Properties of Chemical and Kinetic FreezeOuts in HighEnergy Nuclear Collisions
View this Special IssueResearch Article  Open Access
JiaQi Hui, ZhiJin Jiang, DongFang Xu, "A Description of the Transverse Momentum Distributions of Charged Particles Produced in Heavy Ion Collisions at RHIC and LHC Energies", Advances in High Energy Physics, vol. 2018, Article ID 7682325, 9 pages, 2018. https://doi.org/10.1155/2018/7682325
A Description of the Transverse Momentum Distributions of Charged Particles Produced in Heavy Ion Collisions at RHIC and LHC Energies
Abstract
By assuming the existence of memory effects and longrange interactions, nonextensive statistics together with relativistic hydrodynamics including phase transition are used to discuss the transverse momentum distributions of charged particles produced in heavy ion collisions. It is shown that the combined contributions from nonextensive statistics and hydrodynamics can give a good description of the experimental data in Au+Au collisions at GeV and in Pb+Pb collisions at TeV for and in the whole measured transverse momentum region and for in the region of GeV/c. This is different from our previous work using the conventional statistics plus hydrodynamics, where the describable region is only limited in GeV/c.
1. Introduction
The primary goals of the experimental programs performed in high energy heavy ion collisions are to find the deconfined nuclear matter, namely, the quarkgluon plasma (QGP), which is believed to have filled in the early universe several microseconds after the Big Bang. Therefore, studying the properties of QGP is important for both particle physics and cosmology. In the past decade, a number of bulk observables about charged particles, such as the Fourier coefficients of azimuthangle distributions [1, 2], transverse momentum spectra [3–8], and pseudorapidity distributions [9–12], have been extensively studied in nuclear collisions at both RHIC and LHC energies. These investigations have shown that the matter created in these collisions is in the state of strongly coupled quarkgluon plasma (sQGP), exhibiting a clear collective behavior nearly like a perfect fluid with very low viscosity [13–31]. Therefore, the spacetime evolution of sQGP can be described in the scope of relativistic hydrodynamics, which connects the static aspects of sQGP properties and the dynamical aspects of heavy ion collisions [32].
In our previous work [33], by considering the effects of thermalization, we once used a hydrodynamic model incorporating phase transition in analyzing the transverse momentum distributions of identified charged particles produced in heavy ion collisions. In that model, the quanta of hot and dense matter are supposed to obey the standard statistical distributions and the experimental measurements in Au+Au collisions at and 130 GeV can be well matched up in the region of GeV/c. Known from the investigations in [34, 35], the memory effects and longrange interactions might appear in the hot and dense matter. This guarantees the reasonableness of nonextensive statistics in describing the thermal motions of quanta of hot and dense matter, since this new statistics is just suitable for the thermodynamic system including memory effects and longrange interactions. Hence, in this paper, on the basis of hydrodynamics taking phase transition into consideration, we will use nonextensive statistics instead of conventional statistics to simulate the transverse collective flow of the matter created in collisions.
The nonextensive statistics are also known as Tsallis nonextensive thermostatistics, which were first proposed by Tsallis in 1988 in his pioneering work [36]. This statistical theory overcomes the shortcomings of the conventional statistics in many physical problems with longrange interactions, memory effects, or fractal spacetime constrains [34]. It has a wide range of applications in high energy physics [37–39], astrophysical selfgravitating systems [40], cosmology [41], the solar neutrino problem [42], manybody theory, dynamical linear response theory, and variational methods [43].
In Section 2, a brief description is given on the employed hydrodynamics, presenting its analytical solutions. The solutions are then used in Section 3 to formulate the transverse momentum distributions of charged particles produced in heavy ion collisions in the light of nonextensive statistics and CooperFrye prescription. The last section is about conclusions.
2. A Brief Introduction to the Hydrodynamic Model
The key points of the hydrodynamic model [18] used in the present paper are as follows.
The expansions of fluid follow the continuity equationwhere andis the energymomentum tensor of perfect fluid, where is the metric tensor and is the fourvelocity of fluid with rapidity . The energy density and the pressure of fluid are related by the sound speed via the equation of statewhere is the temperature and is the entropy density of fluid.
Projecting (1) to the direction of giveswhich is the continuity equation for entropy conservation. Projecting (1) to the direction Lorentz perpendicular to leads to the following equation:which means the existence of scalar function satisfyingFrom and Legendre transformation, Khalatnikov potential is introduced bywhich makes the coordinates of transform towhere is the initial temperature of fluid and . In terms of , (4) can be rewritten as the socalled telegraphy equation:
With the expansions of created matter, its temperature becomes lower and lower. When the temperature reduces from initial temperature to the critical temperature , the matter transforms from the sQGP state to the hadronic state. The produced hadrons are initially in the violent and frequent collisions, which are mainly inelastic. Hence, the abundance of identified hadrons is constantly changing. Furthermore, the mean free paths of these primary hadrons are very short. In sQGP, . The evolution of them still satisfies (9) with only difference being the values of . In the hadronic state, . At the point of phase transition, is discontinuous.
The solutions of (9) for the sQGP and hadronic state are, respectively [18],where is a constant determined by fitting the theoretical results with experimental data, is the 0thorder modified Bessel function, and
3. The Transverse Momentum Distributions of Charged Particles Produced in Heavy Ion Collisions
3.1. The Energy of Quantum of Produced Matter
In the nonextensive statistics, there are two basic postulations [36, 40].
(a) The entropy of a statistical system possesses the form ofwhere is the probability of a given microstate among different ones and is a real parameter.
(b) The mean value of an observable is given bywhere is the value of an observable in the microstate .
From the above two postulations, the average occupational number of the quantum in the state with temperature can be written in a simple analytical form [44]:where is the energy of quantum and is its baryochemical potential. for fermions, and for bosons. In the limit of , it reduces to the conventional FermiDirac or BoseEinstein distributions. Hence, the value of reflects the discrepancies of nonextensive statistics from the conventional ones. Known from (15), the average energy of quantum in the state with temperature readswhere is the rapidity of quantum and is its transverse mass with rest mass and transverse momentum .
3.2. The Rapidity Distributions of Charged Particles in the State of Fluid
In terms of Khalatnikov potential , the rapidity distributions of charged particles in the state of fluid can be written as [45]whereis the area of overlap region of collisions, is impact parameter, and is the radius of colliding nucleus. From (8), the expression in the round brackets in (17) becomes
3.3. The Transverse Momentum Distributions of Charged Particles Produced in Heavy Ion Collisions
Along with the expansions of hadronic matter, its temperature becomes even lower. As the temperature drops to kinetic freezeout temperature , the inelastic collisions among hadronic matter stop. The yields of identified hadrons remain unchanged, becoming the measured results. According to CooperFrye scheme [45], the invariant multiplicity distributions of charged particles take the form [18, 45, 46]where and the integrand takes values at the moment of .
Substituting in (19) by of (11), it becomeswherewhere and is the 1storder modified Bessel function.
By using (17) and (20)–(22), we can obtain the transverse momentum distributions of identified charged particles as shown in Figures 1 and 2.
(a)
(b)
(a)
(b)
Figure 1 shows the invariant yields of , , and as a function of in Au+Au collisions at GeV. Figure 2 shows the same distributions in Pb+Pb collisions at TeV. The solid dots represent the experimental data [3, 4]. The solid curves are the results calculated from (20). It can be seen that the theoretical results are in good agreement with the experimental data for and in the whole measured region. For , the theoretical model works well in the region of GeV/c. Beyond this region, the deviation appears as shown in Figure 3, which presents the fittings for in the most peripheral collisions for up to about 4 GeV/c.
(a)
(b)
In analyses, the sound speed in hadronic state takes the value of [46–49]. The critical temperature takes the value of GeV [50]. The kinetic freezeout temperature takes the values of 0.12 GeV for pions and kaons and 0.13 GeV for protons, respectively, from the investigations of [8], which also shows that the average value of baryochemical potential is about 0.019 GeV in Au+Au collisions. For Pb+Pb collisions, takes the value of [8]. The initial temperature in central Au+Au and Pb+Pb collisions takes the values of GeV and GeV, respectively [51].
Tables 1 and 2 list the values of , , , and /NDF in different centrality cuts. It can be seen that decreases with increasing centrality cuts. The value of , affecting the slopes of curves, is slightly larger than 1 for different kinds of charged particles. It is almost irrelevant to centrality cuts, while it increases with increasing beam energies and the masses of charged particles.


The parameter has the same effects as on curves. They all affect the heights of the curves. The fitted in Tables 1 and 2 gives the ratiosThese ratios are in good agreement with the relative abundance of particles and antiparticles given in [3, 4]. This agreement may be due to the fact that the integrand of (20) is the same for particles and antiparticles in case that takes a common constant for these two kinds of particles. Hence, might be proportional to the abundance of corresponding particles.
4. Conclusions
By introducing the nonextensive statistics, we employ the relativistic hydrodynamics including phase transition to discuss the transverse momentum distributions of charged particles produced in heavy ion collisions. The model contains rich information about the transport coefficients of fluid, such as the initial temperature , the critical temperature , the kinetic freezeout temperature , the baryochemical potential , the sound speed in sQGP state , and the sound speed in hadronic state . Except for , the other five parameters take the values either from widely accepted theoretical results or from experimental measurements. As for , there are no widely acknowledged values so far. In this paper, takes the values referring to other studies for the most central collisions, and, for the rest of centrality cuts, is determined by tuning the theoretical results to experimental data. The present investigations show the conclusions as follows:
The theoretical model can give a good description of the experimental data in Au+Au collisions at GeV and in Pb+Pb collisions at TeV for and in the whole measured transverse momentum region and for in the region of GeV/c.
The fitted is close to 1. This might mean that the difference between nonextensive statistics and conventional statistics is small. However, it is this small difference that plays an essential role in extending the fitting region of .
The fitted increases with beam energies and the masses of charged particles. This might mean that the nonextensive statistics are more suitable for these conditions.
The model cannot describe the experimental measurements for in the region of GeV/c for the both kinds of collisions. This might be caused by the hard scattering process [52]. To improve the fitting conditions, the results from perturbative QCD should be taken into account.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by the Shanghai Key Lab of Modern Optical System.
References
 L. Adamczyk, J. K. Adkins, G. Agakishiev et al., “Centrality dependence of identified particle elliptic flow in relativistic heavy ion collisions at √s_{NN} = 7.7–62.4 GeV,” Physical Review C, vol. 93, Article ID 014907, 2016. View at: Google Scholar
 J. Adam, D. Adamova, M. M. Aggarwal et al., “Anisotropic Flow of Charged Particles in PbPb Collisions at √s_{NN} = 5.02 TeV,” Physical Review Letters, vol. 116, Article ID 132302, 2016. View at: Google Scholar
 A. Adare, S. Afanasiev, C. Aidala et al., “Spectra and ratios of identified particles in Au+Au and d+Au collisions at √s_{NN} = 200 GeV,” Physical Review C, vol. 88, Article ID 024906, 2013. View at: Google Scholar
 B. Abelev, J. Adam, D. Adamova et al., “Centrality dependence of π, K, and p production in PbPb collisions at √s_{NN} = 2.76 TeV,” Physical Review C, vol. 88, Article ID 044910, 2013. View at: Google Scholar
 K. Adcox, S. S. Adler, N. N. Ajitanand et al., “Single identified hadron spectra from √s_{NN} = 130 GeV Au + Au collisions,” Physical Review C, vol. 69, Article ID 024904, 2004. View at: Google Scholar
 C. Adler, Z. Ahammed, C. Allgower et al., “Measurement of Inclusive Antiprotons from Au+Au Collisions at √s_{NN} = 130 GeV,” Physical Review Letters, vol. 87, Article ID 262302, 2001. View at: Google Scholar
 K. Adcox, S. Belikov, J. C. Hill et al., “Centrality Dependence of π±, K±, p, and p Production from √s_{NN} = 130 GeV Au+Au Collisions at RHIC,” Physical Review Letters, vol. 88, Article ID 242301, 2002. View at: Google Scholar
 B. I. Abelev, M. Aggarwal, Z. Ahammed et al., “Systematic measurements of identified particle spectra in pp, d+Au, and Au+Au collisions at the STAR detector,” Physical Review C, vol. 79, Article ID 034909, 2009. View at: Google Scholar
 B. Alver, B. Back, M. D. Baker et al., “Chargedparticle 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, Article ID 024913, 2011. View at: Google Scholar
 M. Murray, “Scanning the phases of QCD with BRAHMS,” Journal of Physics G: Nuclear and Particle Physics, vol. 30, no. 8, pp. S667–S674, 2004. View at: Publisher Site  Google Scholar
 M. Murray, J. Natowitz, B. S. Nielsen et al., “Flavor dynamics,” Journal of Physics G: Nuclear and Particle Physics, vol. 35, no. 4, Article ID 044015, 2008. View at: Publisher Site  Google Scholar
 I. G. Bearden, D. Beavis, C. Besliu et al., “Charged Meson Rapidity Distributions in Central Au+Au Collisions at √s_{NN} = 200 GeV,” Physical Review Letters, vol. 94, Article ID 162301, 2005. View at: Google Scholar
 A. Bialas, R. A. Janik, and R. Peschanski, “Unified description of Bjorken and Landau 1+1 hydrodynamics,” Physical Review C nuclear physics, vol. 76, no. 5, Article ID 054901, 2007. View at: Publisher Site  Google Scholar
 C.Y. Wong, “Landau hydrodynamics reexamined,” Physical Review C: Nuclear Physics, vol. 78, no. 5, Article ID 054902, 2008. View at: Publisher Site  Google Scholar
 G. Beuf, R. Peschanski, and E. N. Saridakis, “Entropy flow of a perfect fluid in (1 + 1) hydrodynamics,” Physical Review C: Nuclear Physics, vol. 78, no. 6, Article ID 064909, 2008. View at: Publisher Site  Google Scholar
 T. Csörgo, M. I. Nagy, and M. Csanád, “New family of simple solutions of relativistic perfect fluid hydrodynamics,” Physics Letters B, vol. 663, no. 4, pp. 306–311, 2008. View at: Publisher Site  Google Scholar
 Z. W. Wang, Z. J. Jiang, and Y. S. Zhang, “The investigations of pseudorapidity distributions of final multiplicity in Au+Au collisions at high energy,” Journal of university of Shanghai for science andtechnology, vol. 31, pp. 322–326, 2009. View at: Google Scholar
 N. Suzuki, “Onedimensional hydrodynamical model including phase transition,” Physical Review C nuclear physics, vol. 81, no. 4, Article ID 044911, 2010. View at: Publisher Site  Google Scholar
 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 Site  Google Scholar
 A. Bialas and R. Peschanski, “Asymmetric (1+1)dimensional hydrodynamics in highenergy collisions,” Physical Review C: Nuclear Physics, vol. 83, no. 5, Article ID 054905, 2011. View at: Publisher Site  Google Scholar
 M. Csanád, M. I. Nagy, and S. Lökös, “Exact solutions of relativistic perfect fluid hydrodynamics for a QCD Equation of State,” The European Physical Journal A, vol. 48, no. 11, pp. 173–178, 2012. View at: Publisher Site  Google Scholar
 Z. J. Jiang, Q. G. Li, and H. L. Zhang, “Revised Landau hydrodynamic model and the pseudorapidity distributions of charged particles produced in nucleusnucleus collisions at maximum energy at the BNL Relativistic Heavy Ion Collider,” Physical Review C: Nuclear Physics, vol. 87, no. 4, Article ID 044902, 2013. View at: Publisher Site  Google Scholar
 C. Gale, S. Jeon, and B. Schenke, “Hydrodynamic modeling of heavyion collisions,” International Journal of Modern Physics A, vol. 28, no. 11, Article ID 1340011, 2013. View at: Publisher Site  Google Scholar
 U. Heinz and R. Snellings, “Collective flow and viscosity in relativistic heavyion collisions,” Annual Review of Nuclear and Particle Science, vol. 63, pp. 123–151, 2013. View at: Publisher Site  Google Scholar
 A. N. Mishra, R. Sahoo, E. K. G. Sarkisyan, and A. S. Sakharov, “Effectiveenergy budget in multiparticle production in nuclear collisions,” The European Physical Journal C, vol. 74, article 3147, 2014. View at: Publisher Site  Google Scholar
 Z. J. Jiang, Y. Zhang, H. L. Zhang, and H. P. Deng, “A description of the pseudorapidity distributions in heavy ion collisions at RHIC and LHC energies,” Nuclear Physics A, vol. 941, pp. 188–200, 2015. View at: Publisher Site  Google Scholar
 H. Niemi, K. J. Eskola, and R. Paatelainen, “Eventbyevent fluctuations in a perturbative QCD + saturation + hydrodynamics model: Determining QCD matter shear viscosity in ultrarelativistic heavyion collisions,” Physical Review C: Nuclear Physics, vol. 93, no. 2, Article ID 024907, 2016. View at: Publisher Site  Google Scholar
 J. NoronhaHostler, M. Luzum, and J.Y. Ollitrault, “Hydrodynamic predictions for 5.02 TeV PbPb collisions,” Physical Review C nuclear physics, vol. 93, no. 3, Article ID 034912, 2016. View at: Publisher Site  Google Scholar
 J. S. Moreland and R. A. Soltz, “Hydrodynamic simulations of relativistic heavyion collisions with different lattice quantum chromodynamics calculations of the equation of state,” Physical Review C: Nuclear Physics, vol. 93, no. 4, Article ID 044913, 2016. View at: Publisher Site  Google Scholar
 E. K. G. Sarkisyan, A. N. Mishra, R. Sahoo, and A. S. Sakharov, “Erratum: Multihadron production dynamics exploring the energy balance in hadronic and nuclear collisions (Physical Review D  Particles, Fields, Gravitation and Cosmology (2016) D93 (054046)),” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 93, no. 7, Article ID 079904, 2016. View at: Publisher Site  Google Scholar
 E. K. G. Sarkisyan, A. N. Mishra, R. Sahoo, and A. S. Sakharov, “Centrality dependence of midrapidity density from GeV to TeV heavyion collisions in the effectiveenergy universality picture of hadroproduction,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 94, no. 1, Article ID 011501, 2016. View at: Publisher Site  Google Scholar
 T. Hirano, N. Van Der Kolk, and A. Bilandzic, “Hydrodynamics and flow,” Lecture Notes in Physics, vol. 785, pp. 139–178, 2008. View at: Publisher Site  Google Scholar
 Z.J. Jiang, J.Q. Hui, and Y. Zhang, “The Rapidity Distributions and the Thermalization Induced Transverse Momentum Distributions in AuAu Collisions at RHIC Energies,” Advances in High Energy Physics, vol. 2017, Article ID 6896524, 8 pages, 2017. View at: Publisher Site  Google Scholar
 W. M. Alberico, A. Lavagno, and P. Quarati, “Nonextensive statistics, fluctuations and correlations in high energy nuclear collisions,” The European Physical Journal C, vol. 13, pp. 499–506, 2000. View at: Google Scholar
 M. Biyajima, T. Mizoguchi, N. Nakajima, N. Suzuki, and G. Wilk, “Modified Hagedorn formula including temperature fluctuation: Estimation of temperatures at RHIC experiments,” The European Physical Journal C, vol. 48, no. 2, pp. 597–603, 2006. View at: Publisher Site  Google Scholar
 C. Tsallis, “Possible generalization of BoltzmannGibbs statistics,” Journal of Statistical Physics, vol. 52, no. 12, pp. 479–487, 1988. View at: Publisher Site  Google Scholar  MathSciNet
 T. S. Biro and G. Purcsel, “Nonextensive Boltzmann equation and hadronization,” Physical Review Letters, vol. 95, Article ID 162302, 2005. View at: Publisher Site  Google Scholar
 A. S. Parvan and T. S. Biró, “Equilibrium statistical mechanics for incomplete nonextensive statistics,” Physics Letters A, vol. 375, no. 3, pp. 372–378, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 T. S. Biró, G. G. Barnaföldi, and P. Ván, “Thermodynamic derivation of the tsallis and rényi entropy formulas and the temperature of quarkgluon plasma,” The European Physical Journal A, vol. 49, no. 9, p. 110, 2013. View at: Google Scholar
 A. R. Plastino and A. Plastino, “Information theory, approximate time dependent solutions of Boltzmann's equation and Tsallis' entropy,” Physics Letters A, vol. 193, no. 3, pp. 251–258, 1994. View at: Publisher Site 