Equation of States and Charmonium Suppression in Heavy-Ion Collisions

Abstract

The present article is the follow-up of our work Bottomonium suppression in quasi-particle model, where we have extended the study for charmonium states using quasi-particle model in terms of quasi-gluons and quasi quarks/antiquarks as an equation of state. By employing medium modification to a heavy quark potential thermodynamic observables, viz., pressure, energy density, speed of sound, etc. have been calculated which nicely fit with the lattice equation of state for gluon, massless, and as well massive flavored plasma. For obtaining the thermodynamic observables we employed the debye mass in the quasi particle picture. We extended the quasi-particle model to calculate charmonium suppression in an expanding, dissipative strongly interacting QGP medium (SIQGP). We obtained the suppression pattern for charmonium states with respect to the number of participants at mid-rapidity and compared it with the experimental data (CMS JHEP) and (CMS PAS) at LHC energy (Pb+Pb collisions, = TeV).

1. Introduction

The primary goal of heavy-ion experiment at the RHIC and the LHC is to search a new state of matter, i.e., the Quark Gluon Plasma. To study the properties of the Quark Gluon Plasma (QGP) heavy quarks are considered to be a suitable tool. Initially, the heavy quarks can be calculated in pQCD, which are produced in primary hard N N collisions [1].The charmonia is a bound states of charm () and anticharm (), which is an extremely broad and interesting field of investigation [2]. Charmonium states can have smaller sizes than hadrons (down to a few tenths of a fm) and large binding energies ( MeV) [3]. In ultrarelativistic heavy-ion collisions, it has been realized that early ideas associating with charmonium suppression with the deconfinement transition [4] are less direct than originally hoped for [5–8].

At sufficiently large energy densities, lattice QCD calculations predict that hadronic matter undergoes a phase transition of deconfined quarks and gluons, called Quark Gluon Plasma (QGP). In order to reveal the existence and to analyze the properties of this phase transition several researches in this direction have been done. In the high-energy heavy-ion collision field, the study of charmonium production and suppression is the most interesting investigations, since the charmonium yield would be suppressed in the presence of a QGP due to color Debye screening [4].

In heavy-ion collisions, charmonium suppression study has been carried out first at the Super Proton Synchrotron (SPS) by the NA38 [9–11] and NA60 [12] and then at the Relativistic Heavy-Ion Collider (RHIC) by the PHENIX experiment at = 200 GeV [13]. The suppression is defined by the ratio of the yield measured in heavy-ion collisions and a reference, called the nuclear modification factor [14] and it is considered as a suitable probe to identify the nature of the matter created in heavy-ion collisions. At high temperature, Quantum Chromodynamics (QCD) is believed to be in Quark Gluon Plasma (QGP) phase, which is not an ideal gas of quarks and gluons, but rather a liquid having very low shear viscosity to entropy density () ratio [15–20].

This strongly suggests that QGP may lie in the nonperturbative domain of QCD which is very hard to address both analytically and computationally. Similar conclusion about QGP and perfect fluidity of QGP have been reached from recent lattice studies and from the AdS/CFT studies [20], spectral functions and transport coefficients in lattice QCD [21] and studies based on classical strongly coupled plasmas [22–24], which predict that the equation of state (EoS) is interacting even at [25–30].

The bag model, confinement models, and quasi-particle models are the several models for studying the EoS of strongly interacting quark gluon plasma [31, 32], etc. Here in our analysis we are using quasi-particle debye mass [33] where equation of state was derived with temperature dependent parton masses and bag constant [34, 35], with effective degrees of freedom [36], etc. All of them claim to explain lattice results, either by adjusting free parameters in the model or by taking lattice data on one of the thermodynamic quantity as an input and predicting other quantities. However, physical picture of quasi-particle model and the origin of various temperature dependent quantities are not clear yet [37]. In strongly interacting QGP [38–40], one considers all possible hadrons even at and try to explain nonideal behavior of QGP near . Recently, an equation of state for strongly coupled plasma has been inferred by utilizing the understanding from strongly coupled QED plasma [41] which fits lattice data well. It is implicitly assumed that once the charmonium dissociates, the heavy quarks hadronize by combining with light quarks only [42]. About of the observed ’s are directly produced in a hadronic collisions, the remaining stemming from the decays of and , excited charmonium states. Since each bound state dissociates at a different temperature, a model of sequential suppression was developed, with the aim of reproducing the charmonium suppression pattern in the heavy-ion collision [43–47]. A suppressed yield of quarkonium in the dilepton spectrum, measured in experiments [48, 49], was proposed as a signature of QGP formation. To determine quarkonium spectral functions at finite temperature there are mainly two theoretical lines of studies which are potential models [50–52] and lattice QCD[21, 53–55].

The central theme of our work is that the potential which we are considering in the deconfined phase could have a nonvanishing confining (string) term, in addition to the Coulomb term [56] unlike Coulomb interaction alone in the aforesaid model [32]. By incorporating this potential we had calculated the thermodynamic variables, viz., pressure, energy density, speed of sound, etc. Our results match nicely with the lattice results of gluon [25–28] and 2-flavor (massless) as well as 3-flavor (massless) QGP [57, 58]. There is also an agreement with (2+1) (two massless and one is massive) and 4 flavoured lattice results too. Motivated by the agreement with lattice results, we employ our equation of state (using quasi-particle Debye mass) to study the Charmonium suppression in expanding plasma in the presence of viscous forces. Here in this work we are not considering the bulk viscosity. This issue will be taken into consideration in near future. of prompt and nonprompt has been measured separately by CMS in bins of transverse momentum, rapidity, and collision centrality [14]. We have compared our results with the experimental data (CMS JHEP) [14] and (CMS PAS) [59] in Pb+Pb collision at LHC energy and found is closer to the experimental results.

In our previous work [60], we had calculated the plasma parameter, pressure, energy density, and speed of sound for only 3-flavor QGP and finally studied the sequential suppression for bottomonium states at the LHC energy in a longitudinally expanding partonic system for only because the experimental data is available only for ADS/CFT case. In this present article we have extended our previous work for charmonium states for all 3-flavors by using quasi-particle model in terms of quasi-gluons and quasi quarks/antiquarks as an equation of state. Here, we had considered three values of the shear viscosity-to-entropy density ratio to see the effects of nonzero values of the shear viscosity on the expansion. The first one is from perturbative QCD calculations, where is =0.3 near . The second one is from AdS/CFT studies, where . Finally we consider =0 (for the ideal fluid) for the sake of comparison. These three ratios have been used only for the charmonium states for both EoS1 and EoS2.

The paper is organized as follows. In Section 2 we briefly discuss our recent work on medium modified potential in isotropic medium and we study we study the effective fugacity quasi-particle model (EQPM). In Section 3 we studied binding energy and dissociation temperature of , , and state considering isotropic medium. Using this effective potential and by incorporating quasi-particle debye mass, we have then developed the equation of state for strongly interacting matter and have shown our results on pressure, energy density, speed of sound, etc., along with the lattice data in Section 4. In Section 5, we have employed the aforesaid equation of state to study the suppression of charmonium in the presence of viscous forces and estimate the survival probability in a longitudinally expanding QGP. Results and discussion will be presented in Section 6 and finally, we conclude in Section 7.

2. Medium Modified Effective Potential and Fugacity Quasi-Particle Model

The interaction potential between a heavy quark and antiquark gets modified in the presence of a medium. The static interquark potential plays vital role in understanding the fate of quark-antiquark bound states in the hot QCD/QGP medium. In the present analysis, we preferred to work with the Cornell potential [3, 61] that contains the Coulombic as well as the string part given as Here, is the effective radius of the corresponding quarkonia state, is the strong coupling constant, and is the string tension. The in-medium modification can be obtained in the Fourier space by dividing the heavy-quark potential from the medium dielectric permittivity, aswhere is the Fourier transform of , shown in (1), given as and is the dielectric permittivity which is obtained from the static limit of the longitudinal part of gluon self-energy [62]

Next, substituting (3) and (4) into (1) and evaluating the inverse FT, we obtain r-dependence of the medium modified potential [63]:

In the limiting case , the dominant terms in the potential are the long range Coulombic tail and . The potential will be shown as

Now we employ the Debye mass computed from the effective fugacity quasi-particle model (EQPM) [64, 65] to determine the dissociation temperatures for the charmonium states in isotropic medium computed for EoS1 and EoS2, respectively, and develop the equation of state for strongly interacting matter. The Debye mass, , is defined in terms of the equilibrium (isotropic) distribution function as where is taken to be a combination of ideal Bose-Einstein and Fermi-Dirac distribution functions as [66] and is given by Here, and are the quasi-parton thermal distributions, denotes the number of colors, and the number of flavors.

Now, we obtain quasi particle debye mass for full QCD/QGP medium by considering quasi-parton distributions and EoS1 is the hot QCD [67–69] and EoS2 is the hot QCD EoS [70] in the quasi-particle description [64, 65], respectively.

3. Binding Energy and Dissociation Temperature

Binding energy is defined as the distance between peak position and continuum threshold at finite temperature. The medium modified potential has similar appearance to the hydrogen atom problem [71].Therefore to get the binding energies with medium modified potential we need to solve the Shrödinger equation numerically. The solution of the Schrödinger equation gives the eigenvalues for the ground states and the first excited states in charmonium (, , etc.) and bottomonium (, , etc.) spectra: where is the mass of the heavy quark.

In our analysis, the quark masses , as GeV, GeV, and GeV, as calculated in [72] and the string tension () is taken as .

We listed the values of dissociation temperature in Tables 1 and 2 for the charmonium states , , and for EoS1 and EoS2, respectively, and also have seen that dissociates at lower temperatures as compared to and for both the EoS.

4. Equation of States of Different Flavors in Quasi-Particle Picture

An extensive study of strong-coupled plasma in QED with proper modifications to include colour degrees of freedom and the strong running coupling constant gives an expression for the energy density as a function of the plasma parameter can be written asNow, the scaled-energy density is written in terms of ideal contribution

At sufficiently high temperature one must expect hadrons to melt, deconfining quarks and gluons. The exposure of new (color) degrees of freedom would then be manifested by a rapid increase in entropy density, hence in pressure, with increasing temperature, and by a consequent change in the equation of state (EOS) [15–17]. In this section we will find the pressure, energy density and speed of sound for pure gauge, 2-flavor, 3-flavor, (2+1)-flavor, and 4-flavors QGP for EoS1 and EoS2. To begin with first of all, we will calculate the energy density from (11) and using the thermodynamic relation,and we calculated the pressure as Here is the pressure at some reference temperature . This temperature has been fixed with the values of pressure at critical temperature , , , and for a particular system-pure gauge, 2-flavor, 3-flavor, and 4-flavor QGP, respectively. For the sake of comparison with the results of Bannur EoS we took the same value of critical temperature as used in Bannur Model. Now, the speed of sound can be calculated once we know the pressure and energy density . In Figures 1 and 2, we have plotted the variation of pressure () with temperature () using EoS1 and EoS2 for pure gauge, 2-flavor, and 3-flavor QGP along with Bannur EoS [32] and compared it with lattice results [25–28, 32]. For each flavor, and are adjusted to get a good fit to lattice results in Bannur Model. However, in our calculation we have fixed from the lattice data at the critical temperature for each system as mentioned above, and there is no quantity to be fitted for predicting lattice results as done in Bannur case. Now, energy density , speed of sound , etc., can be derived since we had obtained the pressure, . In Figures 3 and 4, we had plotted the energy density () with temperature () using EoS1 [67–69] and EoS2 for pure gauge, 2-flavor, and 3-flavor QGP along with Bannur EoS [32] and compared it with lattice result [25–28, 32]. We observe that reasonably good fit is obtained without any extra parameters for all three systems. As the flavor increases, the curves shift to left. In Figures 5 and 6, the speed of sound, is plotted for all three systems, using EoS1 and EoS2 for pure gauge, 2-flavor, and 3-flavor QGP along with Bannur EoS [32]. Since lattice results are available for only pure gauge, therefore comparison has been checked for the above-mentioned flavor only. Our flavored results match excellently with the lattice results. We observe that as the flavor increases becomes larger for both EoS1 and EoS2. All three curves show similar behaviour, i.e., sharp rise near and then flatten to the ideal value (). However, in the vicinity of critical temperature, fits or predictions may not be good, especially for energy density and which strongly depends on variations of pressure with respect to temperature . However, except for small region at , our results are very good for all regions of . It is interesting to note that Peshier and Cassing [73] also obtained similar results on the dependence of plasma parameter in quasi-particle model and concluded that QGP behaves like a liquid, not weakly interacting gas. Now for the realistic case u and d quarks have very small masses (5-10 MeV), strange quarks are having masses 150-200 MeV, and charm quark is with mass 1.5 GeV. Let count the effective number of degrees of freedom of a massive Fermi gas. For a massless gas we have, of course, . In Figures 7–10, we have shown our results on (2+1)-flavors and 4-flavors QGP using EoS1 and EoS2 for pure gauge, 2-flavor, and 3-flavor QGP and compared it with Bannur EoS along with lattice data [74, 75] and replotted the variation of and energy density with temperature for all systems. This has concluded that in the massless limit the deviations of pressure from the ideal gas value are larger in the presence of a heavier quark. This is in qualitative agreement with the observations. We also calculate the thermodynamical quantities, viz., pressure, screening energy density (), the speed of sound, etc. to study the hydrodynamical expansion of plasma and finally to estimate the suppression of in nuclear collisions.

Figure 1 

Figure 2 

Figure 3 

Figure 4 

Figure 5 

Figure 6 

Figure 7 

Figure 8 

Figure 9 

Figure 10 

5. Survival Probability of States

To obtain the charmonium survival probability for an expanding QGP/QCD medium in the presence of viscous forces, the solution of equation of motion gives the time , which is estimated when the energy density drops to the screening energy density as where and is . The critical radius is seen to mark the boundary of the region where the quarkonium formation is suppressed, can be obtained by equating the duration of screening to the formation time for the quarkonium in the plasma frame, and is given by The quark-pair will escape the screening region (and form quarkonium) if its position and transverse momentum are such thatThus, if is the angle between the vectors and , then Here we choose in our calculation as used in [76]. Therefore the survival probability for the charmonium in QGP medium can be expressed as [76–78]where is the maximum positive angle [79]. In nuclear collisions, the -integrated inclusive survival probability of in the QGP/QCD medium becomes [21, 80]

6. Results and Discussion

Now we will discuss the physical understanding of charmonium suppression due to screening in the deconfined medium produced in relativistic nucleus-nucleus collisions. This involves a competition of various time-scales involved in expanding plasma. From Tables 1 and 2 we observe that the value of is different for different charmonium states and varies from one EoS to another. If , then there will be no suppression at all; i.e., survival probability, , is equal to 1. With this physical understanding we analyze our results, as a function of the number of participants in an expanding QGP. At RHIC energy, yields have resulted from a balance between annihilation of ’s due to hard, thermal gluons [81, 82] along with colour screening [76, 77] and enhancement due to coalescence of uncorrelated pairs [83–86] which are produced thermally at deconfined medium. A detailed investigation of the scaling properties of suppression as a function of several centrality variables would give valuable insights into the origin of the observed effect [12]. However, recent CMS data do not show a fully confirmed indication of enhancement except for the fact that of the data and shape of rapidity-dependent nuclear modification factor [13, 14, 59, 87] show some characteristics of coalescence production.

In our analysis, we have employed the quasi-particle debye mass to determine the dissociation temperatures for the charmonium states (, , , etc.) in isotropic medium computed in Tables 1 and 2 for EoS1 and EoS2, respectively. On that dissociation temperature we had calculated the screening energy densities, , and the speed of sound which are also listed in Tables 1 and 2 for both EoS1 and EoS2. These values will be used as inputs, to calculate .

We have shown the variation of -integrated survival probability in the range allowed by invariant spectrum of in CMS experiment with at mid-rapidity and compared with the experimental data (CMS JHEP) [14] in Figures 11 and 13 and (CMS PAS) [59] in Figures 12 and 14. For this we had used the values of ’s and related parameters from Tables 1 and 2 using SIQGP equation of state for both EoS1 and EoS2.

Figure 11 

Figure 12 

Figure 13 

Figure 14 

We find that the survival probability of sequentially produced is slightly higher compared to the directly produced and is closer to the experimental results. The smaller value of screening energy density causes an increase in the screening time and results in more suppression to match with the CMS results at LHC. We have also plotted the pressure, energy density, and speed of sound for pure gauge, 2-flavor, 3-flavor, (2+1)-flavors, and 4-flavors QGP for both EoS1 and EoS2 in Figures 1–10 where we have employed QP EoS (QP EoS is the equation of state calculated by using quasi-particle debye mass) along with the Bannur EoS. Here we observe that the results of various equations of states coming by incorporating the quasi-particle Debye mass increase sharply.

7. Conclusion

We studied the equation of state for strongly interacting quark-gluon plasma in the framework of strongly coupled plasma with appropriate modifications to take account of color and flavor degrees of freedom and QCD running coupling constant. In addition, we incorporate the nonperturbative effects in terms of nonzero string tension in the deconfined phase, unlike the Coulomb interactions alone in the deconfined phase beyond the critical temperature. Our results on thermodynamic observables, viz., pressure, energy density, speed of sound, etc., nicely fit the results of lattice equation of state with gluon, massless, and as well massive flavored plasma. In Figures 1–10 we see that the results coming out by using quasi-particle Debye mass increase sharply as the temperature increases. Now by using quasi-particle Debye mass we estimated the centrality dependence of charmonium suppression in an expanding dissipative strongly interacting QGP produced in relativistic heavy-ion collisions as shown in Figures 11–14 for both EoS1 and Eos2. We find that the survival probability of sequentially produced is slightly higher compared to the directly produced and is closer to the experimental results. The smaller value of screening energy density causes an increase in the screening time and results in more suppression to match with the experimental results.

At LHC energies, the inclusive yield contains a significant nonprompt contribution from b-hadron decays [88, 89]. For the lower value of we observe that our predictions are closer to the experimental ones.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Vineet Kumar Agotiya acknowledge the Science and Engineering Research Board (SERB) Project No. EEQ/2018/000181, New Delhi, for financial support. We record our sincere gratitude to the people of India for their generous support for the research in basic sciences.

References

1. M. Cacciari, P. Nason, and R. Vogt, “QCD predictions for charm and bottom quark production at RHIC,” Physical Review Letters, vol. 95, no. 12, Article ID 122001, 4 pages, 2005. View at: Publisher Site | Google Scholar
2. N. Brambilla, S. Eidelman, and B. K. Heltsley, “Heavy quarkonium: progress, puzzles, and opportunities,” The European Physical Journal C, vol. 71, article 1534, 2011. View at: Publisher Site | Google Scholar
3. E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T. M. Yan, “Charmonium: comparison with experiment,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 21, no. 1, pp. 203–233, 1980. View at: Publisher Site | Google Scholar
4. T. Matsui and H. Satz, “ suppression by quark-gluon plasma formation,” Physics Letters B, vol. 178, pp. 416–422, 1986. View at: Publisher Site | Google Scholar
5. X. Zhao and R. Rapp, “Transverse momentum spectra of in heavy-ion collisions,” Physics Letters B, vol. 664, no. 4-5, pp. 253–257, 2008. View at: Publisher Site | Google Scholar
6. R. Rapp, D. Blaschke, and P. Crochet, “Charmonium and bottomonium in heavy-ion collisions,” Progress in Particle and Nuclear Physics, vol. 65, no. 2, pp. 209–266, 2010. View at: Publisher Site | Google Scholar
7. L. Kulberg and H. Satz, https://arxiv.org/abs/0901.3831.
8. P. Barun-Munzinger and J. Stachel, https://arxiv.org/abs/0901.2500.
9. NA38 Collaboration, C. Baglina, A. Bussière et al., “Ψ′ and J/Ψ production in p-W, p-U and S-U interactions at 200 GeV/nucleon,” Physics Letters B, vol. 345, no. 4, pp. 617–621, 1995. View at: Google Scholar
10. NA50 Collaboration, “A new measurement of suppression in Pb-Pb collisions at 158 GeV per nucleon,” The European Physical Journal C, vol. 39, no. 3, pp. 335–345, 2005. View at: Publisher Site | Google Scholar
11. B. Alessandro, C. Alexa, R. Arnaldi et al., “ψ′ production in Pb–Pb collisions at 158 GeV/nucleon,” The European Physical Journal C, vol. 49, no. 2, pp. 559–567, 2007. View at: Publisher Site | Google Scholar
12. NA60 Collaboration, R. Arnaldi et al., “J/ψ production in indium-indium collisions at 158 GeV/nucleon,” Physical Review Letters, vol. 99, Article ID 132302, 2007. View at: Google Scholar
13. PHENIX Collaboration, A. Adare et al., “J/ψ Production versus centrality, transverse momentum, and rapidity in Au + Au collisions at ,” Physical Review Letters, vol. 98, Article ID 232301, 2007. View at: Google Scholar
14. CMS Collaboration, S. Chatrchyan, V. Khachatryan et al., “Suppression of non-prompt J/ψ, prompt J/ψ, and Υ (1S) in PbPb collisions at ,” Journal of High Energy Physics, vol. 5, article no. 63, 2012. View at: Google Scholar
15. STAR Collaboration, J. Adams, M. M. Aggarwal et al., “Experimental and theoretical challenges in the search for the quark–gluon plasma: The STAR Collaboration's critical assessment of the evidence from RHIC collisions,” Nuclear Physics A, vol. 757, no. 1-2, pp. 102–183, 2005. View at: Publisher Site | Google Scholar
16. PHENIX Collaboration, K. Adcox, S. S. Adler et al., “Formation of dense partonic matter in relativistic nucleus–nucleus collisions at RHIC: experimental evaluation by the PHENIX Collaboration,” Nuclear Physics A, vol. 757, no. 1-2, pp. 184–283, 2005. View at: Google Scholar
17. PHOBOS Collaboration, B. B. Back, M. D. Baker, M. Ballintijn et al., “The PHOBOS perspective on discoveries at RHIC,” Nuclear Physics A, vol. 757, no. 1-2, pp. 28–101, 2005. View at: Publisher Site | Google Scholar
18. H.-J. Drescher, A. Dumitru, C. Gombeaud, and J.-Y. Ollitrault, “Centrality dependence of elliptic flow, the hydrodynamic limit, and the viscosity of hot QCD,” Physical Review C: Nuclear Physics, vol. 76, no. 2, Article ID 024905, 2007. View at: Publisher Site | Google Scholar
19. E. Shuryak, “Toward the theory of strongly coupled quark-gluon plasma (sQGP),” Nuclear Physics A, vol. 774, pp. 387–396, 2006. View at: Google Scholar
20. P. K. Kovtun, D. T. Son, and A. O. Starinets, “Viscosity in strongly interacting quantum field theories from black hole physics,” Physical Review Letters, vol. 94, no. 11, Article ID 111601, 2005. View at: Publisher Site | Google Scholar
21. H. Satz, “Quarkonium binding and dissociation: the spectral analysis of the QGP,” Nuclear Physics A, vol. 783, no. 1-4, pp. 249–260, 2007. View at: Google Scholar
22. E. V. Shuryak, “Strongly coupled quark-gluon plasma: the status report,” Continuous Advances in QCD 2006, pp. 3–16, 2007. View at: Google Scholar
23. B. A. Gelman, E. V. Shuryak, and I. Zahed, “Classical strongly coupled quark-gluon plasma. I. Model and molecular dynamics simulations,” Physical Review C: Nuclear Physics, vol. 74, no. 4, Article ID 044908, 2006. View at: Publisher Site | Google Scholar
24. B. A. Gelman, E. V. Shuryak, and I. Zahed, “Classical strongly coupled quark-gluon plasma. II. Screening and equation of state,” Physical Review C: Nuclear Physics, vol. 74, no. 4, Article ID 044909, 2006. View at: Google Scholar
25. G. Boyd, J. Engels, F. Karsch et al., “Equation of state for the SU(3) gauge theory,” Physical Review Letters, vol. 75, Article ID 4169, 1995. View at: Google Scholar
26. G. Boyd, J. Engels, F. Karsch et al., “Thermodynamics of SU(3) lattice gauge theory,” Nuclear Physics B, vol. 469, no. 3, pp. 419–444, 1996. View at: Publisher Site | Google Scholar
27. F. Karsch, “Lattice QCD at high temperature and density,” Lecture Notes in Physics, vol. 583, pp. 209–249, 2002. View at: Google Scholar
28. A. Bazavov, T. Bhattacharya, M. Cheng et al., “Equation of state and QCD transition at finite temperature,” Physical Review D, vol. 80, Article ID 014504, 2009. View at: Google Scholar
29. M. Cheng, N. H. Christ, S. Datta et al., “QCD equation of state with almost physical quark masses,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 77, no. 1, Article ID 014511, 2008. View at: Publisher Site | Google Scholar
30. R. V. Gavai, “Present status of lattice QCD at nonzero T and μ,” Pramana, vol. 67, no. 5, pp. 885–898, 2006. View at: Google Scholar
31. E. Shuryak, “What RHIC experiments and theory tell us about properties of quark–gluon plasma?” Nuclear Physics A, vol. 750, no. 1, pp. 64–83, 2005. View at: Publisher Site | Google Scholar
32. V. M. Bannur, “Strongly coupled quark gluon plasma (SCQGP),” Journal of Physics G: Nuclear and Particle Physics, vol. 32, no. 7, p. 993, 2006. View at: Google Scholar
33. A. Peshier, B. Kämpfer, O. Pavlenko, and G. Soff, “An effective model of the quark-gluon plasma with thermal parton masses,” Physics Letters B, vol. 337, no. 3-4, pp. 235–239, 1994. View at: Publisher Site | Google Scholar
34. P. Lévai and U. Heinz, “Massive gluons and quarks and the equation of state obtained from SU(3) lattice QCD,” Physical Review C, vol. 57, Article ID 1879, 1998. View at: Google Scholar
35. A. Peshier, B. Kämpfer, O. P. Pavlenko, and G. Soff, “Massive quasiparticle model of the SU(3) gluon plasma,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 54, no. 3, Article ID 2399, 1996. View at: Publisher Site | Google Scholar
36. R. A. Schneider and W. Weise, “Quasiparticle description of lattice QCD thermodynamics,” Physical Review C: Nuclear Physics, vol. 64, no. 5, Article ID 055201, 2001. View at: Publisher Site | Google Scholar
37. D. H. Rischke, “The quark–gluon plasma in equilibrium,” Progress in Particle and Nuclear Physics, vol. 52, no. 1, pp. 197–296, 2004. View at: Google Scholar
38. E. V. Shuryak, “Why does the quark–gluon plasma at RHIC behave as a nearly ideal fluid?” Progress in Particle and Nuclear Physics, vol. 53, no. 1, pp. 273–303, 2004. View at: Google Scholar
39. E. V. Shuryak and I. Zahed, “Rethinking the properties of the quark-gluon plasma at T_c < T < 4T_c,” Physical Review C, vol. 70, Article ID 021901, 2004. View at: Google Scholar
40. E. V. Shuryak and I. Zahed, “Understanding nonperturbative deep-inelastic scattering: instanton-induced inelastic dipole-dipole cross section,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 69, Article ID 014011, 2004. View at: Publisher Site | Google Scholar
41. V. M. Bannur, “Equation of state for a non-ideal quark gluon plasma,” Physics Letters B, vol. 362, no. 1-4, pp. 7–10, 1995. View at: Google Scholar
42. W. M. Alberico, A. Beraudo, A. D. Pace, and A. Molinari, “Heavy quark bound states above ,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 72, Article ID 114011, 2005. View at: Publisher Site | Google Scholar
43. S. Gupta and H. Satz, “Final state suppression in nuclear collisions,” Physics Letters B, vol. 283, no. 3-4, pp. 439–445, 1992. View at: Google Scholar
44. S. Digal, P. Petreczky, and H. Satz, “String breaking and quarkonium dissociation at finite temperatures,” Physics Letters B, vol. 514, no. 1-2, pp. 57–62, 2001. View at: Publisher Site | Google Scholar
45. S. Digal, P. Petreczky, and H. Satz, “Quarkonium feed-down and sequential suppression,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 64, no. 9, Article ID 094015, 2001. View at: Publisher Site | Google Scholar
46. F. Karsch, “Deconfinement and quarkonium suppression,” The European Physical Journal C - Particles and Fields, vol. 43, no. 1-4, pp. 35–43, 2005. View at: Google Scholar
47. D. Kharzeev, C. Lourenco, M. Nardi, and H. Satz, “A quantitative analysis of charmonium suppression in nuclear collisions,” Zeitschrift für Physik C, vol. 74, no. 2, pp. 307–318, 1997. View at: Google Scholar
48. G. Borges and NA50 Collaboration, “Charmonia production at SPS energies,” Journal of Physics G: Nuclear and Particle Physics, 32 S381, 2006. View at: Google Scholar
49. PHENIX Collaboration, A. Adare et al., “J/ψ Production in = 200 GeV Cu + Cu collisions,” Physical Review Letters, vol. 101, Article ID 122301, 2008. View at: Google Scholar
50. A. Pineda and J. Soto, “Effective field theory for ultrasoft momenta in NRQCD and NRQED,” Nuclear Physics B- Proceedings Supplements, vol. 64, no. 1-3, pp. 428–432, 1998. View at: Google Scholar
51. N. Brambilla, A. Pineda, J. Soto, and A. Vairo, “Potential NRQCD: an effective theory for heavy quarkonium,” Nuclear Physics B, vol. 566, no. 1-2, pp. 275–310, 2000. View at: Publisher Site | Google Scholar
52. C.-Y. Wong, “Heavy quarkonia in quark-gluon plasma,” Physical Review C: Nuclear Physics, vol. 72, no. 3, Article ID 034906, 2005. View at: Publisher Site | Google Scholar
53. T. Umeda, K. Nomura, and H. Matsufuru, “Charmonium at finite temperature in quenched lattice QCD,” The European Physical Journal C, vol. 39, no. S1, pp. 9–26, 2005. View at: Publisher Site | Google Scholar
54. M. Asakawa and T. Hatsuda, “J/ψ and η_c in the deconfined plasma from lattice QCD,” Physical Review Letters, vol. 92, no. 1, Article ID 012001, 2004. View at: Publisher Site | Google Scholar
55. G. Aarts, C. Allton, M. Oktay, M. Peardon, and J.-I. Skullerud, “Charmonium at high temperature in two-flavor QCD,” Physical Review D, vol. 76, Article ID 094513, 2007. View at: Google Scholar
56. V. Agotiya, V. Chandra, and B. K. Patra, “Dissociation of quarkonium in a hot QCD medium: modification of the interquark potential,” Physical Review C: Nuclear Physics, vol. 80, no. 2, Article ID 025210, 2009. View at: Publisher Site | Google Scholar
57. F. Karsch, “Lattice results on QCD thermodynamics,” Nuclear Physics A, vol. 698, no. 1-4, pp. 199–208, 2002. View at: Publisher Site | Google Scholar
58. E. Laermann and O. Philipsen, “Lattice QCD at finite temperature,” Annual Review of Nuclear and Particle Science, vol. 53, pp. 163–198, 2003. View at: Google Scholar
59. CMS Collaboration, CMS-PAS-HIN-12-014.
60. I. Nilima and V. K. Agotiya, “Bottomonium suppression in nucleus-nucleus collisions using effective fugacity Quasi-Particle model,” Advances in High Energy Physics, vol. 2018, Article ID 8965413, 12 pages, 2018. View at: Google Scholar
61. E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T. M. Yan, “Charmonium: the model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 17, no. 11, pp. 3090–3117, 1978. View at: Publisher Site | Google Scholar
62. A. Schneider, “Debye screening at finite temperature reexamined,” Physical Review D, vol. 66, Article ID 036003, 2002. View at: Google Scholar
63. V. Agotiya, L. Devi, U. Kakade, and B. K. Patra, “Strongly interacting QGP and quarkonium suppression at RHIC and LHC energies,” International Journal of Modern Physics A, vol. 27, no. 2, Article ID 1250009, 2012. View at: Google Scholar
64. V. Chandra, A. Ranjan, and V. Ravishankar, “On the chromo-electric permittivity and Debye screening in hot QCD,” The European Physical Journal A, vol. 40, pp. 109–117, 2009. View at: Publisher Site | Google Scholar
65. V. Chandra, R. Kumar, and V. Ravishankar, “Hot QCD equations of state and relativistic heavy ion collisions,” Physical Review C: Nuclear Physics, vol. 76, no. 5, Article ID 054909, 2007. View at: Publisher Site | Google Scholar
66. M. E. Carrington and A. Rebhan, “Next-to-leading order static gluon self-energy for anisotropic plasmas,” Physical Review D, vol. 79, Article ID 025018, 2009. View at: Google Scholar
67. C. Zhai and B. Kastening, “Free energy of hot gauge theories with fermions through ,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 52, no. 12, pp. 7232–7246, 1995. View at: Publisher Site | Google Scholar
68. P. Arnold and C. Zhai, “Three-loop free energy for pure gauge QCD,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 50, no. 12, pp. 7603–7623, 1994. View at: Publisher Site | Google Scholar
69. P. Arnold and C. Zhai, “Three-loop free energy for high-temperature QED and QCD with fermions,” Physical Review D, vol. 51, article no. 1906, 1995. View at: Google Scholar
70. K. Kajantie, M. Laine, K. Rummukainen, and Y. Schröder, “Pressure of hot QCD up to ,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 67, no. 10, Article ID 105008, 2003. View at: Publisher Site | Google Scholar
71. T. Matsui and H. Satz, “J/ψ Suppression by quark-gluon plasma formation,” Physics Letters B, vol. 178, pp. 416–422, 1986. View at: Google Scholar
72. KEDR Collaboration, V. M. Aulchenko, S. A. Balashov et al., “New precision measurement of the - and -meson masses,” Physics Letters B, vol. 573, pp. 63–79, 2003. View at: Google Scholar
73. A. Peshier and W. Cassing, “The hot nonperturbative gluon plasma is an almost ideal colored liquid,” Physical Review Letters, vol. 94, Article ID 172301, 2005. View at: Publisher Site | Google Scholar
74. F. Karsch, E. Laermann, and A. Peikert, “The pressure in 2, 2+1 and 3 flavour QCD,” Physics Letters B, vol. 478, no. 4, pp. 447–445, 2000. View at: Google Scholar
75. J. Engels, R. Joswig, F. Karsch, E. Laermann, M. Lütgemeier, and B. Petersson, “Thermodynamics of four-flavour QCD with improved staggered fermions,” Physics Letters B, vol. 396, no. 1-4, pp. 210–216, 1997. View at: Publisher Site | Google Scholar
76. M. Chu and T. Matsui, “Pattern of J/ψ suppression in ultrarelativistic heavy-ion collisions,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 37, no. 7, pp. 1851–1855, 1988. View at: Publisher Site | Google Scholar
77. M. Mishra, C. P. Singh, V. J. Menon, and R. K. Dubey, “ suppression in Au + Au collisions at RHIC: Colour screening scenario in the bag model at variable participant numbers,” Physics Letters B, vol. 656, no. 1-3, pp. 45–50, 2007. View at: Google Scholar
78. M. Mishra, C. P. Singh, and V. J. Menon, “J/ψ suppression vs centrality at forward and mid-rapidity in Au+Au collisions at RHIC in colour screening mechanism,” Indian Journal of Physics, vol. 85, no. 6, pp. 849–853, 2011. View at: Google Scholar
79. V. Agotiya, L. Devi, U. Kakade, and B. K. Patra, “Strongly interacting QGP and quarkonium suppression at rhic and LHC energies,” International Journal of Modern Physics A, vol. 27, no. 2, 2012. View at: Google Scholar
80. D. Pal, B. K. Patra, and D. K. Srivastava, “Determination of the equation of state of quark matter from Jψ and Υ suppression at RHIC and LHC,” European Physical Journal C, vol. 17, pp. 179–186, 2000. View at: Publisher Site | Google Scholar
81. X. Xu, D. Kharzeev, H. Satz, and X. Wang, “J /ψ suppression in an equilibrating parton plasma,” Physical Review C: Nuclear Physics, vol. 53, no. 6, pp. 3051–3056, 1996. View at: Publisher Site | Google Scholar
82. B. K. Patra and V. J. Menon, “J/ψ gluonic dissociation revisited: III. Effects of transverse hydrodynamic flow,” The European Physical Journal C, vol. 48, no. 1, pp. 207–213, 2006. View at: Publisher Site | Google Scholar
83. L. Grandchamp, R. Rapp, and G. E. Brown, “In-medium effects on charmonium production in heavy-ion collisions,” Physical Review Letters, vol. 92, no. 21, Article ID 212301, 2004. View at: Publisher Site | Google Scholar
84. A. Andronic, P. Braun-Munzinger, K. Redlich, and J. Stachel, “Statistical hadronization of charm in heavy-ion collisions at SPS, RHIC and LHC,” Physics Letters B, vol. 571, no. 1-2, pp. 36–44, 2003. View at: Publisher Site | Google Scholar
85. R. L. Thews and M. L. Mangano, “Momentum spectra of charmonium produced in a quark-gluon plasma,” Physical Review C: Nuclear Physics, vol. 73, no. 1, Article ID 014904, 2006. View at: Publisher Site | Google Scholar
86. R. L. Thews, “Quarkonium production via recombination,” Nuclear Physics A, vol. 783, no. 1-4, pp. 301–308, 2007. View at: Publisher Site | Google Scholar
87. M. Spousta, “On similarity of jet quenching and charmonia suppression,” Physics Letters B, vol. 767, pp. 10–15, 2017. View at: Google Scholar
88. LHCb Collaboration, R. Aaij, B. Adeva et al., “Measurement of J/Ψ production in pp collisions at = 7 TeV,” European Physical Journal C, vol. 71, article no. 1645, 2011. View at: Google Scholar
89. The CMS Collaboration, V. Khachatryan, A. M. Sirunyan et al., “Prompt and non-prompt J/ψ production in pp collisions at = 7 TeV,” The European Physical Journal C, vol. 71, p. 1575, 2011. View at: Google Scholar

Copyright

Copyright © 2019 Indrani Nilima and Vineet Kumar Agotiya. 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.