Effects of Gravitational Field of a Topological Defect on Statistical Properties of Heavy Quarkonia.

In this paper, we determine the behavior and statistical properties of non-relativistic heavy quarkonia interacting with the extended Cornel potential within a space-time generated by a cosmic-string. We extend the Cornel potential by adding the inverse square potential plus the quadratic potential. We have calculated the energy eigenvalues and the corresponding eigenstates using the Extended Nikiforov-Uvarov (ENU) method. Then, the thermodynamic properties like partition function, entropy, free energy, mean energy and speciﬁc heat capacity are calculated within the space-time of a cosmic-string. In the next step, we investigate the inﬂuence of the cosmic-string parameter on quantum states of heavy quarkonia and their statistical properties.


Introduction
The study of quantum particles under the influence of a potential in a curved space-time with Topological defect has inspired a large number of research in recent years [1,2,3,4,5,6] with different motivations. Kibble [4,5] was the first to state the mechanism which would have led to the appearance of topological defects in the universe.
According to grand unification theories (GUT) [4,5,6], the conditions which reigned in the very early moments of the universe were such that the four fundamental interactions were one [4,5,7]. A few moments after the Big-Bang, the universe began to expand and became less and less hot and its density decreased. [7,8]. From the initial phase where the grand unification symmetries would be realized, a spontaneous breaking of these symmetries would then have taken place and at this moment, stable topological defects were formed.
Thus, a large variety of topological defects have been studied in previous works, namely: point defects [4,5,6,7,8] called monopoles, linear defects or cosmic-strings [4,5,7], surface defects or "domain walls" [9], and also combinations between these different defects [10,11,12]. These faults are often stable; so it is absolutely possible that some of them would have survived, until now [9]. However, cosmic strings are compatible with cosmological models [6] and current observations. Their cosmological effects could be confirmed with current and future work. It is therefore quite logical to give particular interest to the study of quantum systems on curved space-time with cosmic-string background. In [6], the author presents cosmic-strings as the most important topological defect in our universe.
Their effects on the environment are mainly gravitational in nature. [6,10,11,12]. To this end, a quantum particle placed in a gravitational field will be influenced by the topology of the space and by its interaction with the local curvature. From this interaction, it follows that an observer at rest with respect to the particle will see a shift in its spectrum. This would be of considerable interest both from a theoretical and an observational point of view. Apart from the effects due to the gravitational field, other effects can be induced by a cosmic-string, like for example bremsstrahlung process [13], the creation of the (e + , e − ) pair [14], gravitational leasing [15] and the gravitational Aharonov-Bohm effect [16].
The study of thermodynamic properties of quantum systems plays a major role in theoretical High Energy Physics (HEP) and related fields [17]. Usually, quarks and gluons remain confined in hadrons and in particular in the protons and neutrons which form the atomic nucleus. Indeed, Quantum Chromodynamics (QCD), a theory that accounts for the strong interaction predicts that by compressing or heating nuclei, it is possible to create a plasma of quarks and gluons [18,19],a matter with poorly understood thermodynamic properties. In past few decades, some experimental works have been carry out to identify the existence of deconfinement transitions [19] and determine its signatures. Then, it was predicted that the suppression of charmonium 1 meson is a possible signature of the QCD phase transition [18,19,20]. This phenomenon of deconfining matter is in principle observable through the study of the J/ψ particle [20], a meson as heavy as three surrounding protons. The suppression of the cc meson has been announced as being the signature of deconfinement transition, it would be interesting to study in depth the thermodynamic properties of heavy mesons, in this case heavy quarkonia.
In [21], the author obtained thermodynamic properties of the quark-gluon plasma from the constituent quasi particle model of quark-gluon plasma. Then in [22,23,24], thermodynamic properties are investigated using chiral quark models. In this work, we obtain the solutions of the Schrödinger equation produced by the gravitational field of a topological defect, for all values of the orbital angular momentum quantum number l.
The Extended Nikiforov-Uvarov method [25,26] is used to obtain exact analytical solutions of the radial Schrödinger equation in cosmic-string background. Next, we apply the results to obtain the thermodynamic properties of heavy quarkonia from cosmic-string geometry, which is not consider in other works. This paper is organized as follows : in Section 2, the Extended Nikiforov-Uvarov method is reviewed. The bound state solutions for the extended Cornel potential are obtained for the Schrödinger equation in cosmic-string background in Section 3. Then in section 4, thermodynamic properties of heavy quarkonia within cosmic-string geometry are presented. In Section 5 results are discussed, and conclusions are presented in Section 6.

Basic Concepts of the Extended Nikiforov-Uvarov Method
In this section, we briefly present the Extended Nikiforov-Uvarov (ENU) technique, for more details see [25,26,27].
The extended Nikiforov-Uvarov method is obtained by changing the boundary conditions of the standard Nikiforov-Uvarov (NU) method and is used for solving any second order differential equation which has at most four singular points. The equation to be solved is of the form: such that τ (r), σ(r) and σ(r) are polynomials of at most second,third and fourth degree, respectively. By choosing ψ such that: where φ(r) and Y (r) are functions to be determined later. Moreover, considering the above substitution Eq. (1) reduces to an equation of the form, where φ(r) solves the equation And the polynomials of the function Y (r) satisfy the Rodriguez formula [27] Y n (r) = B n ρ(r) In Eq.(6), B n is the normalization factor and ρ is the density function. The function ρ solves the following equation: (σ(r)ρ(r)) ′ = τ (r)ρ(r).
The function π(r) required for this method are given by: where π(r) is a second degree polynomial. The function h n (r) is determined from the equation where τ (r) = τ (r) + 2π(r) and C n is an integration constant. In the order to obtain the eigenvalue solution of the problem by ENU method [25,26,27], the polynomials h(r) and h n (r) are equal and Y (r)=Y n (r) is a particular solution of degree n of Eq.(3). The function φ(r) is defined as a logarithmic derivative given in Eq.(4), and then eigenfunction spectra can be obtained analytically. For more details one can see Refs. [26] and [27].

Schrödinger wave equation in the space-time of a cosmic string
In spherical coordinates, the line element describing the cosmic string space-time [6] is given by where 0 < r < ∞, 0 < θ < π, and 0 < ϕ < 2π, 0 < α = 1 − 4J is the topological parameter of the cosmic string, χ = 4GJ c 3 is the torsion [23] parameter and J denotes the linear mass density [30,31] of the cosmic string. From General Relativity (GR), we know that the values of J varies in the interval J ∈ ]0, 1[. For α −→ 1 and χ −→ 0, the metric given by Eq.(10) reduces to the usual Minkowski metric in spherical coordinates. The metric tensor for the space-time described by Eq.(10) is: with the inverse metric, We adopt the signature (−, +, +, +) for the metric tensor g µν , and its determinant is given by g = det(g µν ) = −α 2 r 4 sin 2 θ, with µ, ν = 0, 1, 2, 3. In the curvilinear coordinates system The Laplace-Beltrami operator of the local coordinates system can be expressed as: Then, considering Eq. (14) and for small values of the torsion parameter χ << 1, the Laplace-Beltrami operator becomes: from this we can write the Hamiltonian operator in Atomic Units (h = c = 1) as: .
is the reduced mass of the quark-antiquark system, where m q and mq are the mass of quark and antiquark respectively [8,26].
The non-relativistic Schrödinger equation in curved background space-time is [16,23] ih Here, we consider bound states of two heavy quarks interacting with the extended Cornel potential [26,32] The first term is a Coulomb-like potential due to one-gluon exchange processes between quark-antiquark [32], the second term is a linear confinement term. The additional part was added to improve quark-antiquark properties [26,32,33]. As V = V (r), the following commutation relations must be satisfied Consequently, L z and L 2 are good quantum numbers to describe the quantum states of the system under the influence of this potential. Inserting the new form of Ψ as and substituting in Eq. (17) gives the following set of second-order differential equations, where δ and m 2 are the separation constants. Using the following boundary condition : We can easily obtain the solution of Eq. (23) as To find the solutions of Eq. (22), we introduce the variable η = cos θ, which map Eq. (22) into To obtain suitable solutions of Eq.(25), it is therefore necessary to analyze the way the solutions behave arround singular points, namely ±η. Then we assume the following form for the solution : where g l is analytic everywhere except at η → ±∞. From Eq.(25) and Eq. (26), and introducing the following generalized quantum numbers, namely l (α) , m (α) = m α , from the cosmic-string geometry, we obtain the following equation: From the expansion of g l (α) (η) into a power series, and inserting in Eq. (27) gives us the following recurrence relation: a n+2 = n(n − 1) + 2(m (α) + 1)n − δ + m (α) (m (α) + 1) (n + 1)(n + 2) a n where n = 0, 1, 2, 3, ... For physically acceptable solutions, the series must be truncated at a certain value of n: l (α) being the generalized angular orbital quantum number. The generalized quantum numbers l (α) and m (α) are not always integers. l (α) = m (α) + n = m α + n = l − 1 − 1 α m, where l = 0, 1, 2, ... Eq. (25) can therefore take the form: where Eq. (32) Let's now turn to the radial equation.
Substituting the proposed potential in Eq. (21) and choosing the separation constant being δ = l (α) (l (α) + 1), the radial equation becomes: that we put in a simple form by introducing the following constants: The functions from the Nikiforov-Uvarov method are : Using equation Eq.(10), π(r) is found as : Assuming that the term under the square root of the above expression becomes quadratic, Let's take the function G(r) in the form G(r) = Ar+B. By comparison between Eq.(39) and Eq.(40), we obtain the following four sets of solutions for the unknowns α 2 , α 1 , α 0 , A and B in terms of the problem parameters: I : Considering Eq.(4), we have two solutions for φ(r) which are given by: The wave function will be physically acceptable if α 1 < 0, α 2 < 0 and α 0 > 0 and then the set II of parameters is the suitable one in the determination of the eigenvalue and the eigenfunction of the problem.
As the functions h(r) and h n (r) are equal, we use Eq.(5) and Eq. (9) to obtain the energy equation as where C n is an integration constant which couples with the parameters of potential. Eq.(43) leads us to have two distinct choices ++ and −−, which are given by I and II as follows The choice of −− and set II results on the relation A + 2(n + 1)α 2 = 0 from which we can write : Using Eq. (35) and Eq.(45), we obtain the expression below: We note that the presence of the cosmic string parameter α breaks the degeneracy in all the states with l = 0.

Thermodynamic properties of the qq system
To consider the thermodynamic properties of heavy quarkonia within cosmic-string framework, the starting point is the partition function [34,35]. From a statistical mechanical point of view, the partition function can be constructed as follow: Here β = k B T , where k B is the Boltzmann constant and T is the absolute temperature of the system.
Once the partition function is obtained, we can have the Helmholtz free energy as: From the Helmholtz free energy we can obtain the other statistical quantities in a straightforward manner.
The entropy of the system can be obtained as Then from the internal energy we can compute the specific heat capacity at constant volume C v as

Results and discussion
In this section, we discuss the effect of cosmic-string geometry on thermodynamic properties and spectrum of heavy quarkonia. Fig.1,2  In Fig.7, we have plotted the canonical partition function for both cc and bb quarkonia in terms of β for different values of the cosmic-string parameter α. It is clearly seen that by decreasing the value of α, all the curves are separated from the classical Minkowski curve (α = 1). For all the values of the topological parameter the behavior is the same but with different magnitudes. Moreover, the splitting increases as α decreases and then the partition function decreases with increasing β which is in agreement with [36,37]. In [36], the author applied the deformed five parameter exponential potential and observed that with increasing β, the canonical partition function for the Minkowskian case decreases monotonically.
In Fig.8, The Helmholtz free energy is plotted as a function of β. The effect of the cosmicstring geometry clearly appears when α takes values different from α = 1 (classical Minkowski space-time). It is found that the Helmholtz function increases monotonically with increasing β, and the curves with α = 1 are separated from the classical Minkowski curve. Moreover, the splitting is important for small values of α. In [34], the author treated quark-gluon plasma as composed of light quarks only, which interact weakly. They observed that the Helmholtz free energy decreases as the temperature increases. In [35], Modarres and Gholizade calculated the Helmholtz function of a neutral particle and observed that the free energy of the system decreases as the temperature increases. In the present model, the behavior of charm and bottom quark matter is in agreement with [34,35,38].
In Fig.9, the internal energy is plotted against β for different values of α. It reveals that, U (β) decreases with increasing β, and its values shift to lower values by increasing the parameter of the topology α.
In [38], the authors studied thermodynamic properties of neutral particles in a cosmic-string background from non-relativistic Schrödinger-Pauli equation. They observed that the internal energy of the system increases as the temperature increases. Thus, the conclusion of the present work for internal energy is the same with recent works [36,37,38].
The entropy S(β) and the specific heat capacity C v (β) are plotted against β respectively in

Conclusion
In this work, we have investigated the effects of the gravitational field of a topological defect on thermodynamical properties of heavy quarkonia. We have used the extended Nikiforov-Uvarov method to solve the radial Schrödinger wave equation in the space-time of a cosmic string, for the potential V (r) = − K r + ar + g r 2 + br 2 . We obtain the radial wave functions of cc and bb mesons as well as the energy eigenvalues, which are shifted from the usual Minkowski energies. From this, we have considered all the statistical quantities using the canonical partition function of the system. The wave functions and the thermodynamic properties were analyzed graphically. It was observed that the peaks of radial functions of probability densities are shifted toward the origin as α decrease. The thermodynamic quantities present a shift compared to the classic limit, this difference becomes more important for small values of the cosmic-string parameter. The partition function shifts to lower values as α decrease, the