Advances in High Energy Physics

Volume 2019, Article ID 9574136, 13 pages

https://doi.org/10.1155/2019/9574136

## Equation of States and Charmonium Suppression in Heavy-Ion Collisions

Department of Physics, Central University of Jharkhand Ranchi, 835 205, India

Correspondence should be addressed to Vineet Kumar Agotiya; moc.liamg@18ayitoga

Received 7 December 2018; Revised 11 March 2019; Accepted 5 May 2019; Published 6 August 2019

Academic Editor: Theocharis Kosmas

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 SCOAP^{3}.

#### 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.