Screening Potential and Bound State Properties
in Strongly Coupled Quark Gluon Plasma
Thesis submitted to the University of Calicut
in partial fulfillment of the requirements
for the award of the degree of
Doctor of Philosophy
in
Physics
Rethika K T
Department of Physics
University of Calicut
Malappuram, Kerala - 673635
November 2021
CERTIFICATE
Certified that the work presented in this thesis entitled ‘Screening Potential and
Bound State Properties in Strongly Coupled Quark Gluon Plasma’ is a bonafide
work done by Mrs. Rethika K T under our guidance for the award of the degree
of Doctor of Philosophy in the Department of Physics, University of Calicut and
that this work has not been included in any other thesis submitted previously for
the award of any degree.
C. U. campus Dr. C. D. Ravikumar Dr. Vishnu Mayya Bannur
Date: (Co-Guide) (Supervising Guide)
DECLARATION
I hereby declare that the work presented in this thesis entitled ‘Screening Po-
tential and Bound State Properties in Strongly Coupled Quark Gluon Plasma’ is
based on the original work done by me under the guidance of Dr. Vishnu Mayya
Bannur (Guide), and Dr. C D Ravikumar (Co-Guide), Department of Physics,
University of Calicut, and has not been included in any other thesis submitted
previously for the award of any degree.
University of Calicut
Date: Rethika K T
Publications and Presentations
Publications in International Journals
1. K. T. Rethika, V. M. Bannur, Screening of a Test Quark in the Strongly
Coupled Quark Gluon Plasma., Few-Body Syst 60, 38 (2019)
2. K. T. Rethika, C. D. Ravikumar, V. M. Bannur, Heavy Quarkonium Prop-
erties at Finite Temperature in Strongly Coupled Quark Gluon Plasma, Few-
Body Syst 62, 10 (2021)
Papers Presented in International / National Conferences
1. K. T. Rethika, V. M. Bannur, Acoustic Waves in the Strongly Coupled
Quark Gluon Plasma, National Conference on Fundamental and Applied
Physics on 12th and 13th November 2019 at University College, Trivandrum
2. K. T. Rethika, C. D. Ravikumar, V. M. Bannur, Temperature Depen-
dent Screening Potential, ESTEEM 2021 on 21st and 22nd January 2021, at
MANIT, Bhopal
3. K. T. Rethika, C. D. Ravikumar, V. M. Bannur, Binding Energies of
Diquarks, International Conference on Advanced Physics 2021 (IEMPHYS
2021) on1st − 3rd April 2021 at IEM, Kolkata
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my supervisor, Dr. Vishnu Mayya
Bannur, who has been the cornerstone of my Ph.D. I would like to thank him
for the initial direction, and the direction provided at every point. Without his
support and guidance, the PhD would have been impossible to accomplish. I am
indebted to my co-guide Dr. C D Ravikumar for giving me support for submitting
the thesis. He has been patient and encouraging. I am grateful to him for his kind
discussions. I express my heartfelt thanks to Dr. Vinodkumar, for the support
during the course of this work.
I thank my co-researchers Dr. Prasanth Jayaprakash and Dr. Sebastian
Koothottil for their creative discussions. Also I thank Dr. Sineeba Ramadas,
Anjana A V, Jisha, Nighilnath who have been helpful in one way or another. I
am thankful to the faculty members and non-teaching staff of the Department of
Physics, University of Calicut.
I express my deep gratitude and love to my family for their immense encour-
agements.
Contents
1 The Quark-Gluon Plasma 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Asymptotic freedom and quark confinement . . . . . . . . . . . . . 5
1.3 The QCD phase diagram . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 The quark-gluon plasma (QGP) . . . . . . . . . . . . . . . . . . . . 9
1.5 The search for QGP . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 The QGP signatures . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6.1 Direct photon production . . . . . . . . . . . . . . . . . . . 12
1.6.2 Dilepton production . . . . . . . . . . . . . . . . . . . . . . 13
1.6.3 Quarkonia suppression . . . . . . . . . . . . . . . . . . . . . 14
1.6.4 Strangeness enhancement . . . . . . . . . . . . . . . . . . . 15
1.6.5 Jet quenching . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6.6 Anisotropic flow . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 Phenomenological models . . . . . . . . . . . . . . . . . . . . . . . 18
1.7.1 MIT Bag Model . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7.2 The quasi particle QGP model (qQGP) . . . . . . . . . . . . 19
1.7.3 Cornell Potential Model . . . . . . . . . . . . . . . . . . . . 20
i
1.7.4 Relativistic Harmonic Oscillator (RHO) Potential
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7.5 The Strongly Interacting Quark-Gluon Plasma
Model (sQGP) . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7.6 Strongly Coupled Quark-Gluon Plasma Model
(SCQGP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Screening of a Test Quark in the Strongly Coupled Quark Gluon
Plasma 31
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Strongly coupled quark-gluon plasma . . . . . . . . . . . . . . . . . 33
2.3 Screening potential in SCQGP through G-QHD approach . . . . . . 35
2.4 Suppression of J/ψ meson . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Temperature dependence of screening potential . . . . . . . . . . . 39
2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Calculation of Binding Energy of Bound States of Quarks in Strongly
Coupled Quark Gluon Plasma 46
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Exact solution of the N-dimensional
Schrödinger equation with the two
body potential in SCQGP at finite
temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.1 Analytical exact iteration method (AEIM) . . . . . . . . . . 48
3.2.2 Power series method (PSM) . . . . . . . . . . . . . . . . . . 53
3.2.3 Nikiforov-Uvarov (NU) Method . . . . . . . . . . . . . . . . 55
ii
3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Binding Energies and Effective Masses of Diquarks in SCQGP 68
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Binding energy of diquarks . . . . . . . . . . . . . . . . . . . . . . 70
4.2.1 Analytical exact iteration method (AEIM) . . . . . . . . . . 70
4.2.2 Power series method (PSM) . . . . . . . . . . . . . . . . . . 70
4.3 Effective mass of diquarks . . . . . . . . . . . . . . . . . . . . . . . 71
4.4 Relativistic Correction . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 73
5 Heavy Quarkonium Properties at Finite Temperature in Strongly
Coupled Quark Gluon Plasma 77
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Binding energy of quarkonium . . . . . . . . . . . . . . . . . . . . . 79
5.2.1 Analytical exact iteration method (AEIM) . . . . . . . . . . 79
5.2.2 Power Series Method (PSM) . . . . . . . . . . . . . . . . . . 80
5.2.3 Nikiforov-Uvarov (NU) method . . . . . . . . . . . . . . . . 80
5.3 Variation of Binding energy of quarkonium with temperature . . . . 81
5.4 Mass spectra of quarkonia . . . . . . . . . . . . . . . . . . . . . . . 91
5.4.1 Analytical exact iteration method (AEIM) . . . . . . . . . . 91
5.4.2 Power Series Method (PSM) . . . . . . . . . . . . . . . . . . 91
5.4.3 Nikiforov-Uvarov (NU) method . . . . . . . . . . . . . . . . 92
5.5 Dissociation temperature of heavy
quarkonium in N-dimensional space . . . . . . . . . . . . . . . . . . 102
5.6 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 103
iii
6 Properties of bc and cs mesons at Finite Temperature in Strongly
Coupled Quark Gluon Plasma 110
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2 Variation of binding energy of bc and cs mesons with temperature . 111
6.3 Mass spectra of quarkonia . . . . . . . . . . . . . . . . . . . . . . . 112
6.4 Dissociation temperature of bc and cs
mesons in N-dimensional space . . . . . . . . . . . . . . . . . . . . 114
6.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 114
7 Conclusions and Future plans 120
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2 Future plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
iv
List of Figures
1.1 The QCD Phase Diagram [12] . . . . . . . . . . . . . . . . . . . . . 7
1.2 Evolution of the universe from Big-Bang up to today [13] . . . . . 10
1.3 Schematic representation of Relativistic Nuclear Collisions (Cour-
tesy to Prof. Steffan Bass) . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Feynman diagrams for direct photon production at leading order via
the QCD Compton scattering process (left) and annihilation (right)
[21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Diagrams contributing to the dilepton production from a QGP: (a)
Born mechanism, (b) gluon-Compton scattering (GCS), (c) vertex
correction, (d) gluon Bremsstrahlung (NLODY), where virtual pho-
tons (wavy lines) split into lepton pairs [23] . . . . . . . . . . . . . 14
1.6 Strange hadron production from QGP [29] . . . . . . . . . . . . . . 16
1.7 Jet quenching in a head-on nucleus-nucleus collision [33] . . . . . . 17
1.8 Elliptic flow in heavy ion collision [38] . . . . . . . . . . . . . . . . . 18
2.1 Variation of electric potential with r for different values of α . . . . 40
2.2 Variation of screening potential φ(r, T ) with temperature T . . . . . 41
v
2.3 The potential φ(r) between the quark and antiquark as a function
of r for different temperature . . . . . . . . . . . . . . . . . . . . . 42
4.1 Dependence of diquark binding energy in units of the constituent
quark mass on temperature . . . . . . . . . . . . . . . . . . . . . . 72
5.1 (a) Variation of J/ψ binding energy on temperature T and (b) Vari-
ation of χc binding energy on temperature T (AEIM) . . . . . . . . 82
5.2 (a) Variation of J/ψ binding energy on temperature T and (b) Vari-
ation of χc binding energy on temperature T (PSM) . . . . . . . . 83
5.3 (a) Variation of J/ψ binding energy on temperature T and (b) Vari-
ation of χc binding energy on temperature T (NU Method) . . . . 84
5.4 (a) Variation of Υ binding energy on temperature T and (b) Vari-
ation of χb binding energy on temperature T (AEIM) . . . . . . . . 85
5.5 (a) Variation of Υ binding energy on temperature T and (b) Vari-
ation of χb binding energy on temperature T (PSM) . . . . . . . . 86
5.6 (a) Variation of Υ binding energy on temperature T and (b) Vari-
ation of χb binding energy on temperature T (NU method) . . . . 87
5.7 Variation of quarkonium binding energy on temperature T for dif-
ferent values of N (AEIM) . . . . . . . . . . . . . . . . . . . . . . 88
5.8 Variation of quarkonium binding energy on temperature T for dif-
ferent values of N (PSM) . . . . . . . . . . . . . . . . . . . . . . . 89
5.9 Variation of quarkonium binding energy on temperature T for dif-
ferent values of N (NU method) . . . . . . . . . . . . . . . . . . . . 90
5.10 The mass spectra of charmonium and bottomonium as a function
of temperature T for 1S and 1P states (AEIM) . . . . . . . . . . . 93
vi
5.11 The mass spectra of charmonium and bottomonium as a function
of temperature T for 1S and 1P states (PSM) . . . . . . . . . . . . 94
5.12 The mass spectra of charmonium and bottomonium as a function
of temperature T for 1S and 1P states (NU method) . . . . . . . . 95
5.13 The mass spectra of charmonium as a function of temperature T for
1P state for different charm quark mass and (b) the mass spectra
of bottomonium as a function of temperature T for 1P state for
different bottom quark mass (AEIM) . . . . . . . . . . . . . . . . . 96
5.14 The mass spectra of charmonium as a function of temperature T for
1P state for different charm quark mass and (b) the mass spectra
of bottomonium as a function of temperature T for 1P state for
different bottom quark mass (PSM) . . . . . . . . . . . . . . . . . . 97
5.15 The mass spectra of charmonium as a function of temperature T for
1P state for different charm quark mass and (b) the mass spectra
of bottomonium as a function of temperature T for 1P state for
different bottom quark mass (NU method) . . . . . . . . . . . . . . 98
5.16 (a)The mass spectra of charmonium as a function of temperature
T for 1S state for different values of N , (b) the mass spectra of
bottomonium as a function of temperature T for 1S state for differ-
ent values of N ,(c) the mass spectra of charmonium as a function
of temperature T for 1P state for different values of N and (d) the
mass spectra of bottomonium as a function of temperature T for
1P state for different values of N (AEIM) . . . . . . . . . . . . . . 99
vii
5.17 (a)The mass spectra of charmonium as a function of temperature
T for 1S state for different values of N , (b) the mass spectra of
bottomonium as a function of temperature T for 1S state for differ-
ent values of N ,(c) the mass spectra of charmonium as a function
of temperature T for 1P state for different values of N and (d) the
mass spectra of bottomonium as a function of temperature T for
1P state for different values of N (PSM) . . . . . . . . . . . . . . . 100
5.18 (a)The mass spectra of charmonium as a function of temperature
T for 1S state for different values of N , (b) the mass spectra of
bottomonium as a function of temperature T for 1S state for differ-
ent values of N ,(c) the mass spectra of charmonium as a function
of temperature T for 1P state for different values of N and (d) the
mass spectra of bottomonium as a function of temperature T for
1P state for different values of N (NU method) . . . . . . . . . . . 101
6.1 Variation of bc binding energy on temperature T . . . . . . . . . . 112
6.2 Variation of cs binding energy on temperature T . . . . . . . . . . 113
6.3 The mass spectra of different states of bc and csmesons as a function
of temperature T for different values of N . . . . . . . . . . . . . . 115
6.4 The mass spectra of bc 1S state as a function of temperature T for
different quark mass at N=3 . . . . . . . . . . . . . . . . . . . . . 116
6.5 The mass spectra of cs 1S state as a function of temperature T at
N=3 for different component quark mass . . . . . . . . . . . . . . . 117
viii
List of Tables
1.1 Properties of quarks according to the standard model [6] . . . . . . 6
4.1 Diquark Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1 The dissociation temperature (TD) with Tc = 175MeV for the
quarkonium states using Mc = 1710MeV and Mb = 5050MeV at
N=3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 The dissociation temperature (TD) with Tc = 175MeV for the
quarkonium states using Mc = 1710MeV and Mb = 5050MeV at
N=4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3 The dissociation temperature (TD) with Tc = 175MeV for the
quarkonium states using Mc = 1710MeV and Mb = 5050MeV at
N=5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.1 The dissociation temperature (TD) with Tc = 175MeV for the bc
meson states using Mc = 1710MeV and Mb = 5050MeV . . . . . . 114
6.2 The dissociation temperature (TD) with Tc = 175MeV for the cs
meson states using Mc = 1710MeV and Ms = 540MeV . . . . . . 114
ix
Preface
Both experimental and lattice results show that the matter formed near and just
above the transition temperature is non-ideal. There are lots of attempts to explain
such a matter using various phenomenological models. The quark-gluon plasma
near Tc is called strongly coupled quark-gluon plasma (SCQGP) and can be treated
as the strongly coupled QED plasma with a few modifications due to color degrees
of freedom. The aim of this work is to study the screening potential and the
properties of bound states in SCQGP.
The thesis is structured as follows
The purpose of chapter 1 is to give a small introduction to QGP. We start with
the theory of quarks and gluons and a brief description of quantum chromody-
namics. We then give an introduction to the different phenomenological models of
QGP.
In chapter 2, we calculate the screening potential of a test quark entering
into the strongly coupled QGP using generalized hydrodynamic equations. Also,
we investigate the temperature dependence of the screening potential. We try
to explain the existence of bound states and the expected J/ψ suppression in
connection with the potential.
In chapter 3, we calculate the binding energy of the bound states of quarks by
solving the N-dimensional radial Schrödinger equation using this potential in three
different methods. i.e., analytical exact iteration method (AEIM), power series
method (PSM), and Nikiforov-Uvarov (NU) method. The energy eigenvalues have
1
been calculated in the N-dimensional space for any state (n, l).
In chapter 4, we study the binding energies of scalar diquarks using AEIM and
PSM and plotted as functions of temperature. Effective masses of diquarks are
calculated using the binding energies at finite temperature in the strongly coupled
quark-gluon plasma.
In chapter 5, the quarkonium properties are studied in two methods. i.e. AEIM
and NU Method. The variations in binding energy and mass spectra with tempera-
ture have been studied in the N-dimensional space. The dissociation temperatures
for different states of quarkonia are calculated in the N-dimensional space. Also, we
discuss the influence of dimensionality number on the dissociation temperatures.
In chapter 6, we study the properties of bc and cs mesons. Variations of
binding energy and mass spectra with temperature have been studied for different
dimensions. The dissociation temperatures for different states of bc and cs mesons
are also calculated.
Chapter 7 summarizes our results and provides conclusion and an outlook re-
garding our future work.
2
Chapter 1
The Quark-Gluon Plasma
1.1 Introduction
In 1897, the electron was discovered by J.J. Thompson using a cathode-ray tube.
This event set the stage for modern particle physics. The proton and neutron
were discovered subsequently. These three particles were initially regarded as el-
ementary particles. But soon there occurred propagation in the inventory of the
so-called elementary particles. In 1937, the study of cosmic rays led to the discov-
ery of muons. A decade later, pions and strange particles were discovered. In the
1950s, the study was shifted from cosmic rays to man-made particle accelerators.
The use of particle accelerators helped to discover a large number of new particles
including mesons of spin higher than zero and baryons of spin higher than half
with various values of charge and strangeness. These new particles are generally
called hadrons. Though unstable, they exhibit similar behavior to protons and
neutrons. These hadrons including the nucleons are transformed into each other
in many complex ways. Now the physicists had to deal with these hadrons and
3
their interactions. The interaction between hadrons came to be known by a new
name, the strong interaction.
The rapidly increasing number of hadrons showed that they could not be ele-
mentary. Studies on the classification mechanism, mass spectra and interactions of
hadrons revealed that the hadrons were made up of more fundamental elementary
particles -the quarks. In the 1960s, M. Gell-Mann and G. Zweig independently
proposed the quark model [1,2]. It was then necessary to understand the quark
dynamics which is responsible for the composition of hadrons and hadronic in-
teractions. The non-abelian gauge theory, first proposed by Yang and Mills [3]
was later found applicable to strong interactions also [4,5]. Gell-Mann named it
as Quantum chromodynamics (QCD). QCD is a gauge theory of strong interac-
tions based on color symmetry group SU(3) in which the non-abelian gauge field
called the gluon (which is a boson), belongs to the octet representation of color
SU(3). The color interactions between quarks are being mediated by the gluons
and hence the binding of quarks occurs. Gluons carry color charges and hence
they can interact with each other even in the absence of quarks.
Quarks fall into the fundamental triplet representation of the color SU(3) group.
Each quark carries a color quantum number which can have three values (red, blue,
green). Properties of quarks according to the standard model are given in Table
1.1 [6]. Six types of quarks have been discovered till now. (up (u), down (d),
strange (s), charm (c), bottom (b), and top (t) quarks). The electric charges for
the up, charm, and top quarks are +2e and for down, strange, and bottom quarks,
3
−1e. All the six quarks have spin 1 and baryon number, B=1 . Experimentally no
3 2 3
colored particles have been detected. So it is proposed that only uncolored particles
(color singlet) are observable. This is the hypothesis of color confinement. Thus
4
three quarks combine to form a colorless baryon and a quark and antiquark are
combined to form a meson.
In 1968, the first experimental evidence of quarks as constituents of hadrons
came from the inelastic electron-proton scattering experiments performed at the
Stanford Linear Accelerator Centre (SLAC). The scattering cross sections provided
evidence of elastic scattering from the point-like objects inside the proton [7]. The
experimental results at SLAC could be explained only if quarks inside a proton
are nearly free as was proposed by Feynman. Now there emerged a paradox.
The strong interaction is powerful to permanently confine the particles within the
hadron. But Feynman suggested that the interaction between the quarks should
be weak enough at short distances so that the quarks can behave as free particles
[8]. This paradox was solved by David Gross, Wilczek, and Politzer.
The theory of strong interactions which came to be called quantum chromo-
dynamics (QCD) was developed in the 1970’s by generalizing the existing gauge
theory of electromagnetic interactions - quantum electrodynamics (QED). The
fundamental difference between QCD and QED is that in QCD, the generators of
the symmetry do not commute with each other. Thus the theory is a non-abelian
gauge field theory, which have some paculiar properties, one of which provided the
solution to the paradox created by the SLAC experiment.
1.2 Asymptotic freedom and quark confinement
For QCD, when the energy-momentum transfer increases, the effective interaction
between quarks decreases, and when these variables tend to infinity, the theory
becomes a free field theory. This property was discovered by David Gross, Wilczek,
5
Table 1.1: Properties of quarks according to the standard model [6]
Quark Charge Mass Baryon number Isospin
up(u) +2/3 ≈ 4MeV 1/3 +1/2
down(d) −1/3 ≈ 7MeV 1/3 −1/2
strange(s) −1/3 ≈ 135MeV 1/3 0
charm(c) +2/3 ≈ 1.5GeV 1/3 0
bottom(b) −1/3 ≈ 5GeV 1/3 0
top(t) +2/3 ≈ 175GeV 1/3 0
and Politzer and it is called the asymptotic freedom [4, 5, 9, 10]. It results from
the anti-screening of the color charges. This property explains the SLAC results
where quarks behave as free particles, though they are confined.
The attractive force between quarks increases with separation and an infinitely
large amount of energy is required to separate them fully. Thus they are per-
manently bound to hadrons. This is called quark confinement, which is another
important property of QCD.
Thus at low energy, there is confinement and at high energy, there is asymp-
totic freedom in QCD. These two properties can be explained using the coupling
constant. The effective running coupling constant in QCD can be written as [11],
12π
αs(µ) = (1.1)
(33− 2nf ) ln (µ2/Λ2QCD)
where nf is the number of active quark flavor and ΛQCD is the QCD scaling
parameter
When the momentum scale µ approaches infinity, αs will go to zero. This
phenomenon is called asymptotic freedom.
6
1.3 The QCD phase diagram
Figure 1.1: The QCD Phase Diagram [12]
The QCD phase diagram gives an idea about the different phases of QCD and
the corresponding phase transitions. Fig. 1.1 [12] shows the QCD phase diagram
as a function of temperature T and baryon chemical potential µB.
The available lattice simulations predict that, there are no phase transitions
and hence no phase boundaries at zero chemical potential and finite temperature.
Calculations also show a phase transition from hadronic phase to QGP for realistic
values of light quark masses (u, d, and s) at a temperature in the range 150-180
7
MeV. The early universe was in this region of small µB. RHIC and LHC also
explore this region.
As µB is increased at zero temperature, there occurs a first-order phase tran-
sition from the hadronic phase to the nuclear matter at about µB ≈ 940MeV .
As µB is increased further, neutron fluidity is occurred as in neutron stars. At
higher µB, high-density QGP is formed. Here there are different phases such as
color superconductor, which is formed due to the condensation of quark cooper
pairs. Many other phases are proposed in the high µB region such as Color Flavor
Locked (CFL) phase and crystalline color superconductor.
In the region of small T and finite µB, the first-order transition line ends at
the point with T = Tc and µB = µc and at which the phase transition is second
order. This point is called the critical point.
At high temperature and/or high baryon density, because of asymptotic free-
dom, the phase of QCD is described in terms of quarks and gluons degrees of
freedom. At sufficiently high temperature (T > ΛQCD, where ΛQCD is the QCD
scaling parameter), the quarks and gluons are weakly interacting. Such a system
can be described using perturbation theory. In the transition region, where the
hadronic description is not valid and perturbation theory is not feasible, then the
non-perturbative nature of QCD should be taken into account. The term strongly
coupled quark-gluon plasma is used to represent the QCD state near the transition
temperature.
At extremely high temperatures, QGP can be considered as an ideal gas of
quarks and gluons. Around the phase transition temperature, the coupling con-
stant is not weak and QGP can be considered as a non-ideal gas of quarks and
gluons.
8
1.4 The quark-gluon plasma (QGP)
With the point that the quarks and gluons are confined within the hadrons, the
important prediction of QCD is that at very high temperature and/or density,
the hadrons dissolve into quarks and gluons. Such a state of deconfined quarks
and gluons is called the quark-gluon plasma (QGP). The properties of QGP are
expected to be extremely different from the hadronic matter at corresponding
energy densities.
The hadronic matter is strongly interacting but according to the asymptotic
freedom of non-abelian QCD, at very short distances and/or high momentum
transfer the constituent quarks and gluons are weakly interacting. Thus the equa-
tion of state should be the same as that of a free system. The number of degrees
of freedom in the hadronic phase consisting of pions is three, but the number of
degrees of freedom in QGP is (16 + 21nf ), with nf the number of flavors.2
The characteristic values of temperature and density at which the transition
from hadronic phase to QGP occurs correspond to Tc = 10
12K and ρc = 10
15g/cm3.
Studies on QGP in laboratory give a clear picture of the early universe. During the
first 10 µs after the big bang, the universe was filled with the QGP phase which
was in the form of a dilute gas of weakly coupled quarks and gluons. Because of
the expansion of the universe, the temperature decreased, neutrons and protons
were formed from QGP, and atomic nuclei were formed further. Fig. 1.2 shows
the evolution of the universe from Big-Bang up to toay [13]. QGP can occur natu-
rally at very high baryon number densities and low temperatures in astrophysical
compact objects (neutron stars). Many research projects are going on to produce
QGP in laboratory conditions (RHIC, LHC).
9
Figure 1.2: Evolution of the universe from Big-Bang up to today [13]
1.5 The search for QGP
QGP is studied experimentally in heavy-ion collisions at Relativistic Heavy Ion
Collider (RHIC) and Large Hadron Collider (LHC) to provide critical data to the
quest of QGP. Powerful accelerators in the heavy-ion colliders make head-on col-
lisions between massive ions (Little bangs) to recreate the situation similar to the
early universe just after the Big Bang. Such heavy-ion collisions were performed for
more than 20 years at sufficiently high energies to produce the deconfined phase.
In 1986, the first attempt to create QGP was done using light nuclei at Alter-
nating Gradient Synchrotron (AGS) at Brookhaven National Laboratory (BNL),
USA with Silicon (Si) and Super Proton Synchrotron (SPS) at CERN, the Euro-
pean Organization for Nuclear Research, Switzerland with Sulfur (S). Later in the
early 1990s, collision with Gold ions (Au) was done at AGS and with Lead ions
(Pb) at SPS. The centre of mass energy per colliding nucleon-nucleon pair at AGS
and at SPS was 4.6 GeV and 17.2 GeV , respectively. In 2000, the evidence of a
new state was announced by CERN SPS. In the same year, RHIC at BNL started
10
operation with four detectors PHENIX, STAR, PHOBOS, and BRAHMS with ten
times larger the centre of mass energy than previously available. In 2010, LHC
at CERN started Pb-Pb collisions at 14 times larger centre of mass energies than
√
RHIC. The analysis of Au-Au collision at SNN = 200GeV at RHIC and Pb-Pb
√
collision at SNN = 2.76TeV at LHC provided evidence for QGP [14]. The first
results from RHIC showed that the QGP is a strongly coupled system which is a
nearly perfect liquid [15]. The LHC results also gave clear evidence for the ideal
liquid nature of QGP seen at RHIC [16, 17]. Fig. 1.3 schematically illustrates the
nuclear collision time evolution.
Figure 1.3: Schematic representation of Relativistic Nuclear Collisions (Courtesy
to Prof. Steffan Bass)
1.6 The QGP signatures
The identification of QGP generation in relativistic heavy-ion collisions has ap-
peared to be difficult [18]. The quarks and gluons in the deconfined plasma could
not be measured directly. The signals of QGP can be extracted from the analysis
of the particles produced at the final state. Some of the techniques for signaling
QGP are discussed next.
11
1.6.1 Direct photon production
QGP can be observed directly through the electromagnetic radiation emitted by
the quarks in the early stage. Since photons can escape the medium without
much interaction, they seem to be a direct signal of QGP and they can provide
information on the properties of QGP. In [19, 20], it is explained how photons
become evidence for the production of QGP in high-energy heavy-ion collisions.
Photons are produced in the QGP medium by processes like annihilation (q+ q →
γ+g ) and Compton scattering ( q+g → γ+q , q+g → γ+q ). Fig. 1.4 shows the
diagrams for the production of direct photons via the QCD Compton scattering
process and annihilation process [21]. Since the photons are produced during the
decay of pions and muons in heavy-ion collisions, it is not easy to measure the direct
photons from the electromagnetic interaction of the constituents of the QGP.
There are many meaningful works for the subtraction of the photons from
the π0 and µ0 background to obtain direct photons. In [22], a measurement of
direct photon production in Pb+Pb collisions at 158 AGeV has been carried out
in the CERN WA98 experiment. A significant direct photon excess is observed at
pT > 1.5GeV/c. To minimize the contribution of photon from the hadron phase,
photon transverse momentum pT must be greater than 2 GeV/c. In the hot QGP,
photons have transverse momentum in the range 2 - 3 GeV/c [23]. Photons at
this pT are also produced by the collision of constituents of the projectile nucleons
with that of target nucleons. In order to get the net photons from the QGP, the
contributions from the other sources must be subtracted.
12
Figure 1.4: Feynman diagrams for direct photon production at leading order via
the QCD Compton scattering process (left) and annihilation (right) [21]
1.6.2 Dilepton production
In QGP, the virtual photons that are produced by the annihilation of quark and
antiquark decay into lepton pairs L+L− (e+e− and µ+µ− ) which are termed as
dileptons. These dileptons pass through the medium towards particle detectors
without any further collisions. Thus they provide critical information about the
interior of the fireball. Due to the effects of the medium, the quark dispersion
relation in QGP changes in such a way that sharp gaps arise in the production
rate of low mass dileptons. This is an important signature for the presence of
deconfinement. Fig. 1.5 illustrates different elementary processes contributing to
the dilepton production [24].
The dilepton mass can be written as,
2 + − 2M = (PL + PL ) (1.2)
Where P+ and P−L L are the four-momenta of the dilepton. The dilepton’s transverse
13
Figure 1.5: Diagrams contributing to the dilepton production from a QGP: (a)
Born mechanism, (b) gluon-Compton scattering (GCS), (c) vertex correction, (d)
gluon Bremsstrahlung (NLODY), where virtual photons (wavy lines) split into
lepton pairs [23]
momenta can be written as,
PT = (P
+ −
L )T + (PL )T (1.3)
where (P+L )T and (P
−
L )T are the transverse momenta of the dilepton.
1.6.3 Quarkonia suppression
Bound states of heavy quarks and their antiquarks are called quarkonia. The
bound state of a charm quark and anti-charm quark (cc) is called charmonium,
and the corresponding bottom and anti-bottom quark pair (bb) is known as bot-
tomonium. They are formed during the initial hard collision and carry information
about the initial stages of the QGP. Their binding energies are large so that they
can survive in the QGP medium. The binding energy of the heavy quarkonium
states is studied in [25,26]. Quarkonium suppression means the decrease in the
production of quarkonia in the region where QGP is formed compared to the case
where no QGP is formed in heavy-ion collision experiments. The main reason for
this is the interaction of quarkonium with the QGP medium.
14
Thus the QGP formation affects the quarkonium suppression. The experimen-
tal evidence of the NA50 collaboration shows the anomalous J/ψ suppression in
Pb+Pb collisions [27]. In [28] the phenomenon of anomalous J/ψ suppression has
been studied in detail. In this thesis, we try to explain the quarkonium suppres-
sion in connection with the screening potential between quarks in strongly coupled
quark-gluon plasma (SCQGP).
1.6.4 Strangeness enhancement
It is one of the most important probes for the detection of QGP. It is the ob-
servation of the abundance of strange particles (hadrons) in the detector. The
strange quarks and antiquarks are easy to be created in a hot and dense QGP
which combine with the other quarks to form a large number of strange hadrons.
Fig. 1.6 shows the strange hadron production from QGP [29]. The strangeness
evolution in QGP [30] can be described by the processes uu → ss , dd → ss and
gg → ss . The gluonic process provides a faster production rate. Thus the yield
of strange and multi strange mesons and baryons has been strongly enhanced in
the presence of QGP [31]. The WA97 collaboration [32] experimentally proved the
+ −
enhancement in the production of multistrange hyperons in particular Ω and Ω
in Pb+Pb collisions at 158AGeV .
1.6.5 Jet quenching
Jet quenching means the modification of the energy of the hadron jet by the
QGP. During nuclei collision, the partons produced with very high transverse
momenta fly off to all possible directions and finally form narrow cones of hadrons
15
Figure 1.6: Strange hadron production from QGP [29]
called jets. The energies of these jets are modified by the QGP medium. The
comparative study of jet energies with the same in the proton-proton collision
may lead to the identification of QGP. Thus the study of jet quenching is an
important tool for understanding the QGP. Fig. 1.7 explains the jet quenching in
a head-on nucleus-nucleus collision [33]. The energy loss can be observed through
the nuclear modification factor RAA. [ ]
dN
1 AAd2pT dyRAA(pT , b) = (1.4)
T (b) dσppAA
d2pT dy
Here the numerator is a single particle momentum distribution of a jet in
the nucleus-nucleus collision and traveling through the thermal medium and the
denominator is the same in the proton-proton collision. TAA(b) is the nuclear
thickness function and b is the impact parameter. If this ratio is less than unity,
it is a measure of jet quenching in the medium [34, 35, 36, 37].
16
Figure 1.7: Jet quenching in a head-on nucleus-nucleus collision [33]
1.6.6 Anisotropic flow
Anisotropic flow provides information about the thermodynamics and collective
flow of the hot and dense medium. In non-central collisions, the pressure and
energy profile are not symmetric in the plane perpendicular to the beam axis. The
large pressure gradient will lead to a larger expansion velocity and asymmetrical
particle emission. This distribution can be explained by Fourier expansion. The
first Fourier coefficient gives the strength of the direct flow and the second Fourier
coefficient is called the elliptic flow. The higher coefficients are for studying the
initial state fluctuations and to obtain the ratio of shear viscosity to energy density
(η/s ) of the hot medium. The azimu[thal momentum[an(isotrop)y]i]s represented as,d3N 1 d2 ∑∞N
E = 1 + 2νn cos n φ− ψ (1.5)
d3p 2πpT dpTdy n=1
where φ is the azimuthal angle of the particle and ψ is that of the reaction
plane in the laboratory frame. 1.8 shows the magnitude of ellipti flow for different
17
particle species [38].
Figure 1.8: Elliptic flow in heavy ion collision [38]
1.7 Phenomenological models
A phenomenological model means a group of predictions about a system according
to a theory. Using a phenomenological model, the properties of a system can
be predicted. These predictions are experimentally verified. There are many
phenomenological models about QGP. Some of them are explained below.
1.7.1 MIT Bag Model
It is a very simple phenomenological model. It was developed by Chodos et al.
[39] in 1974 at the Masachusetts Institute of Technology in Cambridge (USA).
According to this model, quarks are confined to move in a given spatial region
forced by external pressure. The quark confinement can be visualized as an elastic
bag (usually spherical in shape but can be deformable) in which the quarks move
freely around as long as we don’t try to pull them apart. If we try to pull a
quark, then the bag stretches and resists the pulling, B is the pressure exerted by
the vacuum on quarks and gluons. In this model, quarks are considered massless.
Here quark confinement occurs due to the balance of the pressure B exerted on
the bag from outside and the kinetic energy of quarks. B is called the bag pressure
18
which incorporates the non-perturbative effects of QCD (B ≈ 200MeV ). By
compressing the bags against each other phase transition can be occurred. The
pressure and energy in terms of Bag constant are,
37π2
P = T 4 −B(T ) (1.6)
90
37π2
 = T 4 +B(T ) (1.7)
90
The bag model gives an estimation of the critical temperature of the phase
transition. It is very simple and easy to apply. This model could easily explain
the properties of low-lying hadrons. It has some drawbacks too. The physics at the
boundary of the bag is too complicated because of the sharpness of the boundary
of the bag. The size of the bag is too large so that the bags would touch each other
at normal nuclear density. Also, the chiral symmetry breaks at the boundary of
the bag (bag surface).
1.7.2 The quasi particle QGP model (qQGP)
Within the approach of this model, the interacting plasma is considered as a system
of massive quasi-particles. This model was independently proposed by Satz et al.
and Peshier et al. [40, 41]. According to Peshier et al. in gluon plasma, all
thermodynamic quantities can be derived from the partition function.
∞
PV ∑
= − ln(1− e−βk) (1.8)
T
k=0
Where β = 1 , k is the single particle energy of the quasi-gluon,T √
 = k2k +m2(T ) (1.9)
19
Where k is the momentum and m(T ) is the temperature-dependent mass. The
pressure is similar to ideal gas with temperature dependent mass and can be
denoted as Pi.
Later the thermodynamic inconsistency of the quasi-particle model was reme-
died by introducing the temperature-dependent bag pressure term in the expression
of pressure and energy density [42, 43].
P = Pi −B(T ) (1.10)
 = i +B(T ) (1.11)
In [44] the authors proposed the self-consistent quasi particle model without
the external terms like B(T). In this model, all the thermodynamic quantities
can be derived in a self-consistent manner. Such models obtained good results in
agreement with the lattice data.
1.7.3 Cornell Potential Model
The effective potential between quark and antiquark which has the form Coulomb
+ linear potential is termed as Cornell potential. Cornell potential was successful
in studying non-relativistic systems such as mesons. The Cornell potential can be
read as,
a
V (r) = − + br (1.12)
r
where a and b are constants.
At short distances, the Cornell potential becomes V (r) ≈ 1 , which is known
r
as the Coulomb like term and arises from the gluon exchange between quark and
20
antiquark and at large distances V (r) ≈ r, which is the linear term which arises
from the higher-order effects.
In [45], EOS of QGP was derived using the Cornell potential model based on
Mayer’s theory of plasma and the results were compared with the lattice QCD
results of the gluon plasma. EOS for quark-antiquark plasma was derived in
[46] and the results were in agreement with the 2-flavor and 3-flavor lattice QCD
results. Also, the Cornell potential succeeded in explaining the spectra of both
charmonium and bottomonium [47]. In [48], using Cornell potential EOS was
obtained and the dissociation temperatures of quarkonia were found out.
1.7.4 Relativistic Harmonic Oscillator (RHO) Potential
Model
RHO potential model was initially proposed by S. B. Khadkikar and S. K. Gupta
[49]. It has been proved successful in explaining many properties of hadrons. Later
a confinement model was formulated by S. B. Khadkikar and V. Kumar [50]. These
two models are collectively called the RHO model. Energy eigenvalues for quarks
and gluons are given as, √
En = M2q + (2n+ 1)Ωq (1.13)
√
En = Cg(2n+ 1) (1.14)
with n=1,2,3,...
where Mq is the mass of a quark, Ωq is the size parameter which is energy
dependent, and Cg is ’frequency’ of gluon fields.
21
This model was used to find out the hadronic and mesonic masses and the
results were found in agreement with the experimental results. The leptonic width
also agrees with the experimental result. The confinement behavior of this model
has been found smooth compared to the bag model. This model was helpful in
describing many features of hadron spectroscopy, baryon magnetic moments and
properties of glueballs, nucleon antinucleon annihilation, etc.
In [51], the authors found out the EOS of infinite gluon system with classical
and quantum energy eigenvalues and both EOS show T 4 dependence of pressure
and energy density at high pressure.
1.7.5 The Strongly Interacting Quark-Gluon Plasma
Model (sQGP)
Heavy ion experiments and lattice calculations show that the QGP displays strong
interaction between the constituents in the regime T = Tc to T = 3Tc. In [52],
the properties of bound states like colorless and colored pairs of gq, qq, and gg at
T > Tc were studied using the sQGP model. Also, their binding energies were
calculated and zero binding endpoints were located. In the limit of the high cou-
pling constant ( Γ >> 1), the one component plasma shows liquid state properties
[53]. In [53], the transport coefficients – self-diffusion, shear viscosity, and thermal
conductivity were studied. Two types of collective modes (longitudinal and trans-
verse modes) have been understood in OCP and transverse mode arises because
of strong correlations.
22
1.7.6 Strongly Coupled Quark-Gluon Plasma Model
(SCQGP)
In this model, QGP near Tc is proposed as a strongly coupled plasma called the
strongly coupled quark-gluon plasma (SCQGP). At sufficiently high temperatures,
QGP is a quasi-color-neutral gas of colored particles (like quarks and gluons) which
exhibit collective behavior. In this model QGP near Tc can be treated as strongly
coupled QED plasma with appropriate modifications due to the color and flavor
degrees of freedom and quantum effects. In [54], the author derived a modified
equation for energy density as, ( )
 = 2.7 + Uex(Γ) nT (1.15)
with n = 1.1a 3fT .
The coupling constant is defined as the ratio of the average potential energy
to the average kinetic energy of the particles of SCQGP.
〈P.E.〉
Γ = (1.16)
〈K.E.〉
( ) 1
4.4πa 3f
Γ = gcαs(T ) (1.17)
3
2
With af = (16 +
21n )πf .2 90
In SCP, gc = 1 but in SCQGP, it is different for gluon and flavored plasma. in
SCP α 1s = and in SCQGP,137
( ( )
− ln 2 ln (T/Λ )
)
T
6π 3(153 19nf )
αs(T ) = 1−
(33− 2nf ) ln (T/ΛT ) (33− 2nf )2 ln (T/ΛT )
23
After this introduction, in the next chapter (chapter 2), a new screening po-
tential between quarks in SCQGP has been proposed. Also, the temperature
dependence of this potential has been explained. The binding energy of the bound
states of quarks is calculated by solving the N-dimensional radial Schrödinger
equation using this potential in three different methods. i.e., analytical exact it-
eration method (AEIM), power series method (PSM), and Nikiforov-Uvarov (NU)
method in chapter 3. Using the results, effective masses of diquarks and the heavy
quarkonium properties such as binding energies, mass spectra, and dissociation
temperatures are discussed in chapter 4 and chapter 5. Finally, we study the
properties of bc and cs mesons in chapter 6. Chapter 7 summarizes the work
presented in this thesis.
24
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30
Chapter 2
Screening of a Test Quark in the
Strongly Coupled Quark Gluon
Plasma
2.1 Introduction
As discussed in chapter 1, quark-gluon plasma (QGP) is a special kind of plasma
where electric charges in plasma are replaced by the color charges of quarks and
gluons mediating the strong interaction between them. Such a state of matter
can be expected to occur at sufficiently high temperature (≥ 150 MeV) and/or
densities (>10 times nuclear density) [1]. These conditions could be fulfilled in
the early universe after a few microseconds or in the interior of neutron stars
and these conditions are recreated in the laboratory by heating nuclear matter to
temperatures above the quark-gluon confinement temperature with high energy
nuclear collisions at the RHIC at Brookhaven National Laboratory or the LHC at
31
CERN [2]. During these collisions, the hot and dense fireballs are created and they
undergo thermalization, cooling, hadronization, chemical, and thermal freeze-out
into hadrons. The search for QGP can be done by the detection and analysis of
these hadrons, photons, etc. It has been proposed that the QGP near and above
Tc is a strongly coupled coulombic plasma of color charges (SCQGP) [3]. The
strongly coupled quark-gluon plasma is in several ways similar to some kinds of
electromagnetic plasmas containing electrically charged particles that show some
liquid-like or even solid-like behavior [4].
In plasma physics, the screening of charges is considered as one of the im-
portant collective effects. The Debye screening potential can be derived from
the Linearized Poisson’s equation within a homogeneous, isotropic plasma. The
Coulomb potential of a charge Q at rest, in the plasma, is modified into Debye
Huckel or Yukawa potential [5], ( )
Q r
φ(r) = exp − (2.1)
r λD
where λD is the shielding radius of the sphere.
Bound states of heavy quarks, in particular J/ψ mesons, are produced in the
initial hard scattering processes of the collision and will be dissociated in the QGP
due to the screening of the quark potential and break up by the energetic gluons [6].
Hence the J/ψ suppression has been considered as one of the important signatures
for the QGP formation [7]. A J/ψ meson suppression was observed experimentally
[8] and it was interpreted as a major indication of the QGP formation in RHIC
[9].
Experimental data suggest that QGP near to the confinement phase could be
in liquid phase and for a transition from the liquid phase to the solid phase, a
32
Lennard Jones type interaction may be required [10]. In [11] a new attractive
potential between the ions that are shielded by degenerate electrons in quantum
plasmas was reported. A similar screening potential was found for a color charge
at rest by using a polarization tensor beyond the high-temperature limit in [12].
The screening potential of a parton moving through QGP was calculated in [1]
using semi-classical transport theory.
In this chapter, we try to find out the screening potential between quarks in
SCQGP, through generalized quantum hydrodynamic (G-QHD) approach. We
shall use the generalized QHD equations [13] for the quarks in SCQGP, supple-
mented by Poisson’s equation. In G-QHD model, pressure and Bohm’s potential
are included [14]. Here the plasma is of high density, so exchange and correlation
functions are neglected. Pressure is derived from the ground state energy equa-
tion of quarks in SCQGP. Here we demonstrate that the potential around a quark
in strongly coupled quark-gluon plasma resembles the Lennard Jones type poten-
tial which may be responsible for the phase transition of SCQGP. We also try to
explain the expected J/ψ suppression in connection with the screening potential.
2.2 Strongly coupled quark-gluon plasma
Strongly coupled plasma means the plasma where the plasma parameter, which
may be defined as the ratio of average potential energy to average kinetic energy of
the particles is equal to or greater than 1. In strongly coupled quark-gluon plasma,
the interparticle potential energy dominates over the thermal energy of particles.
Thus the coupling parameter for strongly coupled quark-gluon plasma is expected
to be of the order of one [15]. So the theory of SCQGP can be explained using the
33
traditional mathematical descriptions of SCP.
Consider the quark-gluon plasma as a deconfined, quasi-color-neutral system of
quarks, antiquarks, and gluons with color coulombic interactions. It can be treated
in a similar way as the quantum electrodynamic plasma with a few modifications
due to color degrees of freedom. The plasma parameter can be written as,
〈P.E.〉 4 αs
Γ = = 3 a (2.2)
〈K.E.〉 T
where αs is the coupling constant, a is usually referred to as the ion-sphere
1
radius or the Wigner-Seitz radius and can be estimated as a = ( 3gc ) 3 Where
4πnf
gc incorporates the color degrees of freedom, nf stands for the number density
corresponding to the particular flavor and T is the absolute temperature.
Taking the typical value of αs ≈ 0.5 , a ≈ 1fm and near the critical temper-
ature Tc ≈ 200MeV , Γ ≈ 2 which is of the order of 1. Thus QGP near Tc is3
strongly coupled plasma. For a degenerate electron system with number density
2 2
n, we can use the Fermi energy, E = h (3π2F n) 3 instead of T . Since SCQGP is2m
like the strongly coupled quantum electrodynamic plasma, we can write
4 αs
Γ = 3 a = 0.543rs (2.3)
EF
with ( ) 1
3g 3c 4Mqαs
rs =
4πnf 3
where Mq is the quark mass, gc incorporates the color degrees of freedom and nf
stands for the number density corresponding to the particular flavor [16]. For a
quark system consisting of u and d quarks,[the qu]ark mass can be considered as−1
Mq ≈ mp and αs ≈ 0.5. Also we can take 4Mqαs ≈ 1fm, the hadron radius3 3
34
[16]. For SCQGP, ∑2 −nfrsf ∂EG
P = −B (2.4)
3 ∂rs
f=1
Here B is the confinement parameter in the form of MIT Bag constant. The
general formula for the ground state energy per quark is known [16] using the
method of Pade a[pproximants [17]. By integrating this equation we get
dE 8M α2G q
= s − 0.0614r1/2s +(0.0791rs + 0.0)158r
3/2
s − 0.2741(r
2
s + 0.6609r
5/2
dr 9 s )
−4.3951r3s − 44250 log r1/2s − 4.1005 − 294430(0 log r1/2s + 5.019)4]
+505797 log 2(r1/2s + 1.50483)
(2.5)
Eq.(2.5) is obtained from the equation of state of degenerate electron system
with a few modifications. It could be applicable at finite temperature since SCQGP
can be treated in a similar way as the strongly coupled QED plasma. Using
the above-mentioned EOS of QGP, the mass-radius relation of compact stars was
estimated [16], which confirms the experimental result [18].
2.3 Screening potential in SCQGP through G-
QHD approach
For Strongly Coupled QGP, rs ≤ 1 and Γ is of the order of 1 and it can be
considered as collisionless. Thus the interaction energy between the partons will
be greater than the kinetic energy. Here no magnetic effects are considered. Within
the hydrodynamic limit, plasma can be treated as liquid and can be described in
35
terms of macroscopic variables that obey the equations of motion. Various color
components of plasma have the same temperature and hydrodynamic velocities
when local equilibrium is attained.
The G-QHD equations are
∂nf
+∇.(nfu) = 0 (2.6)
∂t
( ) ( ) 1
∂u 4α 2s ∂P
M −1q + u.∇u = ∇φ− gcn
∂t 3 f
∇nf +∇VB (2.7)
∂nf
( ) 1
4α 2∇2 sφ = (nf − nf0)−Qδ(r) (2.8)
3
√
Where VB is the Quantum Bohm potential, V = − 1B √1 ∇2 nf . Let n =2M n fq f
nf0 + nf1, where nf1 << nf0.
Linearize the resultant equations to obtain perturbation density, nf1 and insert
in to Eq.(2.8). The Fourier transformation in space leads to the electric potential
around an isolated quark. Thus we have ∫
Q eik.r
φ(r) = d3k (2.9)
8π3 k2D
Where D is the dielectric constant, r denotes the position relative to the in-
stantaneous position of the test quark and Q is the charge of the test quark. The
inverse dielectric constant can be written as
k2 k4
1 + αk2 k4
= s s2 4 (2.10)D 1 + k2 + α
k
k k4s s
M
Where k2 = ( qω2p)s is the inverse Thomas Fermi screening length and α =
g ∂Pc ∂nf
36
(ω2p )2 measures the importance of quantum recoil effect.
2g ∂Pc ∂nf
∂P can (be calculated by using Eq.(2.4) and (2.5).∂nf
∂P
= 2 − −1/3 −2/3 −5/6 −4/370315.3nf + 0.0023nf − 0.0002nf − 0.1044n∂n ff ) (2.11)
− −7/6 −1/20.0083nf + 17240.6nf + 365329n
−1
f
Inserting Eq.(2.10) in Eq.(2∫.9)(, )
Q 1 + b 1− b
φ(r) = + eik.rd3k (2.12)
16π3 k2 + k2 2+ k + k
2
−
√
Where b = √ 1− and k
2 = k2 (1∓ 1−4α) .
1 4α ± s 2α
The above integral was evaluated for different values of α. The boundary
condition is φ→ 0 at r →∞ . E[q.(2.12) becomes ]
Q
φ(r) = (1 + b)e−k+r + (1− b)e−k−r (2.13)
8πr
For α → 0, the modified Thomas-Fermi screened Coulomb potential can be
recovered.
Q
φ(r) = e−ksr (2.14)
4πr
√
For α→ 1 , we have k+ = k = 2k4 − s and
( )
Q k √s
φ(r) = 1 + √ r e− 2ksr (2.15)
4πr 2
For α = 1, ( )
Q 1
φ(r) = cos (kir) + √ sin (kir) e−krr (2.16)
4πr 3
37
Where
ks
ki =
2
, √
3
kr = ks
2
With α >> 1 the exponential cosi(ne-screened)Coulomb potential is recovered.
Q
φ(r) = cos (ksr∗) e
−ksr∗ (2.17)
4πr
Where r = r∗ 1
(4α) 4
2.4 Suppression of J/ψ meson
The two-body potential associated with the dipole created by two test charges q1
and q2 at positions r1 and r2 respe∫cti(vely can be represented as,
q q eik.(r1−r2) e−
)
ik.(r1−r2)
− 1 2φ(r1 r2) = + d3k (2.18)
16π3 k2D k2D
This two-body potential is equal to the one-body potential given by Eq.(2.9).
For two quarks, the potential shows attraction and thus bound state of quarks is
formed. The existence of colored bound states (eg. diquarks) of partons at rest has
been claimed by analyzing lattice data [19]. For a quark-antiquark system, q1q2 < 0
. Thus the two-body potential is inverted, showing a maximum. This may lead
to short living mesonic resonances and an enhancement of the attraction between
quarks and antiquarks of mesonic states moving through SCQGP [1] which results
in the expected suppression.
38
2.5 Temperature dependence of screening poten-
tial
For deriving Eq.(2.16), the value of αs has been fixed (αs ≈ 0.5) but by considering
the temperature dependence of αs from [20] we get,
( ( )ln 2 ln (T/Λ )T )
6π − 3(153− 19nf )αs(T ) = 1 (2.19)
(33− 2nf ) ln (T/ΛT ) (33− 2n 2f ) ln (T/ΛT )
The screening potential between quarks at finite temperature plays an impor-
tant role in the study of various bound states in Quark Gluon Plasma. Theoret-
ically many forms of screening potential have been proposed [21]. The study of
static quark potential is significant for several reasons. Phenomenologically the
properties of bound states of quarks can be derived from potential models [22]. So
it is necessary to find out the temperature dependence of the potential [23]. Substi-
tuting Eq.(2.19) in Eq.(2.13), we get the temperature-dependent quark potential
as φ(r, T ) .
2.6 Results
In Fig. 2.1, the profiles of the potential given by the Eq.(2.14), (2.16), and (2.17)
for different values of α are displayed. It can be seen that the new short-range
attractive electric potential that resembles the LJ type potential for α = 1 while for
the smaller values of α the attractive potential vanishes. In high density strongly
coupled plasma, with Bohm’s potential, the attractive potential exists only for
α ≥ 1. We have investigated the temperature dependence of screening potential
39
0.004
0.002
Α=0
0.000
Α =1
-0.002
Α=10
-0.004
0 5 10 15 20
ksr
Figure 2.1: Variation of electric potential with r for different values of α
in strongly coupled quark-gluon plasma. The result is displayed in Figures 2.2
and 2.3. The attractive potential for α ≥ 1 is plotted against temperature. For
plotting the graphs, we have taken the density n=0.589, Mq = 1, Q=1 and h̄ = 1
(in the case of QGP, Lorentz Heaviside units are used). The potential shows a
minimum near to the critical temperature Tc. The minimum potential near Tc will
give rise to bound states (diquarks). This potential satisfies both the screening
and the confinement type forces just like the Cornell potential.
In strongly coupled systems with high density, quantum effects are important
[24]. Here the results have been obtained in the non-relativistic domain, but QGP
composes an ultra-relativistic system. Since Coulomb interaction is common in
both cases, it may be concluded that the screening effect is present in SCQGP. The
minimum potential can give rise to bound states (diquarks) if thermal fluctuations
40
ΦHrLksQ
Figure 2.2: Variation of screening potential φ(r, T ) with temperature T
do not destroy them. Here nonabelian effects are included in the expression for
pressure. They cannot be treated separately in this method. Since there is an
attraction even in complex plasmas and in QGP, close to the critical temperature
[15], this seems to be a general feature of weakly as well as strongly coupled
plasmas. Therefore, there will not be any qualitative change of the screening
potential due to nonperturbative or nonabelian effects [1]. For a J/ψ meson, the
binding may become weaker and of shorter range due to screening in QGP [7].
When the screening radius falls below the binding radius, the quark and antiquark
can no longer be bound and the bound state becomes dissociated. The quarkonium
dissociation points therefore determine the temperature and energy density of the
QGP [22].
From direct finite temperature lattice studies, both in quenched and in full
41
0
T=175 MeV
-1 T=220 MeV
T=300 MeV
-2
-3
0.0 0.2 0.4 0.6 0.8 1.0 1.2
r
Figure 2.3: The potential φ(r) between the quark and antiquark as a function of
r for different temperature
QCD as well as in lattice-based potential work, it can be shown that the J/ψ meson
can survive up to T ≥ 2Tc [25]. At temperature above the critical temperature,
QGP is strongly coupled quark-gluon plasma (SCQGP) and the screening potential
explains the expected J/ψ suppression.
42
ΦHrL
Bibliography
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(2009).
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[6] B. K. Patra and D. K. Srivastava, J/ψ suppression gluonic dissociation vs
colour screening, Phys. Lett. B 505, 113 (2001).
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[8] M. Gonin (NA50 Collaboration), Anomalous J/ψ suppression in Pb+ Pb col-
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[11] P. K. Shukla and B. Eliasson, Novel attractive force between ions in quantum
plasmas, Phys. Rev. Lett. 108, 1650070 (2012) .
[12] C. Gale and J. Kapusta, Modification of Debye screening in gluon plasma,
Phys. Lett. B 198, 89 (1987).
[13] N. Crouseilles, P. A. Hervieux, and G. Manfredi, Quantum hydrodynamic
model for the nonlinear electron dynamics in thin metal films, Phys. Rev. B
78, 155412 (2008).
[14] F. D. (Tony) Smith, Jr., Bohm confirmed by non relativistic quark model,
quant-ph TS-TH-98-1 (1998).
[15] M.H Thoma, The quark gluon plasma liquid, J. Phys. G: Nucl. Part. Phys.
31, 539 (2005).
[16] S. Ramadas and V.M. Bannur, SCQGP in compact stars, Eur. Phys. J. C 72,
1975 (2012).
[17] A. Isihara and E.W. Montroll, A note on the ground state energy of an as-
sembly of interacting electrons, Proc. Natl. Acad. Sci. USA 68, 12, 3111 (1971).
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[18] P.B. Demorest, T. Pennucci, S. M. Ransom et al, A two-solar-mass neutron
star measured using Shapiro delay, Nature 467, 1081 (2010).
[19] E.V. Shuryak and I. Zahed, Towards a theory of binary bound states in the
quark-gluon plasma, Phys. Rev. D 70, 054507 (2004).
[20] V. M. Bannur, Strongly coupled quark gluon plasma (SCQGP), J. Phys. G:
Nucl. Part. Phys. 32 993 (2006).
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QGP, Phys. Rev. D 101, 034507 (2020).
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45
Chapter 3
Calculation of Binding Energy of
Bound States of Quarks in
Strongly Coupled Quark Gluon
Plasma
3.1 Introduction
There occurs a phase transition from a confined hadronic phase to a deconfined
partonic phase at critical temperature (Tc), as discussed in section 1.3. This phase
of deconfined quarks and gluons is called quark-gluon plasma (QGP) [1]. At ex-
tremely high temperatures QGP can be considered as an ideal gas of quarks and
gluons because of the very weak interaction. At temperature near Tc, QGP can be
considered as strongly coupled and termed as strongly coupled quark-gluon plasma
(SCQGP) [2].
46
The existence of bound states above the critical temperature can be studied by
using the effective temperature-dependent potential in the QGP. In fact, no strictly
stationary bound state can exist in QGP as the interaction with the particles in
the medium will lead to a definite lifetime for all bound states [3]. The bound state
properties in quark-gluon plasma can be studied using two different approaches.
The first one is a direct calculation using QCD [4] and the other one is the use
of a potential model [5,6,7]. The nonrelativistic interaction potential model can
reproduce the experimentally observed mass spectra of bound states [8,9]. This
approach uses the potential as a function of the relative separation between the
particles and the binding energy can be calculated mathematically by solving the
Schrödinger equation.
N-dimensional non-relativistic radial Schrödinger equation can be solved for
the interaction potential for different bound states in SCQGP. For solving the SE,
the interaction potential should be known. A confining potential is a mathematical
representation of a force that governs the dynamics of the particles in a particu-
lar system. Depending on the nature of interaction among the particles, a large
number of confining potentials have been used. Some of them are Harmonic [10],
Cornell [11], Coulomb [12], Coulomb perturbed [13], Ring-shaped [14], Double-
ring shaped [15], Gaussian, modified Gaussian [16] Energy-dependent harmonic
[17] etc.
There exist a number of exact, approximate, and numerical methods to solve
the SE. In the last chapter, the existence of bound states has been discussed in the
context of new attractive potential between the quarks in SCQGP. In this chapter
we try to find out the binding energy of bound state of quarks in strongly coupled
quark-gluon plasma by solving the N-dimensional Schrödinger equation for two
47
particles interacting via the two-body potential in three different methods i.e.,
analytical exact iteration method (AEIM) [18], power series method via a suitable
ansatz to the wave function (PSM) [13] and the Nikiforov- Uvarov (NU) method
[19].
3.2 Exact solution of the N-dimensional
Schrödinger equation with the two
body potential in SCQGP at finite
temperature
3.2.1 Analytical exact iteration method (AEIM)
The N-dimensional non-relativistic radial Schrödinger equation for two particles
[18, 20] in[teracting via the two-body potential is given by ]
d2 N − 1 d − l(l +N − 2)
( )
+ + 2µ Enl − Φ(r) Ψ(r) = 0 (3.1)
dr2 r dr r2
where n , l and N are the principal quantum number, angular quantum number
and the dimensional number respectively. µ is the reduced mass of two particles,
Mq
µ = 1
Mq2
Mq1 +Mq2
Substituting Ψ(r) = R(r)[ N−1 in Eq.(3.1)r 2 ]
d2 N − 1 d − l(l +N − 2)
( ) R(r)
+ + 2µ Enl − Φ(r) = 0 (3.2)
dr2 r dr r2 N−1r 2
48
[ ]
2 2d l(l +N − 2) + N −4N+3 ( )− 4 + 2µ E
2 2 nl
− Φ(r) R(r) = 0 (3.3)
dr r
where φ(r) is the two-body p(otential represented by t)he following equation [21]
Q 1
φ(r) = cos (k r) + √ sin (k −krri ir) e (3.4)
4πr 3
where
ks
ki =
2
√
3
kr = ks
2
with Q = q1q2
Substituting Eq.[(3.4) in Eq.(3.3) a[n(d rearranging, ) ]
d2R(r) − 2µQ 1= 2µE −krr
2 nl
+ cos (kir) + √ sin (kir) e
dr 4πr 3 ] (3.5)
2
l(l +N − 2) + N −4N+3
+ 4 R(r)
r2
The analytical exact iteration method (AEIM) assumes the wave function as
[18], ( )
R(r) = fn(r) exp gl(r) (3.6)
with,
f(n(r) = 1), n = 0 (3.7)
(n)
fn(r) = Π r − αi , n = 1, 2, 3, ..... (3.8)
−1gl(r) = αr2 − βr + δ ln r, α > 0, β > 0 (3.9)
2
49
From Eq.(3.6), [ ]
d2Rnl 2 fn”(r) + 2g
′(r)f ′
= g ”(r) + g (r) + l n
(r)
l l Rnl(r) (3.10)
d2r fn(r)
At n=0, substituting Eq.(3.7) and Eq.(3.9) in Eq.(3.10) , equating with Eq.(3.5)
and comparing the pow√√er[s of r o[n both sides, we get√√ ]]2µQ k2k 2 3 3− i r kik k kα = + √r − r − √i (3.11)
4π 2 2 3 6 6 3
[ ]
√√
2 2
[ 2µQ −
ki − k√ikr + kr
√ 4π 2 3 2β = √ [ ]] (3.12)2µQ k2− kr k k2 3 32 i i√r − kr − k+ √i
4π 2 2 3 6 6 3
√
1 ± 1 N
2 − 4N + 3
δ = + l(l +N − 2) + (3.13)
2 (4 ) 4
2
EN
Q
0l = −
ki β α
kr + √ − + (1 + 2δ) (3.14)
4π 3 2µ 2µ
[
( ) [ ]
]2
Q − k
2 2
i kikr kr
[ − √ +[ 4π 2 3 2N Q kiE0l = −kr + √ − ]]4π 3
Q k2− i kr k k2 − k3
3
+ i√r r√ −
k√i
π 2
√[ 2] 3
6 6 3
√√ [ ]2µQ k2 3− i kr + k 2 3i√kr − kr − k√i
4π 2 2 3 6 6 3 ( √ )
+ 1± 1 + 4l(l +N − 2) +N2 − 4N + 3 + 1
2µ
(3.15)
At n=1, substituting Eq.(3.8) and Eq.(3.9) in Eq.(3.10), equating with Eq.(3.5)
and comparing the powers of r, we get
50
( ) ( )
N Q k
2
− √i − β αE1l = kr + + 1 + 2(δ + 1) (3.16)4π 3 2µ 2µ
[ [ ]]( ) 2Q − k2i − k√ik k2r + r[ [4π 2 3 2N Q kiE1l = −kr + √ − ]]4π 3 2 3
Qπ − ki kr k 2 3+ i√kr − kr − k√i
√√[
2 ]2 3 6 6 3√√ [ ]2µQ k2− kr k k2 3 3i + i√r − kr − k√i
4π 2 2 3 6 6 3 ( √ )
+ 3± 1 + 4l(l +N − 2) +N2 − 4N + 3 + 1
2µ
(3.17)
Similarly at n=2, ( ) ( )
2
EN
Q
2l = −kr + √
ki − β α+ 1 + 2(δ + 2) (3.18)
4π 3 2µ 2µ
[ ]
( ) [ ] 2Q 2 2− ki − k√ikr + kr[ [ 4π 2 3 2N Q − √kiE2l = kr + − ]4π ]3
Q k2 3i kr
2 3
√ − +
ki√kr − kr − k√i
[ π 2√ 2 3
6 6 3
√√ [ ]]2µQ k2− i k 3r 2+ ki√kr − k3r − k√i
4π 2 2 3 6 6 3 ( √ )
+ 5± 1 + 4l(l +N − 2) +N2 − 4N + 3 + 1
2µ
(3.19)
The iteration can be repeated many times and the exact energy formula at
finite temperature can be(written as,) ( )
Q k β2i α
ENnl = −kr + √ − + 1 + 2(δ + n) (3.20)4π 3 2µ 2µ
51
[
( ) [ ]
]2
Q − k
2
i − k√ik
2
r + kr
4π 2 3 2
N Q − √kiEnl = kr + − [ ]4π [ ]3
Q − k
2kr k k2 − k3i i√r r − k
3
√ + √
i
√[
π 2 2] 3 6 6 3√√ [ ]2µQ 2− ki kr + ki√k2 − k3r r − k3√i
4π 2 2 3 6 6 3 ( √ )
+ 2(n+ 1) + 1 + 4l(l +N − 2) +N2 − 4N + 3
2µ
(3.21)
where
ks
ki =
2
√
3
kr = ks
2
M 2
2 q
ωp
ks = g ∂Pc ∂nf
By considering the temperature dependence of αs from the literature [2] we
get,
( ( )ln 2 ln (T/Λ )T )
6π − 3(153− 19nf )αs(T ) = 1
(33− 2nf ) ln (T/ΛT ) (33− 2n )2f ln (T/ΛT )
Eq.(3.21) gives the expression for the binding energy of the bound state of
quarks.
52
3.2.2 Power series method (PSM)
In N-dimensional space, the Schrödinger equation can be written as Eq.(3.1). In
order to find out the solution to Eq.(3.1) ansatz for the wave function is taken as,
Ψ(r) = exp(−ηr2 − δr)G(r) (3.22)
Here η and δ are positive parameters that are to be determined in terms of potential
parameters. G(r) is assumed in series form as,
∑∞
G(r) = a rn+ln (3.23)
n=0
Where a0 and a1 are non zero expansion coefficients. Now substituting Eqs.(3.22)
and (3.23) in Eq.(3.1) and collecting the coefficients of like powers of r, we get
∑ [∞ ( )
an (n+ l)(n+ l −(1) + (N − 1)(n+ l)− l(l +N − 2)
n+l−2
n=0 )r
( + − δ − δ(n+ l)− −
2µQ
δ(N ( 1)− r
n+l−1
4π
2µQ k ))2 − i+ δ 2η(n+ l)− 2(η(N − 1) + 2µE − − k + √ ) rn+l( r ) (3.24)4π 32
− − 2µQ kr − k k k
2
i r
+ 4ηδ 2η √ − i( ) r4π 2 3 2 ]
n+l+1
2µQ( 3 2 2 3 )
+ 4η2 − − kr k√ikr − ki kr − k+ √i rn+l+2 = 0
4π 6 2 3 2 6 3
Equating each coefficient of r to zero, we get the following set of relation
(n+ l)(n+ l +N − 2)− l(l +N − 2) = 0 (3.25)
53
− − 2µQδ(n+ l +(N) = 0 (3.26)4π )
2µQ k2 kik k
2
r
4ηδ − 2η − r − √ − i = 0 (3.27)
( 4π 2 3 2 )
3
2 − 2µQ −kr k√ik
2 k2k k3r
4η + r − i − √i = 0 (3.28)
4π (6 2 3 ) 2 6 3
− 2 2µQ ki2µE = δ + −kr + √ + 2η(n+ l +N − 1) (3.29)
4π 3
In order to find out the eigenvalues from Eq.(3.29), the ansatz parameters η and
δ should be known which can be obtained from Eq.(3.28) and (3.26) respectively.
√ ( )
2µQ k3 k 2 2 3− r √ikr − kη = + i kr − k√i (3.30)
16π 6 2 3 2 6 3
2µQ
δ = − (3.31)
4π(n+ l +N)
After substituting the values of η and δ the energy eigenvalue can be written as,
( )
2
N −( 2µQ ) Q − √kiEnl = 2 + kr +4π 3
√ 4π(n+( l +N) ) (3.32)
Q k3− r k
2 2 3
+ 2 + √ikr − ki kr − k√i (n+ l +N − 1)
32πµ 6 2 3 2 6 3
Eq.(3.32) gives the expression for the binding energy of the bound state of quarks.
54
3.2.3 Nikiforov-Uvarov (NU) Method
a) A brief description of NU method
Consider a second-order differential equation of the form
′′ τ̃(s) ′ σ̃(s)Ψ (s) + Ψ (s) + Ψ(s) = 0 (3.33)
σ(s) σ2(s)
where σ(s) and σ̃(s) are polynomials of maximum second degree and τ̃(s) is a
polynomial of maximum first degree with an appropriate s = s(r) coordinate
transformation.
If
Ψ(s) = Φ(s)χ(s) (3.34)
Substituting Eq.(3.34) in Eq.(3.33)
′ ′ τ̃(s)
( ) σ̃(s)
2Φ (s)χ (s)+Φ(s)χ′′(s)+Φ′′(s)χ(s)+ Φ(s)χ′(s)+Φ′(s)χ(s) + Φ(s)χ(s) = 0
σ(s) σ2(s)
Rearranging the terms we get,
(
′′ ′ τ̃(s)
) (
′ ′ τ̃(s) σ̃(s)
)
φ(s)χ (s)+ 2Φ (s)+ Φ (s) χ (s)+ Φ′′(s)+ Φ′(s)+ Φ(s) χ(s) = 0
σ(s) σ(s) σ2(s)
Dividing throgh out by Φ(s)
(2Φ′(s) τ̃(s)) (Φ′′(s) τ̃(s) Φ′′′ ′ (s) σ̃(s) )χ (s) + + χ (s) + + + χ(s) = 0
Φ(s) σ(s) Φ(s) σ(s) Φ(s) σ2(s)
The coefficient of χ′(s) is taken in the form τ(s)/σ(s), where τ(s) is a polynomial
55
of degree atmost one. Then
2Φ′(s) τ̃(s) τ(s)
+ = (3.35)
Φ(s) σ(s) σ(s)
This happens only if
Φ′(s) π(s)
= (3.36)
Φ(s) σ(s)
Therefore
2π(s) τ̃(s) τ(s)
+ =
σ(s) σ(s) σ(s)
That is
2π(s) + τ̃(s) = τ(s)
Thus
1( )
π(s) = τ(s)− τ̃(s)
2
and
τ(s) = τ̃(s) + 2π(s); τ ′(s) < 0 (3.37)
The term Φ′′(s)/Φ(s) in the coefficient of χ(s) can be written as
Φ′′(s) (Φ′(s))′ (Φ′(s))2
= +
Φ(s) Φ(s) Φ(s)
Using Eq.(3.36),
Φ′′(s) (π(s))′ (π(s))2
= + (3.38)
Φ(s) σ(s) σ(s)
Now the coefficient of χ(s) is transformed in to a more suitable form,
Φ′′(s) τ̃(s) Φ′(s) σ̃(s) σ(s)
+ + = (3.39)
Φ(s) σ(S) Φ(s) σ2(s) σ2(s)
56
Substituting Eq.(3.38) in Eq.(3.39),
π(s)σ′(s) π′(s) π2− (s) π(s)τ̃(s) σ̃(s) σ(s)+ + + + =
σ2(s) σ(s) σ2(s) σ2(s) σ2(s) σ2(s)
Therefore ( )
σ(s) = σ̃(s) + π2(s) + π(s) τ̃(s)− σ(s) + π′(s)σ(s)
Substituting the right hand sides of Eq.(3.35), Eq.(3.38) and Eq.(3.39), Eq.(3.33)
reduces to an equation of hypergeometric type as
χ′′
τ(s) ′ σ(s)(s) + χ (s) + χ(s) = 0 (3.40)
σ(s) σ2(s)
with
Φ(s)
σ(s) = π(s) (3.41)
Φ′(s)
As a consequence of the algebraic transformations above, the functional form of
Eq.(3.33) is protected in a systematic way. If the polynomial σ(s) in Eq.(3.40) is
divisible by σ(s), we can write
σ(s) = λσ(s)
Where λ is a constant. Then Eq.(3.40) is reduced to,
σ(s)χ′′(s) + τ(s)χ′(s) + λχ(s) = 0 (3.42)
57
To determine the polynomial π(s),
( )
σ(s) = σ̃(s) + π2(s) + π(s) τ̃(s)− σ(s) + π′(s)σ(s) = λσ(s)
That is ( )
σ̃(s) + π2(s) + π(s) τ̃(s)− σ(s) −Kσ(s) = 0
with
K = λ− π′(s) (3.43)
The solution of this quadratic equation for Π(s) gives
√
′ ( )2σ (s)− τ(s) σ′(s)− τ(s)
π(s) = ± − σ(s) +Kσ(s) (3.44)
2 2
where π(s) is a polynomial of first degree. The values of K in the square root
of Eq.(3.44) can be calculated if the expressions under the square root are square
of expressions. This is possible if its discriminant is zero. χ(s) = χn(s) is a
polynomial of nth degree which satisfies the hypergeometric equation, in order to
obtain an eigen value solution through the NU method, a relation between λ and
λn must be set up which is given by,
′ n(n− 1)λ = λ ′′n = −nτ (s)− σ (s), n = 0, 1, 2.... (3.45)
2
χn(s) is a hypergeometric type function whose polynomial solutions are given by
the Rodrigue’s relation
B dnn
χn(s) = (σ
′′(s)ρ(s)) (3.46)
ρn dsn
58
where Bn is a normalization constant and ρ(s) is the weight function that
satisfies the equation
d τ(s)
ω(s) = ω(s);ω(s) = σ(s)ρ(s) (3.47)
ds σ(s)
b) Solution of N-dimensional Schrödinger equation at finite temperature
The Schrödinger equation for two particles interacting via a spherically symmetric
potential φ(r) in N- dimensional space is given by Eq.(3.1), where φ(r) is given by
Eq.(3.4). Now the N-dimensional Schrödinger equation can be written as,
[ [
d2 Q [( 1 ) ]
+ 2µ E − cos (k]ir) + √ sin (kir) e
−krr
dr2 4πr ] 3 (3.48)2
l(l +N − 2) + N −4N+3
− 4 R(r) = 0
2µr2
or [ ( [ ( ) ( )
d2 Q 1 k k2 k k 2− − i i r k+ 2µ E + k + √ + r − √ − i r
dr(2 r4π r ) ]3 2 3 2 )]
3 2
−kr k k
2 k2i r i k
3
r ki 2 l(l +N − 2) +
N −4N+3
+ + √ − − √ r − 4 R(r) = 0
6 2 3 2 6 3 2µr2
(3.49)
Putting r = 1 , equation becomes
x
59
[ ( [ ( )
d2 2x d 2µ Q ki
( + + )E − ( x+ −k + √dx2 rx2 dx x4 4π 3 ) ]
k2r − k
2
√ikr − ki 1 k
3 k k2 2 3−)]r √i+ + + r −
ki kr − k√i 1 (3.50)
2 3 2 x 6 2 3 2 6 3 x2
2
l(l +N − 2) + N −4N+3
− 4 x2 R(x) = 0
2µ
or [ [
d2 2x d 2µ C D
+ + E − Ax−]B − −dx2 x2 dx x4 x x2 (3.51)
2
l(l +N − 2) + N −4N+3 ]
− 4 x2 R(x) = 0
2µ
where
Q
A(= 4π )
Q ki
B = −kr + √
4(π 3 )
Q k2r − k√ikr − k
2
C = i
( 4π 2 3 2 )
Q k3 2 2 3
D = − r k√ikr − ki kr − k+ √i
4π 6 2 3 2 6 3
Now expand C and D2 in a power series around the characteristic radius r0 of thex x
meson up to the second-order, Setting y = x − δ, where δ = 1 . Now we expand
r0
C and D2 in a power series around y(= 0 [19] as,x x )
C 3 3x x2
= C − + (3.52)
x δ δ2 δ3
and ( )
D 6 − 8x 3x
2
= D + (3.53)
x2 δ2 δ3 δ4
60
Substi[tuting Eq.(3.52) an[d (3.53) in Eq.(3.51(), )
d2 2x d 2µ 3 2
+ + E − Ax−B − C − 3x x+
dx2( x2 dx x4 ) δ δ2 δ3]] (3.54)2
− 6 − 8x 3x
2 l(l +N − 2) + N −4N+3
D + − 4 x2 R(x) = 0
δ2 δ3 δ4 2µ
or we can write
[ [ ]]
d2 2x d 2µ
+ + −D1 +D 22x−D3x R(x) = 0 (3.55)
dx2 x2 dx x4
where [ ]
− 3C 6DD1 = E −B − −
δ δ2
3C 8D
D2 = −A+ +
δ2 δ3
2
C 3D l(l +N − 2) + N −4N+3
D3 = + +
4
δ3 δ4 2µ
Comparing Eq.(3.33) and Eq.(3.55) we get,
τ(s) = 2x
σ(s) = x2
( )
σ(s) = 2µ −D1 +D2x−D3x2
By following NU method,
√
π = 2a− 2bx+ (K + 2c 21)x (3.56)
61
where
a = 2µD1
b = 2µD2
c1 = 2µD3
The constant K is chosen such as the discriminant of the function under the
square root is zero i.e
∆ = 4b2 − 8a(K + 2c1) = 0
b2 = 2a(K + 2c1)
b2
K = − 2c1
2a
2a√− bxπ = (3.57)
2a
Thus
± 2(2√a− bx)τ = 2x (3.58)
2a
We choose the positive sign in the above equation for bound state solutions.
2b
τ ′ = 2− √ (3.59)
2a
By using Eq.(3.43), we obtain
b2 − bλ = 2c1 − √ (3.60)
a 2a
( )
− − √2bλn = n 2 − n(n− 1) (3.61)
2a
62
From Eq.(3.45),
λ = λn
( )
b2 − − √b2c1 = −n 2− √
2b − n(n− 1)
a 2a 2a
b2 − √b (1 + 2n) = −n2 − n+ 2c1
2a( 2a )2
√b − (2n+ 1) 1= + 2c
2a 2 √ 14
√b − (2n+ 1) 1= + 2c1
2a 2 √4
√b (2n+ 1) ± 1= + 2c1
2a( 2 √ 4 )2
b2 (2n+ 1)
= ± 1 + 2c1
2a 2 4
( b2a = √ )2
2 (2n+1) ± 1 + 2c
2 4 1
( 2µD2D = 21 √ )2
(2n+ 1)± 1 + 8µD3
( )
3C 6D 2µ −A+ 3C + 8D 2− 2 3E +B + + =
2 ( √ ( δ δδ δ ))22
± C 3D l(l+N−2)+
N −4N+3
(2n+ 1) 1 + 8µ + + 4
δ3 δ4 2µ
The energy eigen values at finite temperature in N-dimensional space can be writ-
63
ten as
( )
− 3C 8D 2N 3C 6D 2µ A+ +Enl = B+ + −( √ δ2 δ3δ δ2 ( ))2
(2n+ 1)± 1 + 8µC + 24µD 23 4 + 4 l(l +N − 2) + N −4N+3δ δ 4
(3.62)
Eq.(3.62) gives the expression for the binding energy of the bound state of quarks.
3.3 Conclusion
In this work, we have used the screening attractive potential to study the forma-
tion of bound states of quarks in a simple manner. Using this attractive potential,
Schrodinger’s equation is solved for getting the binding energy of the bound state of
quarks in strongly coupled quark-gluon plasma in three different methods. These
three methods incorporate no different Physics. But they are three ways for ob-
taining the results by solving SE. These are done in order to compare the numerical
results such as binding energy, mass spectra and dissociation temperatures.
64
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67
Chapter 4
Binding Energies and Effective
Masses of Diquarks in SCQGP
4.1 Introduction
In [1], it is proposed that near and above Tc there are multiple bound states of
quasi-particles which are very important for the thermodynamics of the QGP.
At T ≥ Tc, it has been argued that there is no color confinement and there is
Coulomb-like interaction at small distances with a Debye type screening at large
distances [2].
The deconfined quarks and gluons pair up non-perturbatively in order to lower
the energy of the system. This may lead to the formation of a system of two
quarks called diquark [3] or multi quark system consisting of more number of
quarks. These multi-quark systems can be studied using different models such as
the MIT bag model [4] and the Potential model [5]. Out of all possible multi-
quark states, the study of diquarks is more important in the context of QGP
68
formed in RHIC experiments [6]. In 1964 GellMann first mentioned the possibility
of diquarks. A diquark is any system of two quarks considered collectively. The
idea of diquarks has been used in hadron physics for describing many properties of
hadrons [3]. This study is also important in the context of astrophysical conditions
[7], especially in cosmological contexts involving early phases of the evolution of the
universe [8]. The question about the existence of diquarks as quasi-bound states
in cold quark matter was emerged in [6], in which the authors have claimed that
the quark-diquark matter was energetically more favorable than the free quark
matter. The idea of the possible existence of quasi-bound states of quarks and
gluons including diquarks with the temperature in the range Tc ≤ T ≤ 4Tc was
put forward by Shuryak et al. [1]. In [9] it was shown that the Λc/D
0 ratio
could be enhanced by a factor of 4-8 than the case without diquarks in quark-
qluon plasma. Diquark correlations play an important role inside baryons. Two
quarks bound in a color anti triplet configuration can couple with a single quark to
form a color singlet baryon. Recent studies of baryons as bound states of a quark
and a confined diquark are done [10]. Some lattice simulations also indicate the
existence of correlations in the diquark channel. A diquark in its ground state has
positive poarity and may be a scalar (Spin 0) or an axial vector (Spin 1). It has
been pointed out that the production of scalar diquarks is much larger than the
vector diquarks. Evidence for scalar diquarks has been confirmed by lattice QCD
simulation [11].
In chapter 2, a new screening potential is proposed and in chapter 3, the bind-
ing energy of the bound states of quarks is calculated by solving the Schrodinger’s
equation using this potential in three different methods i.e., analytical exact iter-
ation method (AEIM), power series method (PSM) and Nikiforov - Uvarov (NU)
69
method. In this chapter, we calculate the binding energies of scalar diquarks in two
methods i.e., analytical exact iteration method (AEIM) and power series method
(PSM). The scalar diquark masses are calculated using the ground state binding
energy of the bound states in strongly coupled quark-gluon plasma (SCQGP).
Also, the relativistic correction is done to the light diquark mass.
4.2 Binding energy of diquarks
4.2.1 Analytical exact iteration method (AEIM)
Using Eq.(3.21), the binding energy of diquarks in the strongly coupled quark-
gluon plasma at N=3 can be wr[itten [as,( ) ]
]2
k2 2
[ Q4π −
i − k√ikr + kr
[ 2 3 2QEnl = − √kikr + − ]4π ]3
Q 2− k kr 2 3i k
3
√ +
ki√kr − kr − √i
[ π 2√ 2] 3
6 6 3
[ ] (4.1)√√ 2µQ 2 3− k 2 3i kr + ki√kr − kr − k√i
4π 2 2 3 6 6 3 ( √ )
+ 2(n+ 1)± 1 + 4l(l + 1)
2µ
4.2.2 Power series method (PSM)
Using Eq.(3.32), the binding energy of diquarks in the strongly coupled quark-
gluon plasma at N=3 can be written as, ( )
2µQ2 Q
Enl = −√ ( 2 + −
ki
kr + √
(4π(n+ l + 3)) 4π 3 ) (4.2)
Q 3 2 2 3
+ 2 −kr k√ikr − ki kr − k+ √i (n+ l + 2)
32πµ 6 2 3 2 6 3
70
where Q = q1q2
The ground state binding energies of different scalar diquarks in two methods
were given in Fig. 4.1 as functions of temperature.
4.3 Effective mass of diquarks
Using the binding energy, the mass of diquark can be calculated as [12],
M∗D = Mq1 +Mq2 + Enl (4.3)
where Mq1 and Mq2 are the masses of constituent quarks, M
∗
D is the effective
mass of the diquark and Enl is the binding energy given by Eq.(4.1) and (4.2).
Effective masses of scalar diquarks were calculated using the ground state
binding energy of the diquarks in two methods at temperature 1.05Tc, where
Tc = 175MeV and the result is given in Table 3.1. Since the heavy diquark
states are weakly bound, the nonrelativistic treatment could be completely justi-
fied. A similar conclusion would be reached for light qq pairs. Since the binding is
weak, relativistic correction may be excluded. But here we try to do the relativistic
correction in a simple manner.
4.4 Relativistic Correction
The potential used above was evaluated for static charges without spin and there-
fore it does not include the effects of particle velocities and spin. The relativistic
71
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
1.0 1.2 1.4 1.6 1.8 2.0
TTc
(a) Dependence of diquark binding energy in units of the con-
stituent quark mass on temperature, from bottom to top,(i)[qq]0
(ii) [qs]0 (iii) [ss]0 (iv) [qc]0 (v) [cc]0 (vi) [qb]0 (vii) [sb]0 (viii) [bb]0.
The single cross represents the relativistic correction to the binding
energy of [qq]0 diquark at temperature 1.05Tc (q=u,d) (AEIM)
0.0
-0.1
-0.2
-0.3
-0.4
1.0 1.2 1.4 1.6 1.8 2.0
TTc
(b) Dependence of diquark binding energy in units of the con-
stituent quark mass on temperature, from bottom to top,(i)[qq]0
(ii) [qs]0 (iii) [ss]0 (iv) [qc]0 (v) [cc]0 (vi) [qb]0 (vii) [sb]0 (viii) [bb]0.
The single cross represents the relativistic correction to the binding
energy of [qq]0 diquark at temperature 1.05Tc (q=u,d) (PSM)
Figure 4.1: Dependence of diquark binding energy in units of the constituent quark
mass on temperature
72
B.E.HMq1+Mq2L B.E.HMq1+Mq2L
Table 4.1: Diquark Masses
Quark Content AEIM PSM
[qq]0 0.5072 ( 0.4609) 0.5343 ( 0.4954)
[qs]0 0.7023 0.8124
[ss]0 0.8991 0.9972
[qc]0 1.3711 1.7319
[sc]0 2.5588 2.3914
[sb]0 5.4453 5.5172
[qb]0 5.2437 5.3313
[cc]0 2.9431 3.1144
[cb]0 6.9690 6.8582
[bb]0 10.0144 10.0373
effect can be included by the substitution of the effective Coulomb coupling by [2]
αs → αs(1− β1β2) (4.4)
where β1 and β2 are the velocities (in units of c) of both the particles. Since the
velocities are always opposite in C.M., the coupling increases. Let < v2 >= 0.12
for light quarks at T = 1.05 [2]. If we include this correction to the potential we
Tc
get larger binding. By including this correction at T ≈ Tc, the light quarks become
relativistic. The relativistically corrected light diquark mass is written in brackets
in Table 4.1. If the particle velocity becomes the speed of light this correction
doubles the coupling.
4.5 Results and Discussion
In the current chapter, we have investigated the binding energies of some scalar
diquarks. In Fig. 4.1, the light diquark binding energy looks considerable than the
heavy diquark and light-heavy diquarks. This shows the relatively large possibility
73
for the existence of light diquarks near the transition temperature. The diquarks
can be described as elementary excitations stimulating the many-body interactions
and behave like scalar bosons [13]. Two quarks should be antisymmetric in color
so that a total color singlet state is obtained. Recent lattice works have found
that the bound states can persist to at least T = 2Tc [2]. This new potential
also supports this condition. In general, diquark masses have contributions from
the constituent masses, the confining and spin-dependent part of the quark-quark
interaction [14]. These contributions are additive in a simple potential model [15].
The spin-dependent contribution may include a contribution from flavor-spin and
color-spin dependent interaction terms. The masses are expected to get the largest
contribution from the constituent mass term and the confinement part [14], the
spin-dependent part is related to the mass splitting of the S=0, S=1 diquarks and
it is not considered here as we have reported only the scalar diquark masses.
74
Bibliography
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at Tc < T < 4Tc, Phys. Rev. C 70, 021901 (2004).
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quark-gluon plasma, Phys. Rev. D 70, 054507 (2004).
[3] D.B. Lichtenberg, L.J. Tassie, and P. J. Keleman, Quark-diquark model of
baryons and SU(6), Phys. Rev. 167, 1535 (1968).
[4] R. L. Jaffe and F. E. Low, Connection between quark-model eigenstates and
low-energy scattering, Phys. Rev. D 19, 2105 (1979).
[5] J. Weinstein and N. Isgur, qqqq system in a potential model, Phys. Rev D 27,
588 (1983).
[6] J. F. Donoghue and K.S .Sateesh, Diquark clusters in the quark-gluon plasma,
Phys. Rev. D 38, 360 (1988).
[7] D. Blaschke, S. Fredriksson, H. Grigorian, and A.M. Oztas, Diquark conden-
sation effects on hot quark star configurations, Nucl. Phys. A 736, 203 (2004).
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[8] S. K. Karn, R. S. Kaushal, and Y. K. Mathur, On diquark clusters in a quark-
gluon plasma, Z. Phys. C 72, 297 (1996).
[9] S. H. Lee, K. Ohnishi, S. Yasui, I.K. Yoo, and C. M. Ko, Λc enhancement
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(2008).
[10] J. Liao and E. V. Shuryak, Polymer chains and baryons in a strongly coupled
quark-gluon plasma, Nucl. Phys. A, 775, 224 (2006).
[11] C. Alexandrou, Ph. de Forcrand, and B. Lucini, Evidence for diquarks in
lattice QCD, Phys. Rev. Lett. 97, 222002 (2006).
[12] M. Abu-Shady, T. A. Abdel-Karim, and E. M. Khokha, Binding energies and
dissociation temperatures of heavy quarkonia at finite temperature and chem-
ical potential in the N-dimensional space, Advances in High Energy Physics,
Volume 2018, 7356843 (2018).
[13] A. Chandra, A. Bhattacharya, and B. Chakrabarti, Temperature dependent
diquark and baryon masses, J. Mod. Phys. 4, 945 (2013).
[14] M. Hess, F. Karsch, E. Laermann, and I. Wetzorke, Diquark masses from
lattice QCD, Phys. Rev. D 58, 111502(R) (1998) .
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76
Chapter 5
Heavy Quarkonium Properties at
Finite Temperature in Strongly
Coupled Quark Gluon Plasma
5.1 Introduction
The bound states of a heavy quark and its antiquark are generally referred to
as quarkonia. Other than the vector ground states J/ψ and Υ both the cc and
bb systems give rise to many other stable states of different quantum numbers.
The stable charmonium spectrum consists of 1S scalar ηc and vector J/ψ, three
1P states χc (scalar, vector, and tensor) and 2S vector state ψ
′. The stable bot-
tomonium spectrum contains Υ, χb, Υ
′, χ′b and Υ” [1]. Heavy quarkonium has
received more theoretical attention because heavy quark states are dominated by
short-distance physics and can be treated using heavy quark effective theory [2].
Both experimental and theoretical studies suggest that the dissociation tempera-
77
tures of heavy quarkonia are slightly smaller than twice the critical temperature
[3]. Because of the large mass, the velocity of heavy quarks is smaller and hence
the spectrum of the quarkonia comprised of heavy quarks can be estimated using
potential-based nonrelativistic treatments [4]. It has been argued that the screen-
ing above the critical temperature is strong enough to lead to the dissociation
of the J/ψ state which may be considered as the signal of QGP formation [5].
Charmonium suppression and screening may be related as in [6].
Various potential models have been used to study the quarkonium properties
at finite temperature, the first work in this field was done by Karsch, Mehr, and
Satz [7]. Heavy quarkonium suppression has been an interesting subject after the
theoretical work of Matzui and Satz [5]. Later Digal, Petreczky, and Satz reported
the dissociation temperatures of quarkonia in hadron and QGP phases [8, 9]. Re-
cent research successfully attempted to estimate the mass spectra of charmonium
and bottomonium mesons and to determine the binding energy and dissociation
temperature of heavy quarkonia [10, 11, 12]. In [4, 11, 12] the authors tried to
solve the Schrödinger equation at finite temperature for the charmonium and bot-
tomonium states using an effective temperature-dependent potential. The binding
energy of the heavy quarkonium states is studied in [13, 14]. At finite chemical po-
tential and in a small temperature region the dissociation of quarkonia states was
studied in a deconfined medium of quarks and gluons [15]. In [8, 9, 16, 17], where
the color singlet free energy was directly inserted into the Schrödinger equation,
the dissociation temperature of J/ψ was found to be 1.10 Tc [8] and 0.99 Tc [16],
and all other charmonium states were melting below the transition temperature,Tc.
In [18], where a parameterization of the lattice color singlet potential was used,
the dissociation temperature of Tc was calculated at 2 Tc and also the dissocia-
78
tion temperatures of different states of charmonium and bottomonium were also
reported. In [19, 20, 21], dissociation temperatures were obtained using spectral
analysis.
In this chapter, we use the solution of the N-dimensional radial Schrödinger
equation with generalized two-body potential proposed in chapter 2 at finite tem-
perature using the analytical exact iteration method (AEIM), Power Series Method
(PSM), and Nikiforov-Uvarov (NU) method to determine the binding energy of
heavy quarkonia. We try to find out the mass spectra and dissociation tempera-
tures of heavy quarkonium states. Also, the influence of the dimensionality number
has been investigated on the binding energy and mass spectra of heavy quarkonium
at finite temperature.
5.2 Binding energy of quarkonium
5.2.1 Analytical exact iteration method (AEIM)
For quarkonium, Mq = Mq = m and Q = −q2
Using Eq.(3.21),
[
( ) [ ]
]2
Q 2− ki − k√ikr + k
2
r
4π 2 3 2
Q ki
ENnl = −kr + √ − [ ]4π [ ]3
Q − k
2 2 3 3
i kr + ki√kr − kr − k√i
π 2 2 3 6 6 3
√√√√[ [ ]]mQ 2− ki kr + ki√k2 3 3r − kr − k√i
4π 2 2 3 6 6 3 ( √ )
+ 2(n+ 1) + 1 + 4l(l +N − 2) +N2 − 4N + 3
m
(5.1)
79
where
ks
ki =
2
√
3
kr = ks
2
M ω2
2 q pks = g ∂Pc ∂nf
5.2.2 Power Series Method (PSM)
Using Eq.(3.32),
( ( )N − mQ2 ) Q √kiEnl = 2 + −kr +4π 3
√ 4π(n+(l +N) ) (5.2)
Q k3 k k2 2 3i
+ 2 − r + √r − ki kr − k√i (n+ l +N − 1)
16πm 6 2 3 2 6 3
5.2.3 Nikiforov-Uvarov (NU) method
Using Eq.(3.62),
( )2
N 3C 6D ( √ m −A+ 3C (+ 8DE = B + + − δ2 δ3n,l )δ δ2 ) 2
± 4mC 12mD − N2(2n+ 1) 1 + −4N+3
δ3
+ 4 + 4 l(l +N 2) +δ 4
(5.3)
Eq.(5.1), Eq.(5.2), and Eq.(5.3) give the expressions for binding energy of
quarkonium.
80
5.3 Variation of Binding energy of quarkonium
with temperature
By considering the temperature dependence of αs from the literature [22] we get,
( ( )
6π 3(153− 19n ) ln 2 ln (T/ΛT )
)
f
αs(T ) = 1−
(33− 2nf ) ln (T/ΛT ) (33− 2nf )2 ln (T/ΛT )
The binding energy of quarkonia such as charmonium and bottomonium is
calculated in the N-dimensional space for any state at finite temperature using
Eq.(5.1), Eq.(5.2), and Eq.(5.3). Here m = Mc for charmonium and m = Mb
for bottomonium mesons. The variation of binding energy with temperature for
different states of quarkonia is shown in Figures 5.1 - 5.9.
81
500
400
300
200
1.0 1.5 2.0 2.5
TTc
(a)
700
600
500
400
300
1.0 1.5 2.0 2.5
TTc
(b)
Figure 5.1: (a) Variation of J/ψ binding energy on temperature T and (b) Varia-
tion of χc binding energy on temperature T (AEIM)
82
B.E.HMeVL B.E.HMeVL
500
450
400
350
300
250
0.5 1.0 1.5 2.0 2.5 3.0 3.5
TTc
(a)
700
650
600
550
500
450
400
0.5 1.0 1.5 2.0 2.5 3.0 3.5
TTc
(b)
Figure 5.2: (a) Variation of J/ψ binding energy on temperature T and (b) Varia-
tion of χc binding energy on temperature T (PSM)
83
B.E.HMeVL B.E.HMeVL
800
700
600
500
400
300
200
1.5 2.0 2.5 3.0
TTc
(a)
800
700
600
500
400
300
200
1.5 2.0 2.5 3.0
TTc
(b)
Figure 5.3: (a) Variation of J/ψ binding energy on temperature T and (b) Varia-
tion of χc binding energy on temperature T (NU Method)
84
B.E.HMeVL B.E.HMeVL
300
250
200
150
100
50
0
0.5 1.0 1.5 2.0 2.5
TTc
(a)
350
300
250
200
150
100
50
0
0.5 1.0 1.5 2.0 2.5
TTc
(b)
Figure 5.4: (a) Variation of Υ binding energy on temperature T and (b) Variation
of χb binding energy on temperature T (AEIM)
85
B.E.HMeVL B.E.HMeVL
300
250
200
150
100
50
0
0.5 1.0 1.5 2.0 2.5
TTc
(a)
350
300
250
200
150
100
50
0
0.5 1.0 1.5 2.0 2.5
TTc
(b)
Figure 5.5: (a) Variation of Υ binding energy on temperature T and (b) Variation
of χb binding energy on temperature T (PSM)
86
B.E.HMeVL B.E.HMeVL
200
150
100
50
0
1.0 1.5 2.0 2.5 3.0 3.5
TTc
(a)
200
150
100
50
0
1.0 1.5 2.0 2.5 3.0 3.5
TTc
(b)
Figure 5.6: (a) Variation of Υ binding energy on temperature T and (b) Variation
of χb binding energy on temperature T (NU method)
87
MHMeVL MHMeVL
700 N=3 250N=4
600 N=5 200
N=3
N=4
500 150 N=5
400 100
300
200 50
0
1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5
TTc TTc
(a) J/ψ (b) Υ
N=3 350
800 N=4 N=3N=5 300
250 N=4N=5
600 200
400 150100
200 50
0
1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5
TTc TTc
(c) χc (d) χb
Figure 5.7: Variation of quarkonium binding energy on temperature T for different
values of N (AEIM)
88
B.E.HMeVL B.E.HMeVL
B.E.HMeVL B.E.HMeVL
700 N=3 250N=4
600 N=5 200
N=3
N=4
500 150 N=5
400 100
300
200 50
0.5 1.0 1.5 2.0 2.5 00.5 1.0 1.5 2.0 2.5
TTc TTc
(a) J/ψ (b) Υ
800 350N=3
700 N=4 300 N=3N=5 N=4
600 250 N=5
500 200
400 150
300 100
200 50
0.5 1.0 1.5 2.0 2.5 00.5 1.0 1.5 2.0 2.5
TTc TTc
(c) χc (d) χb
Figure 5.8: Variation of quarkonium binding energy on temperature T for different
values of N (PSM)
89
B.E.HMeVL B.E.HMeVL
B.E.HMeVL B.E.HMeVL
800 N=3 200 N=3
700 N=4 N=4N=5
600 150 N=5
500 100
400
300 50
200 0
1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 3.5
TTc TTc
(a) J/ψ (b) Υ
250
N=3 N=3
800 N=4 N=4N=5 200 N=5
600 150
400 100
50
200 0
1.5 2.0 2.5 3.0
 1.0 1.5 2.0 2.5 3.0 3.5T Tc TTc
(c) χc (d) χb
Figure 5.9: Variation of quarkonium binding energy on temperature T for different
values of N (NU method)
90
B.E.HMeVL B.E.HMeVL
B.E.HMeVL B.E.HMeVL
5.4 Mass spectra of quarkonia
For calculating the mass of heavy quarkonium the following relation is used [23].
M = 2Mq + E
N
nl (5.4)
5.4.1 Analytical exact iteration method (AEIM)
Substituting Eq.(5.1) into Eq.(5.4), the mass spectra for different states of heavy
quarkonia as a function of tempera[ture[is given by, ]]2
( ) Q k2 2− i − k√ikr + kr4π 2 3 2Q ki
M = 2m+ − kr + √ − [ ]
4π [ ]3
Q k2− kr k k2 3
3
i k
√ +
i√r − kr − √i
[ π 2 6√ 2 3 6 3√√ [ ]]mQ k2− i kr + k 2 3 3i√kr − kr − k√i
4π 2 2 3 6 6 3 ( √ )
+ 2(n+ 1) + 1 + 4l(l +N − 2) +N2 − 4N + 3
m
(5.5)
5.4.2 Power Series Method (PSM)
Substituting Eq.(5.2) into Eq.(5.4), the mass spectra for different states of heavy
quarkonia as a function of temperature is given by,
( ( )mQ2M = 2m− ) Q − ki2 + kr + √4π 3
√ ( 4π(n+ l +N) ) (5.6)
Q 3 2 2 3
+2 −kr k√ikr − ki kr − k+ √i (n+ l +N − 1)
16πm 6 2 3 2 6 3
91
5.4.3 Nikiforov-Uvarov (NU) method
Substituting Eq.(5.3) into Eq.(5.4), the mass spectra for different states of heavy
quarkonia as a function of temperature is given by,
3C 6D ( √ ( )m −A+ 3C 22 + 8D3M = 2m+B+ + − δ
δ δ2 ( δ ))2
(2n+ 1)± 1 + 4mC + 12mD 23 4 + 4 l(l +N − 2) + N −4N+3δ δ 4
(5.7)
The mass spectra of different states of heavy quarkonia as function of temper-
ature are plotted in Fig. 5.10 - 5.18.
92
4100
4000
3900
3800
1P
3700
1S
1.0 1.5 2.0 2.5 3.0 3.5
TTc
(a)
10 400
10 350
10 300
10 250
10 200
1P
10 150 1S
0.5 1.0 1.5 2.0
TTc
(b)
Figure 5.10: The mass spectra of charmonium and bottomonium as a function of
temperature T for 1S and 1P states (AEIM)
93
MHMeVL
MHMeVL
4000
3900
3800
3700
1P
3600 1S
0.5 1.0 1.5 2.0 2.5
TTc
(a)
10 180
10 170
10 160
10 150
10 140 1S
10 130 1P
0.5 1.0 1.5 2.0 2.5 3.0 3.5
TTc
(b)
Figure 5.11: The mass spectra of charmonium and bottomonium as a function of
temperature T for 1S and 1P states (PSM)
94
MHMeVL
MHMeVL
4300
4200
4100
4000
3900
3800
3700 1P
1S
3600
1.5 2.0 2.5 3.0
TTc
(a)
10 240
10 220
10 200
10 180
10 160
1P
10 140
1S
10 120
1.0 1.5 2.0 2.5 3.0 3.5
TTc
(b)
Figure 5.12: The mass spectra of charmonium and bottomonium as a function of
temperature T for 1S and 1P states (NU method)
95
MHMeVL
MHMeVL
4200
4000
3800
m=1710 MeV
3600
m=1600 MeV
1.0 1.5 2.0 2.5 3.0 3.5
TTc
(a)
11 000
10 500
m=5050 MeV
10 000
9500 m=4700 MeV
0.5 1.0 1.5 2.0 2.5
TTc
(b)
Figure 5.13: The mass spectra of charmonium as a function of temperature T for
1P state for different charm quark mass and (b) the mass spectra of bottomonium
as a function of temperature T for 1P state for different bottom quark mass (AEIM)
96
MHMeVL
MHMeVL
4200
4000
3800
3600 m=1710 MeV
3400 m=1600 MeV
0.5 1.0 1.5 2.0 2.5
TTc
(a)
11 000
10 500
m=5050 MeV
10 000
9500 m=4700 MeV
0.5 1.0 1.5 2.0 2.5
TTc
(b)
Figure 5.14: The mass spectra of charmonium as a function of temperature T for
1P state for different charm quark mass and (b) the mass spectra of bottomonium
as a function of temperature T for 1P state for different bottom quark mass (PSM)
97
MHMeVL
MHMeVL
4200
4000
3800
m=1710 MeV
3600
m=1600 MeV
3400
1.5 2.0 2.5 3.0
TTc
(a)
10 200
m=5050 MeV
10 000
9800
9600
m=4700 MeV
9400
1.0 1.5 2.0 2.5 3.0 3.5
TTc
(b)
Figure 5.15: The mass spectra of charmonium as a function of temperature T for
1P state for different charm quark mass and (b) the mass spectra of bottomonium
as a function of temperature T for 1P state for different bottom quark mass (NU
method)
98
MHMeVL
MHMeVL
4300 10 400
4200 N=3N=4 10 350 N=3N=44100 N=5 N=5
4000 10 300
3900 10 250
3800 10 200
3700 10 150
1.0 1.5 2.0 2.5 3.0 3.5 0.5 1.0 1.5 2.0
TTc TTc
(a) (b)
4400 N=3 10 600
N=4
4200 N=5 10 500
10 400 N=3
4000 N=4N=510 300
3800 10 200
1.0 1.5 2.0 2.5 3.0 3.5 0.5 1.0 1.5 2.0 2.5
TTc TTc
(c) (d)
Figure 5.16: (a)The mass spectra of charmonium as a function of temperature T
for 1S state for different values of N , (b) the mass spectra of bottomonium as
a function of temperature T for 1S state for different values of N ,(c) the mass
spectra of charmonium as a function of temperature T for 1P state for different
values of N and (d) the mass spectra of bottomonium as a function of temperature
T for 1P state for different values of N (AEIM)
99
MHMeVL MHMeVL
MHMeVL MHMeVL
4200
N=3 10 240 N=3
4100 N=4 N=4
N=5 10 220 N=5
4000
10 200
3900
10 180
3800
10 160
3700
10 140
3600
0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5
TT TTc c
(a) (b)
4300
10 260
4200 N=3 N=3
N=4 10 240 N=4
4100 N=5 N=5
10 220
4000
10 200
3900
10 180
3800
10 160
3700
10 140
3600
0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5
TT TTc c
(c) (d)
Figure 5.17: (a)The mass spectra of charmonium as a function of temperature T
for 1S state for different values of N , (b) the mass spectra of bottomonium as
a function of temperature T for 1S state for different values of N ,(c) the mass
spectra of charmonium as a function of temperature T for 1P state for different
values of N and (d) the mass spectra of bottomonium as a function of temperature
T for 1P state for different values of N (PSM)
100
MHMeVL MHMeVL
MHMeVL MHMeVL
4300
4200 N=3 10 300 N=3N=4
4100 N=4N=5 10 250 N=54000
3900 10 200
3800
3700 10 150
3600
1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 3.5
TT TTc c
(a) (b)
N=3 10 350
4200 N=4 N=3N=5 10 300 N=4
N=5
4000 10 250
3800 10 200
3600 10 150
1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 3.5
TTc TTc
(c) (d)
Figure 5.18: (a)The mass spectra of charmonium as a function of temperature T
for 1S state for different values of N , (b) the mass spectra of bottomonium as
a function of temperature T for 1S state for different values of N ,(c) the mass
spectra of charmonium as a function of temperature T for 1P state for different
values of N and (d) the mass spectra of bottomonium as a function of temperature
T for 1P state for different values of N (NU method)
101
MHMeVL MHMeVL
MHMeVL MHMeVL
5.5 Dissociation temperature of heavy
quarkonium in N-dimensional space
When the binding energy of a quarkonium falls below the temperature, that state
is weakly bound and it can be destroyed by thermal fluctuations by transferring
energy. The dissociation temperature is defined as the smallest temperature where
no resonance structure can be observed in reality. The value of binding energy
need not be reached zero for the dissociation of quarkonia. There are a lot of
previous works which have determined the dissociation temperatures for different
states of quarkonia. In [24] the dissociation temperatures were calculated from
the thermal width. In Ref. [4] the upper bound for the dissociation temperature
of quarkonium states was found by using the condition Γ(T ) ≥ 2Ebin(T ). In [12]
the upper bound and lower bound for the dissociation temperatures have been
calculated by the conditions Ebin = T and Ebin = 3T respectively. In [10, 25] the
condition of vanishing binding energy have used for calculating the dissociation
temperature for different states of heavy quarkonium. In this chapter, we use
the condition Ebin = T to find out the dissociation temperatures for the different
states of quarkonium. We have obtained the dissociation temperatures for different
states of quarkonium using AEIM and NU method and are summarized in Table
5.1, Table 5.2, and Table 5.3.
102
Table 5.1: The dissociation temperature (TD) with Tc = 175MeV for the quarko-
nium states using Mc = 1710MeV and Mb = 5050MeV at N=3
State J/ψ χc ψ
′ Υ χb Υ
′
TD(AEIM) 1.47Tc 1.62Tc 1.62Tc 0.77Tc 0.82Tc 0.82Tc
TD(PSM) 1.46Tc 1.61Tc 1.62Tc 0.79Tc 0.83Tc 0.83Tc
TD(NU) 1.72Tc 1.81Tc 1.84Tc 1.04Tc 1.06Tc 1.07Tc
Table 5.2: The dissociation temperature (TD) with Tc = 175MeV for the quarko-
nium states using Mc = 1710MeV and Mb = 5050MeV at N=4
State J/ψ χc ψ
′ Υ χ ′b Υ
TD(AEIM) 1.54Tc 1.71Tc 1.71Tc 0.8Tc 0.85Tc 0.85Tc
TD(PSM) 1.55Tc 1.72Tc 1.73Tc 0.84Tc 0.85Tc 0.85Tc
TD(NU) 1.77Tc 1.82Tc 1.86Tc 1.07Tc 1.08Tc 1.09Tc
5.6 Results and Discussions
Fig. 5.1 and Fig. 5.4 show the variation of the binding energy of heavy quarkonia
with temperature for 1S and 1P states respectively, where binding energy is cal-
culated using the analytical exact iteration method. Fig. 5.2 and Fig. 5.5 show
the variation of the binding energy of heavy quarkonia with temperature for 1S
and 1P states respectively, where binding energy is calculated using the Power
Series method. Fig. 5.3 and Fig. 5.6 show the variation of the binding energy of
heavy quarkonia with temperature for 1S and 1P states respectively, where bind-
ing energy is calculated using the Nikiforov-Uvarov method. It has been observed
that the binding energy becomes weaker with increasing temperature. This result
Table 5.3: The dissociation temperature (TD) with Tc = 175MeV for the quarko-
nium states using Mc = 1710MeV and Mb = 5050MeV at N=5
State J/ψ χ ′c ψ Υ χb Υ
′
TD(AEIM) 1.63Tc 1.81Tc 1.81Tc 0.83Tc 0.88Tc 0.88Tc
TD(PSM) 1.61Tc 1.83Tc 1.82Tc 0.83Tc 0.82Tc 0.88Tc
TD(NU) 1.8Tc 1.85Tc 1.87Tc 1.08Tc 1.09Tc 1.1Tc
103
agrees with the literature [4, 10]. Fig. 5.7, Fig. 5.8, and Fig. 5.9 show the depen-
dence of binding energy on dimensionality number (N). It is clear from the figure
that the binding energy increases with the dimensionality number. This result also
agrees with [10].
Fig. 5.10, Fig. 5.11, and Fig. 5.12 show the mass spectra of heavy quarkonia
with temperature for 1S and 1P states. It is clear from the figures that the mass
spectra decrease with increasing temperature for 1S and 1P states. The values of
the 1P state are larger than the values of the 1S state. This also agrees with [10].
Fig. 5.13, Fig. 5.14, and Fig. 5.15 show the behavior of mass spectra of J/ψ
state and Υ state with temperature for two different values of constituent quark
mass. The increase of constituent quark mass leads to increasing mass spectra of
charmonium for 1S state. Thus the mass spectra of heavy quarkonia increase with
increasing quark mass. This also agrees with the previous findings [11]. Fig. 5.16,
Fig. 5.17, and Fig. 5.18 show the variation of mass spectra with dimensionality
number. The increase in dimensionality number increases the mass spectra as
in [10]. The values of binding energy and mass spectra of different quarkonium
states agree with the previous reports for the lower dimension. Comparing these
values with the experimental results [26], it can be concluded that out of the
three methods, the Nikiforov-Uvarov method has the minimum error. As the
dimensionality number increases, these values also increase.
In Table 5.1, we have summarized the dissociation temperatures for different
states of cc and bb at N=3. Considering the results obtained via AEIM, we found
that due to color screening the bottomonia states dissolve at temperatures near
the critical temperature and the charmonia states exist up to higher temperatures.
The dissociation temperature of J/ψ, χ ′c, and ψ states agree with [10]. The dis-
104
sociation temperature of Υ′′ agrees with [8,9]. The dissociation temperature of χb
agrees with [9, 17]. The value of dissociation temperature of Υ is slightly lesser
than that reported in some literature [4, 10]. The results obtained through PSM
are comparable with that obtained via AEIM. It is observed that the dissociation
temperature becomes slightly higher when calculated through NU method. The
dissociation temperature of J/ψ state agrees with [27]. The dissociation temper-
ature of χb agrees with [3]. In Table 5.2 and Table 5.3, we have summarized the
dissociation temperatures for different states of cc and bb at N=4 and N=5. The
above three tables display the effect of dimensionality number on the dissociation
temperature of different quarkonium states. It is clear from the tables that the
increasing dimensionality number leads to increasing dissociation temperature for
different quarkonium states.
105
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109
Chapter 6
Properties of bc and cs mesons at
Finite Temperature in Strongly
Coupled Quark Gluon Plasma
6.1 Introduction
In RHIC experiments the production and survival of heavy quarkonia have been
considered as a tool to test the deconfinement for a long time [1]. Initially, these
studies were focused on charmonia and bottomonia only. During the last few
years, a rapid increase in the number of new mesonic states has been observed by
various collider experiments leading to a new focus on hadron spectroscopy [2,3,4].
In p-p collisions, production of cc and bb appears to be more favorable whereas
in A-A collisions production of Bc meson (bc or cb) is more relevant [5]. The
bottom-charmed meson family is the only quarkonium bound state consisting of
two heavy quarks with different flavors (cb for positively charged channel and bc for
110
negatively charged channel) and due to this special feature, we can include these
in the category of heavy-light mesons [6]. It is the heaviest and latest discovered
double heavy meson predicted by the standard model [6]. Unlike the charmonium
and bottomonium states the Bc mesons do not annihilate into gluons and thus
these states are very stable [7]. Recently LHCb presented evidence for three new
excited charm-strange mesons [8,9,10]. These states are important due to their
narrow widths and low masses [11]. experimental information on the spectrum of
the cs meson states is limited.
In this chapter, we try to find out the binding energies and mass spectra of bc
and cs mesons at finite temperature. For studying the temperature dependence of
binding energy and mass spectra of bc and cs mesons, we shall use a nonrelativistic
treatment because of the large constituent quark masses. We also calculate the
dissociation temperatures of these mesons in strongly coupled quark-gluon plasma
for different dimensionality numbers.
6.2 Variation of binding energy of bc and cs mesons
with temperature
The binding energy of bound state of quarks is calculated in the N-dimensional
space for any state at finite temperature using the Nikiforov-Uvarov method [12]
which is given by Eq.(3.62).
Here Mq1 = Mb and Mq2 = Mc for bc meson and Mq1 = Mc and Mq2 = Ms for
cs meson.
The variation of binding energy with temperature for different states of bc and
111
cs mesons is shown in Figures 6.1 and 6.2.
300
1S
250 1P
2S
200
150
100
50
0
1.0 1.5 2.0 2.5 3.0 3.5
TTc
Figure 6.1: Variation of bc binding energy on temperature T
6.3 Mass spectra of quarkonia
For calculating the mass of bc meson the following relation is used [12].
Mbc = Mb +Mc + E
N
nl (6.1)
For calculating the mass of cs meson,
Mcs = Mc +Ms + E
N
nl (6.2)
Substituting from Eq.(3.62) in to Eq.(6.1) and (6.2), the mass spectra for dif-
112
B.E.HMeVL
600
1S
500 1P
2S
400
300
200
100
0
1.0 1.5 2.0 2.5 3.0
TTc
Figure 6.2: Variation of cs binding energy on temperature T
ferent states of bc meson as a function of temperature is given by,
3C 6D
Mbc = Mb +Mc +B + +
δ δ
( (
2 )
√ 22µ −A+ 3C + 8D− δ2 δ3 ) (6.3)2
(2n+ 1)± 1 + 8µC + 24µD 23 4 + 4(l(l +N − 2) + N −4N+3)δ δ 4
and the mass spectra for different states of cs meson as a function of temperature
is given by,
3C 6D
Mcs = Mc +Ms +B + +
δ δ(2( )√ 22µ −A+ 3C 8D− δ2 + δ3 ) (6.4)2
2
(2n+ 1)± 1 + 8µC3 + 24µD4 + 4(l(l +N − 2) + N −4N+3)δ δ 4
113
B.E.HMeVL
Table 6.1: The dissociation temperature (TD) with Tc = 175MeV for the bc meson
states using Mc = 1710MeV and Mb = 5050MeV
State 1S 1P 2S
TD(N = 3) 1.23Tc 1.25Tc 1.25Tc
TD(N = 4) 1.24Tc 1.26Tc 1.26Tc
TD(N = 5) 1.25Tc 1.27Tc 1.27Tc
TD(N = 3)[5] 1.90Tc 1.05Tc 1.04Tc
Table 6.2: The dissociation temperature (TD) with Tc = 175MeV for the cs meson
states using Mc = 1710MeV and Ms = 540MeV
State 1S 1P 2S
TD(N = 3) 1.33Tc 1.35Tc 1.36Tc
TD(N = 4) 1.34Tc 1.36Tc 1.37Tc
TD(N = 5) 1.35Tc 1.36Tc 1.38Tc
6.4 Dissociation temperature of bc and cs
mesons in N-dimensional space
Here we use the condition Ebin = T to find out the dissociation temperatures [13]
for the different states of bc and cs mesons. The dissociation temperatures for
different states are summarized in Tables 6.1 and 6.2.
6.5 Results and Discussions
Fig. 6.1 and Fig. 6.2 show the variation of binding energy of bc and cs mesons with
temperature. As the temperature increases, the binding becomes weaker. This
result agrees with the literature [14]. Fig. 6.3 shows the variation of mass spectra
114
7050 N=3 7050 N=3N=4
7000 N=4N=5 7000 N=5
6950 6950
6900 6900
6850 6850
6800 6800
1.0 1.5 2.0 2.5 3.0 3.5 1.0 1.5 2.0 2.5 3.0 3.5
TTc TTc
(a) bc 1S state (b) bc 1P state
7100
7050 N=3 2800
N=3
N=4 N=4
7000 N=5 2700 N=5
6950 2600
6900 2500
6850 2400
6800 2300
1.0 1.5 2.0 2.5 3.0 3.5 1.0 1.5 2.0 2.5 3.0
TTc TTc
(c) bc 2S state (d) cs 1S state
2900 2900
N=3 N=3
2800 N=4 2800 N=4
2700 N=5 2700 N=5
2600 2600
2500 2500
2400 2400
2300 2300
1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0
TTc TTc
(e) cs 1P state (f) cs 2S state
Figure 6.3: The mass spectra of different states of bc and cs mesons as a function
of temperature T for different values of N
115
MHMeVL MHMeVL MHMeVL
MHMeVL MHMeVL MHMeVL
7200
7000
M =1710 MeV
c M b=5050 MeV
6800
6600
M =1600 MeV  M =4700 MeV
c b
6400
6200
6000
5800 Mc=1400 MeV Mb =4300 MeV
1.0 1.5 2.0 2.5 3.0 3.5
TTc
Figure 6.4: The mass spectra of bc 1S state as a function of temperature T for
different quark mass at N=3
of bc and cs mesons with temperature for different dimensionality numbers. From
the figure, it is clear that the mass spectra decrease with increasing temperature
and increase with the increasing dimensionality number. Fig. 6.4 and Fig. 6.5 show
the behavior of mass spectra of bc and cs mesons with temperature for different
constituent quark masses. The mass spectra increase with increasing quark mass
which is in agreement with the previous findings [14]. In Table 6.1 and Table
6.2, the dissociation temperatures for different states of bc and cs mesons are
summarized. We have found that both bc and cs mesonic states can dissociate at
temperatures above the critical temperature (Tc < TD < 2Tc). Also, the results
are compared with [5]. It is clear from the table that the increasing dimensionality
number leads to increasing dissociation temperatures of bc and cs mesonic states.
116
MHMeVL
2800
2600
2400 Mc=1710 MeV Ms=540 MeV
2200
2000
1800 Mc=1209 MeV Ms=419 MeV
1.0 1.5 2.0 2.5 3.0
TTc
Figure 6.5: The mass spectra of cs 1S state as a function of temperature T at N=3
for different component quark mass
117
MHMeVL
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119
Chapter 7
Conclusions and Future plans
Recent experiments show clear evidence for the strongly coupled nature of QGP
at temperatures near the transition temperature. In this regime, QGP seems to be
close to a strongly coupled Coulombic plasma with an effective coupling constant
≈ 0.5-1 and multiple bound states of quasiparticles. These bound states play
an important role in the thermodynamics and transport properties of the QGP.
Recent lattice works have found that the charmonium states and the mesonic
bound states of light quarks can survive upto T ≤ 2Tc. In this work, we have
conducted a study to explore the properties of bound states in strongly coupled
quark-gluon plasma.
7.1 Conclusions
The major outcomes of the present work are summarized as follows.
ˆ The screening potential of a color charge entering into the high-density
strongly coupled quark-gluon plasma is calculated using the hydrodynamic
120
approach.
ˆ The screening potential is plotted as a function of r for different values of
α. For smaller values of α the attractive potential vanishes. For α ≥ 1 , the
screening is remarkable. A minimum in the potential shows the formation
of bound states.
ˆ The temperature dependence of the screening potential between quarks in
the high density strongly coupled quark-gluon plasma is investigated. A
minimum in the potential near Tc shows the formation of bound states. The
screening potential may lead to the expected suppression of J/ψ mesons.
ˆ In this work, we have calculated the screening attractive potential to study
the formation of bound states of quarks in a simple manner.
ˆ We report the solution of the N-dimensional radial Schrödinger equation in
which two-body potential in SCQGP is generalized. The energy eigenval-
ues have been calculated in N-dimensional space for any state with quan-
tum numbers (n, l). The binding energy for the bound states of quarks is
calculated by solving Schrödinger’s equation using this attractive potential
in three different methods. i.e., analytical exact iteration method (AEIM),
power series method (PSM), and Nikiforov-Uvarov (NU) method. Thus three
expressions for the binding energy of the bound state of quarks have been
obtained.
ˆ Using these results, in this work, we have calculated the masses of diquarks
using the attractive screening potential in the high-density strongly coupled
quark-gluon plasma.
121
ˆ The results are applied for studying the properties of quarkonia (charmonium
and bottomonium). The effect of temperature on binding energy and mass
spectra of heavy quarkonia are studied. The dependence of binding energies
of 1S and 1P charmonium and bottomonium on temperature are studied.
The result is in agreement with the existing literature.
ˆ The effect of dimensionality number on binding energy and mass spectra is
also studied and compared it with previous works.
ˆ We have determined the dissociation temperature for different states of heavy
quarkonia at finite temperature and compared with the previous works.
ˆ Also, the variations in binding energy and mass spectra of bc and cs mesons
with temperature are studied and we reported the dissociation temperatures
of bc and cs mesons.
ˆ By comparing the numerical results with experimental values, we can observe
that out of the three mathematical methods, the NU method has minimum
error.
7.2 Future plans
In the present work, we proposed a new attractive potential and properties of
bound states are studied using this potential.
ˆ In this work, only the screening effect is considered. So in future, the present
work can be extended to study all the collective effects in strongly coupled
quark-gluon plasma including waves and instabilities.
122
ˆ Recent trends indicate that the properties of heavy and light mesons can be
studied by solving the relativistic Schrödinger equation. So in future, the
relativistic Schrödinger equation can be solved within the framework of the
new screening potential.
ˆ It is possible to investigate the transport coefficients of SCQGP.
ˆ Also, the magnetized QGP can be studied in this model.
123