1 Introduction

It has been concluded from the ultra-relativistic heavy-ion collision experiments at RHIC and LHC that the quark-gluon plasma (QGP), which was created, behaves more like a perfect fluid than a non-interacting ultra-relativistic gas of quarks (anti-quarks) and gluons. The reason for this is because the QGP have a strong collective characteristic that may be measured in terms of the flow harmonics. In addition, the other significant signals based on experimental data, quarkonia suppression has also been proposed as a direct indicator of QGP creation in the collider studies. The tests show that it highlights the plasma properties of the medium, such as Landau damping, colour screening, and energy loss, see Ref. ^[1] and references therein.

At finite temperature, the evolution of the Schrodinger equation (SE) is vital. In order to study the creation of a hot quark-gluon plasma, Matsui and Satz ^[2] calculated the charmonium’s J/Ψ radius. The properties of quarkonia in a thermal QCD medium in the background of strong magnetic field is studied using SE. Using a temperature-dependent potential deduced from lattice gauge computations. Wong ^[4] has investigated the binding energies and wave functions of heavy quarkonia in quark-gluon plasma. As a result, the study shows that the model with the modified Q-Q potential produces dissociation temperatures that are consistent with spectral function analyses. Also, the N-radial Schrodinger equation is analytically solved. The Cornell potential is extended to finite temperature and/or chemical potential as in Refs. ^{[5- 9]} in which the energy eigenvalues and the wave functions are calculated.

A heavy quark potential at finite temperature was created by correctly introducing a dielectric function that encodes the effects of a deconfined medium to the full Cornell potential, not just its Coulomb part. This means that the heavy quark potential calculation has been expanded to include plasma with limited momentum-space anisotropy. For this case, specifically, the exact component has been calculated. As long as the quark-gluon plasma exhibits local momentum-space anisotropies for any finite shear viscosity, study of anisotropic plasma in the momentum space is necessary. These momentum-space anisotropies can continue for a very long period and can be rather significant depending on the strength of the shear viscosity, especially early on or close to the plasma’s edges. This holds true for both strong and weak coupling shear viscosity levels, and increasing viscosity causes maximum momentum-space anisotropies to grow ^{[10- 14]}.

Recently, the fractional calculus has attracted attention in the different fields of physics. In high energy physics, the description of heavy-quarkonium energy spectra and complex phenomena of the standard model as in Ref. ^[15], in which, the author used the conformable fractional derivative to express the fractional radial SE in the N-dimensional space for the extended Cornell potential by using extended Nikiforov-Uvarov (ENU) method to the fractional domain. In Ref. ^[16] the fractional form of the NU method is applicable in order to solve fractional radial SE with its applications on variety of potentials such as the oscillator potential, Woods-Saxon potential, and Hulthen potential. The generalized fractional derivative ^{[17, 18]} is suggested which is successfully applied in calculating quarkonium properties and molecular chemical properties as in Refs. ^{[19- 22]}.

The aim of the present work is to study the dissociation of quankonium in a hot-dense medium, in which the baryonic chemical potential is included in the framework of the generalized fractional derivative which are not considered in the recent works. The generalized fractional of the Nikiforov-Uvarov (CF-NU) method is applied to obtain the analytic solutions of the N-dimensional radial SE, then the results are applied on the investigation of binding energy and dissociation temperature.

The paper is organized as follows: In Sec. 2, the GF-NU method is briefly explained. In Sec. 3, The energy eigenvalue and wave function are calculated in the N-dimensional space using GF-NU method. In Sec. 4, the results are discussed. In Sec. 5, the summary and conclusion are presented.

2 The generalized fractional NU method

In this section, the GF-NU method is briefly given to solve the generalized fractional of differential equation which takes the following form (see Refs. ^{[16, 17, 23]}, for details)

where σ(s) and σ~(s) are polynomials of maximum second degree of α and 2α, respectively and τ-(s) is a polynomial of maximum degree of α where

where

where 0<α≤1 and 0<β≤1. Substituting by Eqs. 2) and (3) into (1), we obtain

where, τ-f(s)=1-αs-ασ(s)+I-2τ-(s),σf(s)=s1-ασ(s),σ~f(s)=I-2σ~(s). To find the particular solution of Eq. (4) by separation of variables, if one deals with the transformation

it reduces to an equation of hypergeometric type as follows

where

and

χ(s)=χn(s) which is a polynomial of n degree which satisfies the hypergeometric equation, taking the following form

where B _n is a normalization constant and ρ(s) is a weight function which satisfies the following equation

and

the πf(s) is a polynomial of first degree. The values of K in the square-root of Eq. (12) is possible to calculate if the expressions under the square root are square of expressions. This is possible if its discriminate is zero.

3 The Generalized Fractional of Schrödinger Equation

In this section, the SE is solved in the longitudinal-traverse plane as in Refs. ^{[3, 13]} and references therein so, the SE for two particles interacting via the potential Vr,ξ in the N-dimensional space, where r is inter-particle distance, is given by ^[24]

where L,N, and μ are the angular momentum quantum number, the dimensionality number and the reduced mass for the quarkonium particle (for charmonium μ=mc/2 and for bottomonium μ=mb/2), respectively. Setting the wave function Ψ(r)=R(r)r(1-N/2), the following radial SE is obtained

where Vr,ξ is the interaction potential given in Eq. (23). By substituting by Eq. (23) into Eq. (28), we obtain

By taking r =1/x, Eq. (29) takes the following form

The expansion of a ₁/x in a power series around the characteristic radius r ₀ of meson up to the second order is used as in Ref. ^[13]. The following equation is obtained

where,

To transform Eq. (31) into a fractional form, one uses dimensionless form by taking y = Ax where A equals 1 GeV.

where

By using Refs. ^{[16, 17]} one can put Eq. (33) in the following form:

and substituting Eqs. (2) and (3) into (35), we obtain

where

Hence, the Eq. (36) satisfies Eq. (4). Therefore, Eq. (12) takes the following form after substituting by Eq. (37),

The constant K is chosen such as the function under the square root has a double zero, i.e., its discriminant equals zero. Hence,

Substituting by Eq. (39) into Eq. (38), we obtain

The positive sign in Eq. (38) is determined as in Ref. ^[15]. By using Eq. (8), we obtain

and using Eq. (14), we obtain

From Eq. (9), λ=λn. The energy eigenvalues of Eq. (36) in the N-dimensional space are

with

The radial of wave function takes the following form

where C _nL is the normalization constant that is determined by ∫‍RnLrα2dr=1. We note that the radial wave function in Eq. (45) does not explicitly depend on the number of dimensions. Hence, ∫‍RnLr2dr=1 remains unchanged.

4 Discussion of results

In this section, the above results are applied to the quarkonium masses. The quarkonium mass considering fraction-order and dimensionality in the hot-dense medium is ^[29]

where m is quarkonium bare mass for the charmonium or bottomonium mesons. By using Eq. (42), we write Eq. (45) as in Eq. (46). One can obtain the quarkonium masses at zero temperature by taking T=μb=ξ=0⇒ D(T,μb) at α=β=1 and N = 3. Therefore, Eq. (46) takes the following form

Eq. (47) coincides with Ref. ^[30], in which the authors obtained the quarkonium mass at zero temperature and α=β=1 at N = 3.

In the present analysis, various quantities have been obtained, and the results are plotted while considering the weak anisotropy in the hot QCD plasma with the fixed critical temperature T _c = 0.17 GeV. We considered ζ=-0.3 for prolate and ζ=0.3 for oblate, whereas for the isotropic case, we have ζ=0. As discussed earlier, the dissociation temperature has been obtained by employing the following criterion: The temperature at which the binding energy equals and causes dissociation of quarkonia is the dissociation temperature ^[25]. In Fig. 1, we note that the binding energy Eb=Vr→∞-Enl decreases with increasing temperature at finite baryonic chemical potential ub=0.3GeV and α=β=0.5. In addition, the curves shift to upper values by increasing anisotropic parameter ζ that is agreement with Ref. ^[1], in which authors investigated the behavior of binding energy in the hot medium without including baryonic potential. In Fig. 2, a comparison between the classical model at α=β=1 and the fractional model at α=β=0.5, we note that the binding energy takes a similar behavior in which the binding energy decreases with increasing temperature. Also in the fractional model, we note that the binding energy shifts to lower values and thus, the fractional parameter will be affected on the dissociation temperature in hot-dense medium. A similar situation takes place for bottominum energy in the fractional model at α=β=0.5 and in the classical model at α=β=1 as in Figs. 3 and 4. In Fig. 5, the binding energy for charmonium is plotted in 3D as a function of temperature and baryonic chemical potential in the classical model at α=β=1. We note that the effect of temperature on the binding temperature is more affected in comparison with the effect of baryonic chemical potential. We note that the binding energy decreases slightly by increasing baryonic chemical potential. A similar situation is obtained in the fractional model at α=β=0.5 but we note that the binding has a smaller values. In Figs. 7 and 8 the binding energy of bottomonium is plotted in 3D. We note that the temperature is more effect on the binding energy than the baryonic chemical potential in the classical fractional model.

Figure 1 The binding energy of 1S state for CC- is plotted as a function of ratio tempearture for different values of ζ at ub=0.3 GeV in the fractional model α=β=1/2.  

Figure 2 The binding energy of 1S state for CC- is plotted as a function of ratio tempearture for the classical model at α=β=1 and the fractional model α=β=1/2 at u _b = 0.3 GeV and ζ=0.3.  

Figure 3 The binding energy of 1S state for bb- state is plotted as a function of ratio tempearture for different values of ζ at u _b = 0.3 GeV in the fractional model at α=β=1/2.  

Figure 4 The binding energy of 1S state for bb- is plotted as a function of ratio tempearture for the classical model at α=β=1 and the fractional model α=β=1/2 at u _b = 0.3 GeV and   ζ=0.3.  

Figure 5 The binding energy of 1S state for CC- is plotted as a function of T and u _b at α=β=1 and ζ=0.3.  

Figure 6 The binding energy of 1S state for CC- is plotted as a function of T and u _b at α=β=0.5 and ζ=0.3.  

Figure 7 The binding energy of 1S state for bb- is plotted as a function of T and u _b at α=β=1 and ζ=0.3.  

Figure 8 The binding energy of 1S state for bb- is plotted as a function of T and u _b at α=β=0.5 and ζ=0.3.  

The dissociation temperature for both the classical model and the fractional model is shown in Tables I and II.

For the 1S and 2S of charmonium and bottominum, we observe that the dissociation temperature increases as the anisotropic parameter increases. Additionally, we observe that 2S state has smaller values than those in the 1S state. The fractional model is used for obtaining the results in Table II to compute the dissociation temperature at α=β=0.5. We see in Figs. 2 and 4 that the fractional parameter decreases the binding energy cause the dissociation temperature is lower than the classical model at α=β=1. This result is qualitatively consistent with Ref. ^[1], where the authors used another technique to determine the dissociation temperature by the temperature at which twice the binding energy (real part) equals the thermal width. Additionally, the results of the fractional model qualitatively agree with those of Ref. ^[25], in which the authors determined the dissociation temperature by the temperature at which it equals the real part of the binding energy, leading quarkonia to dissociate.

5 Summary and Conclusion

In this study, we examined the binding energy and dissociation temperature in the fractional nonrelativistic model (FNM). The NU technique is analytically used to solve the FNM. The fractional model of the eigenvalues of the energy and wave functions is obtained. As a result, we determined the special cases that were consistent with the classical model. Additionally, we soon realize that the fractional form’s binding energy is lower than it was in the classical model; as a result, the dissociation temperature in a hot-dense medium decreases in comparison to the classical model. We found the dissociation temperature was agreement with other studies, such as Ref. ^[25]. Also, we note that the effect of baryonic chemical potential is slightly less pronounced on binding energy. This finding is not considered in other works in the framework of the fractional model. In addition, the excited states are obtained for charmonium and bottominum.

Finally, we found that the fate of heavy-quarkonia states in the hot-dense QCD medium is significantly influenced by both the anisotropy and the hot-dense QCD medium effects present in EoS when fractional calculus is taken into account.