. Back, M. D. Baker, D. S. Barton et al., “Centrality dependence of charged particle multiplicity at midrapidity in Au+Au collisions at SNN=130 GeV,” Physical Review C, vol. 86, no. 16, pp. 3500–3505, 2001. View at Google Scholar
  16. B. B. Back, B. B. Back, D. S. Barton et al., “Energy dependence of particle multiplicities in central Au+Au collisions,” Physical Review Letters, vol. 88, no. 2, Article ID 022302, 2002. View at Publisher · View at Google Scholar
  17. B. B. Back, M. Ballintijn, M. D. Baker et al., “Centrality dependence of the charged particle multiplicity near midrapidity in Au+Au collisions at SNN=130 and 200 GeV,” Physical Review C, vol. 65, no. 6, Article ID 061901, 2002. View at Publisher · View at Google Scholar
  18. B. B. Back, M. D. Baker, D. S. Barton et al., “Collision geometry scaling of Au+Au pseudorapidity density from SNN=19.6 to 200 GeV,” Physical Review C, vol. 70, no. 2, Article ID 021902, 2004. View at Publisher · View at Google Scholar
  19. A. Toia, “For the ALICE Collaboration. Bulk properties of Pb–Pb collisions at SNN=2.76 TeV measured by ALICE,” Journal of Physics G, vol. 38, no. 12, Article ID 124007, 2011. View at Publisher · View at Google Scholar
  20. D. M. Röhrscheid and G. Wolschin, “Centrality dependence of charged-hadron pseudorapidity distributions in PbPb collisions at energies available at the CERN Large Hadron Collider in the relativistic diffusion model,” Physical Review C, vol. 86, no. 2, Article ID 024902, 2012. View at Publisher · View at Google Scholar
  21. E. Abbas, L. Aphecetche, A. Baldisseri et al., “Centrality dependence of the pseudorapidity density distribution for charged particles in Pb–Pb collisions at SNN=2.76 TeV,” 2013, http://arxiv.org/abs/1304.0347.
  22. E. K. G. Sarkisyan and A. S. Sakharov, “Relating multihadron production in hadronic and nuclear collisions,” The European Physical Journal C, vol. 70, no. 3, pp. 533–541, 2010. View at Publisher · View at Google Scholar
  23. E. K. G. Sarkisyan and A. S. Sakharov, “Multihadron production features in different reactions,” AIP Conference Proceedings, vol. 828, pp. 35–41, 2006. View at Publisher · View at Google Scholar
  24. R. J. Glauber, “High-energy collision theory,” in Lectures of Theoretical Physics, W. E. Brittin and L. G. Dunham, Eds., vol. 1, pp. 315–414, Interscience, New York, NY, USA, 1959. View at Google Scholar · View at MathSciNet
  25. J. Dias de Deus and J. G. Mihano, “A simple evolution equation for rapidity distributions in nucleus-nucleus collisions,” Nuclear Physics A, vol. 795, no. 1–4, pp. 98–108, 2007. View at Publisher · View at Google Scholar
  26. S. S. Adler, S. Afanasiev, C. Aidala et al., “Systematic studies of the centrality and SNN dependence of the dET/dη and dNch/dη in heavy ion collisions at midrapidity,” Physical Review C, vol. 71, no. 3, Article ID 034908, 2005. View at Publisher · View at Google Scholar
  27. B. Alver, B. B. Back, D. S. Barton et al., “Charged-particle multiplicity and pseudorapidity distributions measured with the PHOBOS detector in Au+Au, Cu+Cu, d+Au, and p+p collisions at ultrarelativistic energies,” Physical Review C, vol. 83, no. 2, Article ID 024913, 2011. View at Publisher · View at Google Scholar
  28. G. Wolschin, “Pseudorapidity distributions of produced charged hadrons in pp collisions at RHIC and LHC energies,” EPL, vol. 95, no. 6, Article ID 61001, 2011. View at Publisher · View at Google Scholar · View at Scopus
  29. E. V. Shuryak, “Multiparticle production in high energy particle collisions,” Yadernaya Fizika, vol. 16, no. 2, pp. 395–405, 1972. View at Google Scholar
  30. P. Carruthers, “Heretical models of particle production,” Annals of the New York Academy of Sciences, vol. 229, pp. 91–123, 1974. View at Publisher · View at Google Scholar
  31. M. Gazdzickia, M. Gorensteinc, and P. Seybothe, “Onset of deconfinement in nucleus-nucleus collisions: review for pedestrians and experts,” Acta Physica Polonica B, vol. 42, p. 307, 2011. View at Publisher · View at Google Scholar
  32. G. A. Milekhin, “On the hydrodynamic theory of multiple production of particles,” Soviet Physics, vol. 35, no. 8, pp. 682–684, 1959. View at Google Scholar · View at MathSciNet
  33. E.-Q. Wang, Y.-H. Chen, B.-C. Li, and F.-H. Liu, “Dependences of charged particle pseudorapidity distributions on impact parameter and center-of-mass energy in nucleus-nucleus collisions at relativistic energies,” Indian Journal of Physics, vol. 87, no. 8, pp. 793–801, 2013. View at Publisher · View at Google Scholar
  34. S. Chatrchyan, V. Khachatryan, A. M. Sirunyan et al., “Measurement of the pseudorapidity and centrality dependence of the transverse energy density in Pb–Pb collisions at SNN=2.76 TeV,” Physical Review Letters, vol. 109, no. 15, Article ID 152303, 2012. View at Publisher · View at Google Scholar
  35. A. Tawfik and H. Magdy, “Hadronic equation of state and speed of sound in thermal and dense medium,” 2012, http://arxiv-web3.library.cornell.edu/abs/1206.0901.
  36. A. Tawfik and T. Harko, “Quark-hadron phase transitions in the viscous early universe,” Physical Review D, vol. 85, no. 8, Article ID 084032, 2012. View at Publisher · View at Google Scholar · View at Scopus
  37. Sz. Borsányi, G. Endrodi, Z. Fodor et al., “QCD equation of state at nonzero chemical potential: continuum results with physical quark masses at order mu2,” 2012, http://arxiv.org/abs/1204.6710.