Advances in High Energy Physics

Volume 2018, Article ID 8965413, 12 pages

https://doi.org/10.1155/2018/8965413

## Bottomonium Suppression in Nucleus-Nucleus Collisions Using Effective Fugacity Quasi-Particle Model

Centre for Applied Physics, Central University of Jharkhand, Ranchi 835 205, India

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

Received 20 April 2018; Accepted 26 June 2018; Published 12 July 2018

Academic Editor: Chun-Sheng Jia

Copyright © 2018 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

We have studied the equation of state and dissociation temperature of bottomonium state by correcting the full Cornell potential in isotropic medium by employing the effective fugacity quasi-particle Debye mass. We had also calculated the bottomonium suppression in an expanding, dissipative strongly interacting QGP medium produced in relativistic heavy-ion collisions. Finally we compared our results with experimental data from RHIC 200GeV/nucleon Au-Au collisions, LHC 2.76 TeV/nucleon Pb-Pb, and LHC 5.02 TeV/nucleon Pb-Pb collisions as a function of number of participants.

#### 1. Introduction

At the Relativistic Heavy-Ion Collider (RHIC) situated at Brookhaven National Laboratory (BNL), heavy-ion collisions have been studied. After the pioneer work done in the direction of suppression by Matsui and Satz, and some other development of the potential models, suppression was observed by both SPS and RHIC [1]. Due to the Debye screening of the Quantum Chromo-Dynamic (QCD) potential between the two heavy quarks, quarkonia suppression was originally claimed to be an unambiguous signal of the formation of a quark-gluon plasma (QGP). Quarkonia suppression was suggested to be a signature of the QGP and we can measure the suppression ( as well as ), both at RHIC and at the LHC.

In heavy-ion collisions to determine the properties of the medium formed in A+A collisions and p + p collisions, the A+A collision deviates from simple superposition of independent p + p collisions. This deviation is quantified with the nuclear modification factor (). This factor is the ratio of the yield in heavy-ion collisions over the yield in p + p collisions, scaled by a model of the nuclear geometry of the collision. The value of =1 indicates no modification due to the medium. We can say that the probe of interest is suppressed in heavy-ion collisions if is less than 1. A quarkonia meson that forms on the outside surface will not dissociate regardless of the temperature of the medium because it does not have a chance to interact with it. This is why we never see a that is equal to zero. The suppression can also be affected by the QGP, the formation time of the quarkonia meson, and the QGP lifetime as well. For instance, a high quarkonia meson could have a formation time long enough that it actually does not see the QGP at all and thus is not suppressed.

In the early days most of the interests were focused on the suppression of charmonium states [1–3] of collider experiments at SPS and RHIC, but several observations are yet to be understood;* namely,* the suppression of (1S) does not increase from SPS to RHIC, even though the centre-of-mass energy is increased by fifteen times. The heavy-ion program at the LHC may resolve those puzzles because the beam energy and luminosity are increased by ten times that of the RHIC. Moreover the CMS detector has excellent capabilities for muon detection and provides measurements of (2S) and the family, which enables the quantitative analysis of quarkonia. That is why the interest may be shifted to the bottomonium states at the LHC energy.

A potential model for the phenomenological descriptions of heavy quarkonium suppression would be quite useful inspite of the progress of direct lattice QCD based determinations of the potential. The large mass of heavy quarks and their small relative velocity make the use of nonrelativistic quantum mechanics justifiable to describe the quarkonia in the potential models. This is one of the main goals of this present study that argues for the modification of the full Cornell potential as an appropriate potential for heavy quarkonium at finite temperature. QGP created at RHIC have a very low viscosity to entropy ratio, i.e., [4–9], and in the nonperturbative domain of QCD, with temperature close to , the quark matter in the QGP phase is strongly interacting.

In the present paper, we shall employ quasi-particle model for hot QCD equations of state [10, 11] to extract the Debye mass [12] which is obtained in terms of quasi-particle degrees of freedom. We first obtained the medium modified heavy quark potential in isotropic medium and estimate the dissociation temperature. Here, we have used the viscous hydrodynamics to define the dynamics of the system created in the heavy-ion collisions. We have included only the shear viscosity and not included the bulk viscosity. We will look the issue of bulk viscosity in near future.

Our work is organized as follows. In Section 2, we briefly discuss our recent work on medium modified potential in isotropic medium. In Sections 2.1 and 2.2 we study the real and imaginary part of the potential in the isotropic medium and effective fugacity quasi-particle model (EQPM) in Section 2.3. 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, and speed of sound along with the lattice data. In Section 4, we have employed the aforesaid equation of state to study the suppression of bottomonium in the presence of viscous forces and estimate the survival probability in a longitudinally expanding QGP. Results and discussion will be presented in Section 5 and finally, we conclude in Section 6.

#### 2. Medium Modified Effective Potential in Isotropic Medium

We can obtain the medium modification to the vacuum potential by correcting its both Coulombic and string part with a dielectric function encoding the effect of deconfinement [25]:

Here the functions, and , are the Fourier transform (FT) of the dielectric permittivity and Cornell potential, respectively. After assuming as distribution ( ) we evaluated the Fourier transform of the linear part asWhile putting , we can write the FT of the linear term asThus the FT of the full Cornell potential becomes

To obtain the real and imaginary parts of the potential, we put the temporal component of real and imaginary part in terms of retarded (or advanced) and symmetric parts in the Fourier space in isotropic medium which finally gives

Let us now discuss the real and imaginary part of the potential modified using the above define and along with effective fugacity quasi-particle model (EQPM) in the next subsections.

##### 2.1. Real Part of the Potential in the Isotropic Medium

Now using the real part of retarded (advanced) propagator in isotropic medium, we getwhere the real part of the dielectric permittivity (also given in [26–28]) becomesNow using (6) and real part of dielectric permittivity (7) in (1), we getSolving the above integral, we findwhere . In the limit , we have

##### 2.2. Imaginary Part of the Potential in the Isotropic Medium

To obtain the imaginary part of the potential in the QGP medium, the temporal component of the symmetric propagator in the static limit has been considered, which reads [29, 30]Now the imaginary part of the dielectric function in the QGP medium isAfterwards, the imaginary part of the medium potential is easy to obtain owing to the definition of the potential (1) as done in [31]:After performing the integration, we findwhere .

##### 2.3. Effective Fugacity Quasi-Particle Model (EQPM)

In our calculation, we use the Debye mass for full QCD:Here, is the QCD running coupling constant, () and is the number of flavors, the function has the form , and is the quasi-gluon effective fugacity and is quasi-quark effective fugacity. These distribution functions are isotropic in nature. These fugacities should not be confused with any conservations law (number conservation) and have merely been introduced to encode all the interaction effects at high temperature QCD. Both and have a very complicated temperature dependence and asymptotically reach to the ideal value unity [11]. The temperature dependence of and fits well to the form given below:Here and , , , and are fitting parameters, for both EOS1 and EOS2. Here, EoS1 is the hot QCD [13–15] and EoS2 is the hot QCD EoS [16] in the quasi-particle description [10, 11], respectively. Now, the expressions for the Debye mass can be rewritten in terms of effective charges for the quasi-gluons and quarks aswhere and are the effective charges given by the equations:

In our present analysis we had used the temperature dependence of the quasi-particle Debye mass, , in full QCD with to determine charmonium suppression in an expanding, dissipative strongly interacting QGP medium. This quasi-particle Debye mass, , has the following form:

#### 3. Binding Energy and Dissociation Temperature

To obtain the binding energies with heavy quark potential, we need to solve the Schrödinger equation numerically. In the limiting case discussed earlier, the medium modified potential resembles to the hydrogen atom problem [1]. 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, we have fixed the critical temperature ( = ) and have taken the quark masses , as = 4.5 GeV, = 5.01 GeV, and GeV, as calculated in [32], and the string tension () is taken as . Let us now proceed to the computation of the dissociation temperatures for the above-mentioned quarkonia bound states.

As we know, dissociation of a quarkonia bound state in a thermal QGP medium will occur whenever the binding energy, , of the said state will fall below the mean thermal energy of a quasi-parton. In such situations, the thermal effect can dissociate the quakonia bound state. To obtain the lower bound of the dissociation temperatures of the various quarkonia states, the (relativistic) thermal energy of the partons will be 3 . The dissociation is supposed to occur whenever

’s for the sates , , and with the dissociation temperature are listed in Tables 1 and 2 for EoS1 and EoS2, respectively. We observe that (on the basis of temperature dependence of binding energy) dissociates at lower temperatures as compared to and for both the equations of state.