ANISOTROPIC QUARK GLUON PLASMA: DISSOCIATION OF QUARKONIUM STATES

Abstract

Quantum Chromodynamics (QCD) is the gauge theory of strong interactions which de- scribes the interaction among quarks and gluons by the exchange of gluons. QCD has two important features: the first one states that the interaction among quarks and gluons becomes stronger as their mutual distances increase and tends to be very large as their distances reach about the size of a hadron, known as confinement. This explains why the quarks and gluons are always confined inside hadrons. The second one is the asymptotic freedom which means that the interaction between quarks and gluons becomes exceedingly small as their mutual distances approach asymptotically zero. This asymptotic limit can be envisaged thermodynamically either at large temperatures and/or densities, where the interactions, which confine quarks and gluons inside hadrons, become sufficiently weak and forms a medium of deconfined quarks and gluons, known as Quark Gluon Plasma (QGP). Since last few decades the relativistic heavy ion collision experiments at CERN, BNL. have made enormous efforts to create and study the properties of QGP. As soon as QGP is created, it cools rapidly by expanding and emitting various radiations to a hadron gas at the confining point Tc before freeze-out and finally the hadrons leave the fireball and reach the detectors. Since the lifetime of QGP is very small and also the free quarks cannot be detected, a direct detection of QGP in experiments is not possible. One of the most interesting signature of QGP is the quarkonium suppression proposed by Matsui and Satz, where the properties of quarkonia at finite temperature be used as an important tool for the study of QGP formation in heavy ion collisions. Recently the concept of momentum anisotropy created at the early stage of collisions is resurrected due to the asymptotic free expansion in the longitudinal (beam) direction, thus it becomes worthwhile to study the effect of momentum anisotropy on the properties of quarkonium states, which is the central theme of this work. The thesis consists of seven chapters. Chapter 1: Introduction The introductory chapter starts with a brief introduction to QCD and our understanding about QGP. Therefore, various theoretical efforts, e.g. lattice QCD and hydrodynamics are presented briefly in order to understand the properties of QGP and various signatures are discussed to probe the QGP. We then give an introduction to the thermal field theory vi in both imaginary and real time formalism, to make the platform of our work. We then reviewed the kinetic theory for hot QGP, transport equation in both QED and QCD plasma and the dispersion relation by decomposing the self-energy into longitudinal and transverse components. Thereafter the concept of anisotropy and how it appeared etc is introduced. For the sake of our thesis work, we have also reviewed the recent theoretical devel- opments, such as the holographic correspondence between gravity and gauge theory to understand the shift in the field due to RHIC discovery of QGP as a strongly coupled liquid, Color Glass Condensate (CGC) etc. Finally, we have presented the layout of our research work reported in the present thesis. Chapter 2: Quarkonium in Vacuum and Medium This chapter starts with a brief introduction to the bound states of heavy quarks and its anti-quarks, known as quarkonia and then surveys the present understanding about the properties of quarkonia both at zero and finite temperature. There are mainly two theoret- ical studies to know the properties of quarkonia: the first one is the potential model studies where the heavy quark bound states have been handled by the non-relativistic Schr¨odinger wave equation. Recently there has been theoretical improvements in the understanding of how to arrive at potential models in the framework of Effective Field Theories (EFTs), viz. potential nonrelativistic QCD (pNRQCD) by integrating out the successive scales as- sociated with the heavy quark mass and the momentum exchange. The second one is the lattice QCD that provides the most straightforward way to determine spectral functions. Thus this chapter depicts the different theoretical developments to study the properties of quarkonia both in vacuum as well as in medium. Chapter 3: Quarkonium in Hot and Anisotropic QCD Medium in the Frame- work of Kinetic Theory This chapter deals how the properties of quarkonium states change in a medium, which exhibits a local anisotropy in momentum space, using the kinetic theory approach. To en- code the momentum anisotropy in the medium, we first obtain the gluon self-energy tensor using the linear response theory and derive the potential by correcting both the Coulombic and linear terms in the Cornell potential, not the Coulomb term alone as usually done in the literature, by the static limit of hard-loop resumed gluon propagator. The potential obtained is found to be screened less than in isotropic medium, as a result the quarkonium vii states become more tightly bound. In addition, the anisotropy in the momentum space introduces a characteristic angular dependence in the potential and as a consequence the quark pairs aligned in the direction of anisotropy are bound stronger than those aligned in the perpendicular direction. Thus the potential in anisotropic medium becomes nonspher- ical in contrast to the spherically symmetric potential in isotropic medium. Therefore one cannot simply obtain the energy eigenvalues by solving the radial part of the Schrodinger equation alone because the radial part is no longer sufficient due to the angular dependence in the potential. In the weak-anisotropy limit, the anisotropic correction is small and thus can be treated as a perturbation. So, using the first-order perturbation theory, we esti- mate the shift in energy eigen values due to the small anisotropic correction to the energy eigen values from the spherically-symmetric part in isotropic medium and determine their dissociation temperatures. Chapter 4: Complex Potential in Real-time Formalism: Dissociation of Quarko- nium States Nowadays the dissociation of heavy quarkonia is understood to be not due to the Debye screening of the potential alone, rather it is overtaken by the thermal width obtained from the imaginary part of the potential. Therefore, we explored the dissociation of quarko- nia by a complex potential which is obtained by correcting both the perturbative and non-perturbative terms of the Q¯Q potential at T = 0 through the dielectric function in real-time formalism. The presence of the confining (non-perturbative) term even above the transition temperature makes the real-part of the potential more stronger and hence the quarkonia become more bound. The confining term also enhances the magnitude of imaginary-part, which in turn increases the thermal width as compared to the medium- contribution of the perturbative term alone. These observations result an increase in the dissociation temperatures of quarkonia. Finally we extend our calculation to a medium, exhibiting local momentum anisotropy, like the kinetic theory approach employed in the previous chapter. The presence of anisotropy makes the real-part of the potential stronger but the imaginary-part is weakened slightly, overall the anisotropy makes the dissociation temperatures higher, compared to isotropic medium. Chapter 5: Gauge-Gravity Duality and Quarkonia in a Moving Thermal QCD Medium viii Till now we have discussed how does the properties of quarkonia change in a thermal medium by a temperature-dependent potential between a static Q and ¯ Q. However these studies are limited to a medium, which is weakly-coupled and static as well. Nowadays the medium produced at RHIC and LHC experiments is understood as a strongly-coupled liquid, unlike a weakly-interacting gas. Moreover an interest has been renewed to study the potential either for a static Q¯Q pair in a moving medium or for a moving Q¯Q pair in a static medium. However the limitations of the weak-coupling regime and the additional scales associated with the motion of the pair complicate the calculations in effective field theories (EFTs). Thus in this chapter we resort for the holographic correspondence to calculate the potential for a moving Q¯Q pair in a strongly-coupled medium. However, the earlier calculations were performed in a pure AdS black hole background, where the dual gauge theory is conformal, i.e. does not depend on the energy scale but the QCD depends on the energy scale. Therefore we work on a metric in the gravity side, viz. OKS-BH geometry with an UV cap, whose dual in the gauge theory side runs with the energy and hence proves to be a better background for thermal QCD and obtain the potential by extremizing the action, known as Nambu-Goto action. The potential obtained has confining terms both in vacuum as well as in medium, in addition to the Coulomb term alone reported in the earlier calculations in AdS/CFT literature. We found that as the velocity of the pair is increased the screening of the potential becomes weaker. The important observation of our work is that the potential develops an imaginary part beyond a critical separation of the heavy quark pair. Thus, with the imaginary part of the potential, we have estimated the thermal width for the ground and first excited states and found that non-zero rapidities lead to an increase of thermal width, which therefore implies that the moving quarkonia are dissociated more efficiently than the static ones. ix Chapter 6: Electrical conductivity Since the study of transport coefficients of strongly interacting matter got impetus after the discovery of perfect fluid created at ultra-relativistic heavy ion collision experiments so in this chapter we deal with one of the transport coefficients, namely electrical conductiv- ity as an additional probe of the anisotropy of the medium, apart from the properties of quarkonia. The infinitesimal fluctuations or external fields cause the system to depart from its equilibrium for a brief time. The response of the system to such type of fluctuations or external fields is essentially described by the transport coefficients, e.g. the shear and bulk viscosities, the speed of sound etc. Electrical conductivity of QCD medium has recently become important due to the strong electric field created in the collision zone of heavy- ion experiments. The large electric field affects the behavior of the medium and its effect depends on the magnitude of electrical conductivity, which is, in turn responsible for the production of electric current generated by the quarks in the early stage of the collision. The momentum anisotropy is also produced in the early stage of the collision and lasts for, at least 2 fm. Therefore, it is worthwhile to incorporate the effect of momentum-space anisotropy in the calculation of the electrical conductivity. For the purpose, relativistic Boltzmann’s kinetic equation has been solved in the relaxation-time approximation to ob- tain the electrical conductivity, where the in-medium properties have been incorporated in the distribution function by the quasiparticle description. We have compared our results with the different lattice as well as other model calculations. Further, we extend our model at finite chemical potential. Chapter 7: Conclusion In this chapter, we present a summary and conclusion drawn from this work and provide some insights in this area for future research work.