#### Abstract

In this study, we investigated the impact of a topological defect () on the properties of heavy quarkonia using the extended Cornell potential. We solved the fractional radial Schrödinger equation (SE) using the extended Nikiforov-Uvarov (ENU) method to obtain the eigenvalues of energy, which allowed us to calculate the masses of charmonium and bottomonium. One significant observation was the splitting between nP and nD states, which attributed to the presence of the topological defect. We discovered that the excited states were divided into components corresponding to , indicating that the gravity field induced by the topological defect interacts with energy levels like the Zeeman effect caused by a magnetic field. Additionally, we derived the wave function and calculated the root-mean radii for charmonium and bottomonium. A comparison with the classical models was performed, resulting in better results being obtained. Furthermore, we investigated the thermodynamic properties of charmonium and bottomonium, determining quantities such as energy, partition function, free energy, mean energy, specific heat, and entropy for P-states. The obtained results were found to be consistent with experimental data and previous works. In conclusion, the fractional model used in this work proved an essential role in understanding the various properties and behaviors of heavy quarkonia in the presence of topological defects.

#### 1. Introduction

The thorough description of hadron properties has become a significant issue in particle and nuclear physics. There have been many attempts to improve chiral quark models to calculate hadron properties, such as [1–5]. Additionally, these models have been extended to quark-gluon plasma in hot or dense mediums, as discussed in [6–8].

By the end of the 1970s, in the existence of grand unification theories (GUT), Kibble predicted that a sequence of spontaneous symmetry breaks would occur in our universe during the cooling phase after the Big Bang [9, 10]. Phase transitions were present in conjunction with these symmetry breaks. According to the Kibble-introduced mechanism, these phase transitions would typically have produced some topological flaws. These flaws are areas of space where energy is concentrated at a very high density. Their spatial dimensions define their natures. These flaws produce the GUT’s structure of the collection of empty spaces that the Higgs fields can access. Numerous topological defects, including point defects known as monopoles [9, 11], linear defects or cosmic strings [9, 10, 12], surface defects or domain walls, and combinations of these defects have been examined in the past. Since topological flaws are frequently persistent, it is quite likely that some of them have persisted, possibly even up to the present. Cosmology can detect topological flaws. The dynamics of our universe would be swiftly dominated by monopoles and domain barriers since they are so enormous, which is completely at odds with the observations. Cosmological consequences of cosmic strings could be proven by present and future research, and they are consistent with current observations. The geometry of cosmic strings is flat everywhere except for the symmetry axis, making them the most notable topological defects [9, 11, 13]. They were created early in the universe’s history. The variety of theories produced from general relativity theory is a very strong incentive to investigate how particles behave on these geometrical structures, despite the lack of any fundamental evidence for their existence. The importance of quantum systems in space-times with linear defect geometry should, therefore, receive special attention, as they are thought to constitute the most significant topological flaw in our universe [12]. We can list, for instance, the compression of matter during a moving string’s period and temperature variations in the cosmic microwave background (CMB) as key effects of the hypothetical presence of cosmic strings. A cosmic string’s gravitational field has peculiar characteristics. A quantum particle at rest near a static, straight line with infinite dimensions would not be drawn to it and would not experience any local gravitational fields as a result. This indicates that close to a cosmic string, space is flat locally [11]. It is not a worldwide flatness, though; it is local. A cosmic string can have a range of effects on quantum particle dynamics because of its particular shape, including the generation or destruction of the (e+, e-) pair [14] and the bremsstrahlung process [15] in the vicinity of a static nucleus. We become interested in the characteristics of hadrons in such spaces under the influence of a central field as a result of these cosmic string-induced fluctuations of quantum observables.

Theoretical high-energy physics (HEP) and related sciences heavily rely on the study of the thermodynamic features of quantum systems [16]. Quarks and gluons typically stay contained in hadrons, especially in the protons and neutrons that make up the atomic nucleus. A plasma of quarks and gluons with poorly understood thermodynamic properties can be produced by compressing or heating nuclei. A theory that explains the strong interaction is called quantum chromodynamics [17, 18]. Several experimental studies have been conducted in recent years to discover the presence of deconfinement transitions [18]. A bound state of charm and anticharm quarks called the , which is suppressed, was then suggested as a potential indicator of the QCD phase change [17, 18]. In-depth research on the thermodynamic characteristics in the case of heavy quarkonia would be intriguing, given that the suppression of the meson has been identified as the signature of the deconfinement transition. In [19], the authors employed the quark-gluon plasma component quasiparticle model to derive the thermodynamic parameters of the system. Then, utilizing chiral quark models, thermodynamic properties are examined in [20, 21].

With an appropriate transformation, the Schrödinger equation could be transformed into a generalized hypergeometric equation. Consequently, the NU approach can be applied to this type of second-order differential equation with effectiveness. In Ref. [22], a spherically harmonic oscillatory ring-shaped potential is proposed, and its exact complete solutions are presented by the Nikiforov-Uvarov method. As in Ref. [23], the bound states of the Schrödinger equation for a second Pöschl-Teller-like potential are obtained exactly using the Nikiforov-Uvarov method. It is found that the solutions can be explicitly expressed in terms of the Jacobi functions or hypergeometric functions.

In this paper, we derive the SE solutions caused by the gravitational field of a for all values of the orbital angular momentum quantum number . The radial Schrödinger equation in the cosmic string background is solved analytically using the generalized fractional extended NU method. Then, we obtain the results to calculate the mass of charmonium and bottomonium, the root-mean radii, and the thermodynamic properties of heavy quarkonia from cosmic string geometry which are not considered in previous works in the framework of fractional nonrelativistic quark models.

This paper is organized as follows: Section 2 reviews the generalized fractional derivative of the ENU method. We obtain the solution for the SE in the cosmic string background for the extended Cornel potential with the generalized fractional derivative of the extended NU method in Section 3. The mass of quarkonia, root-mean radii, and thermodynamic properties of charmonium and bottomonium inside linear defect geometry are described in Section 4. Conclusions are provided in Section 5.

#### 2. The Generalized Fractional Derivative of the Extended Nikiforov-Uvarov (GFD-ENU) Method

ENU method is a generalization of the Nikiforov-Uvarov method. Both are frequently used in quantum physics to determine the eigenvalue and eigenfunctions of the Schrödinger or Dirac equations, as well as any other equations that need to be transformed into a hypergeometric form for a full analysis; for further information, see Refs. [24, 25]. To demonstrate the method viability, the NU was successfully applied in a few practical examples in Ref. [26], where it was generalized to conformable fractional derivative. The goal of this section is to expand the ENU within the confines of the GFD. Take into account the generalized fractional differential equations shown in the standard below [27]: where , , and are polynomials with degrees of no more than second, third, and fourth, respectively. Then, using GFD [28], we can write where and . From Eqs. (2) and (3), we get

By comparing Eqs. (1) and (4), we get the fractional parameters Then, we obtain the standard equation (GFD-ENU):

We assume that the following transforms Eq. (6):

Eq. (6) is reduced to a hypergeometric equation. Then, we get where archive

is the hypergeometric function that has a polynomial that archives the Rodrigues relation: where is the constant for normalization and is the weight function and satisfies the relation The function is defined as

is a polynomial of at most 2 degrees, and based on this, the determination of is important in obtaining . is defined by relation where

We obtain the eigenvalues of energy from Eq. (10) and Eq. (14).

#### 3. Cosmic String Space-Time of Heavy Quarkonia

In spherical coordinates, the line element that shows the linear defect space-time [10] is obtained by (, , , and ) where , , , is the topological parameter of the cosmic string, is the torsion [29] parameter, and represents the cosmic string’s linear mass density. From general relativity (GR), we note that the values of vary in the interval [29, 30].

For and , the metric obtained by Eq. (16) reduces to the usual Minkowski metric in spherical coordinates [31, 32].

The metric tensor for the space-time given by Eq. (16) is

With the inverse metric,

We choose the signature for the metric tensor , and its determinant is obtained by , with . In the system of curvilinear coordinates such that , , and , the 3-dimensional interior Euclidian space’s metric tensor is

The Laplace-Beltrami (LB) operator of the local coordinate system may be described as

Next, considering Equation (20) and for small values of the torsion parameter , the LB operator is

The Hamiltonian operator in natural can be written from this. as:

is the reduced mass, where are the mass of quark and antiquark [26, 33].

Now, in curved cosmic string space-time, the nonrelativistic radial SE is presented in detail [12, 29]. where with and the quantum number for generalized angular orbits is It is not necessarily the case that the generalized quantum numbers and are integers:, where .

Suppose two heavy quarks in a bound state like the and with the potential consisting of the Cornell potential plus harmonic potential. The linear part is responsible for quark confinement at large distances, while the Coulomb part dominates at short distances. This potential has been extensively studied in both relativistic and nonrelativistic quantum mechanics and has attracted a great deal of attention in particle physics in which the additional part was added to improve confinement force, and we found that it gave a good result compared with other works and experimental results [34].

To put Eq. (23) in the dimensional fractional form, we let , , and , where GeV. is separation constant . Then we get where , , , , and .

By using Eq. (3), we get where , , , , , and . where , , , , and .

By comparing Eq. (6) and Eq. (27), we get

Then, we get where . We choose a linear function that produces the functions under the root in the above equation to be quadratic . Then,

Comparing Eq. (29) and Eq. (30), we obtain

From Eq. (10), we obtain

From Eq. (14), we obtain

We have four different combinations of sign choices ++, +−, −+, and −−. We chose ++ to obtain the eigenvalue of energy and the eigenfunction as in Ref. [26]. We obtain the energy eigenvalue from Eq. (33) and Eq. (34):

From Eq. (9), we obtain the function : where , , and are obtained from Eq. (31). From Eq. (12), we obtain the function :

Then, we obtain the function :

From Eq. (7), we obtain the fractional radial wave function: where is the normalization constant.

#### 4. Results and Discussions

##### 4.1. Special Cases

The classical case is obtained ; then, . The eigenvalues of energy and wave function in the classical model are

##### 4.2. Mass of Heavy Quarkonia

We obtain the mass of both charmonium and bottomonium from the relation

In Table 1, the mass of has been calculated for 1S, 2S, 3S, 4S, 1P, 2P, and 1D for different values of topological defect . We have calculated the Schrödinger equation by using the generalized fraction (ENU) method, and we obtained the mass of under the effect . The potential parameters , , and are fitted using Eq. (41) where , , and , and the quark mass is obtained from Refs. [26, 35]. We obtained the mass of in two models. The classical model at with topological defect (). We obtained a good result compared with previous works. In addition, some of the states of charmonium are close with experimental data. We calculated the total error which equals to 0.018% in the classical model. In the fractional model, the total error is smaller than the classical model which equals to 0.016%. In the present work, the topological defect parameter takes values between . The topological defect produces a splitting in the mass spectrum of the meson to break the degeneracy in the nP and nD states, which was absent in earlier studies. In Ref. [36], the authors studied the -radial Schrödinger equation by using the asymptotic iteration method for the quark-antiquark interaction potential, and they calculated the mass spectra of heavy quarkonia, so we calculated the total error of this work which equals to 0.9513%. In Ref. [37], the authors solved the Schrödinger equation by using the Nikiforov-Uvarov method for the sum of a harmonic, a linear, and a Coulomb interaction potential, and they calculated the mass spectra of heavy quarkonia, so we calculated the total error of this work which equals to 0.11065%. In Ref. [38], the authors calculated the -radial Schrödinger equation by using the power series iteration method for the quark-antiquark interaction potential, and they calculated the mass spectra of heavy quarkonia, so we calculated the total error of this work that equals to 0.11152%. In Ref. [39], the authors studied the -radial Schrödinger equation by using an exact analytical iteration method for the trigonometric Rosen-Morse of the quark-antiquark interaction potential, and they calculated the mass spectra of heavy quarkonia in which we found the total error that equals to 0.05788%. Therefore, we obtained improved results in comparison with [36–39].

In Table 2, the mass of has been calculated for 1S, 2S, 3S, 4S, 1P, 2P, and 1D for different values of topological defect . We have calculated the Schrödinger equation by using the generalized fraction ENU method, and we get the mass of under effect . The potential parameters , , and are fitted using Eq. (41), where , , and , and the quark mass is obtained from Refs. [26, 35]. We obtained the mass of in two models. The classical model at with topological defect (). We obtained a good result compared with previous works and some of the states of bottomonium that are close to experimental data. We calculated the total error that equals to 0.006% in the classical model which is smaller than previous works. In the fractional model (, ) at , the total error is smaller than the classical model. The topological defect produces a splitting in the mass spectrum of the meson to break the degeneracy in the nP and nD states, which was not considered in earlier studies. In Ref. [36], they studied the -radial Schrödinger equation by using the asymptotic iteration method for the quark-antiquark interaction potential, and they calculated the mass spectra of heavy quarkonia. We found the total error of this research which equals to 0.484%. In Ref. [38], the authors calculated the -radial Schrödinger equation by using the power series iteration method for the quark-antiquark interaction potential, and they calculated the mass spectra of heavy quarkonia. We calculated the total error of this work that equals to 0.0137%. In Ref. [40], the authors studied the -radial Schrödinger equation by using the asymptotic iteration method with the Cornell potential, and we found the total error in the results which equals to 0.028%. Therefore, we obtained improved results in comparison with [36, 38, 40].

In Figure 1, the levels are different from the Minkowski levels, at of P-states of charmonium. In Figure 1(a), we note that the splitting increases as decreases compared to Figure 1(b). This finding is in agreement with Ref. [41]. In addition, we note that the effect of different values of also appeared on the splitting. Since , the degenerated levels appear.

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In Figure 2, we note the P-states of charmonium when . The topological defect does not affect the system and no splitting appears. This is in agreement with Ref. [41], but the generalized fractional parameter plays a role where in Figure 2(a), we take and in Figure 2(b). We note that in Figure 2(b), the curve is higher than the curve in Figure 2(a).

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In Figure 3, we observe the P-states of charmonium when . Since complex eigenenergy appears by the negative sign of , some states do not see it. This is in agreement with Ref. [41]. Also, the generalized fractional parameter plays a role where in Figure 3(a), we take and in Figure 3(b). We note that in Figure 3(b), the curve is higher than the curve in Figure 3(a).

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In Figure 4, the levels are different from the Minkowski levels, as we can see with of P-states of bottomonium. In Figure 4(a), we note that the splitting increases as decreases compared to Figure 4(b) since we took in Figure 4(a) and in Figure 4(b). This is in agreement with Ref. [41]. We note that fractional parameters act on the splitting mass where the degenerated levels appear.

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In Figure 5, we note the P-states of bottomonium when . The topological defect does not affect the system and no splitting appears. This is in agreement with Ref. [41], but the generalized fractional parameter plays a role where in Figure 5(a), we take and in Figure 5(b). We note that in Figure 5(b), the splitting mass is higher than its values in Figure 5(a). In Figure 6, at , the P-states of bottomonium were noted. Some states do not see this since complex eigenvalues of energy appear by the negative sign of . This is in agreement with Ref. [41]. Also, the generalized fractional parameter plays a role where in Figure 6(a), we take and in Figure 6(b). We note that in Figure 6(b), the splitting mass is higher than its values in Figure 6(a).

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##### 4.3. The Radial Expectation Values

The radial expectation values were given by the relation

In Table 3, the radial mean value ((GeV)^{-1}) for various levels of charmonium and topological defect causes to split nP and nD states. The mean values were calculated with the normalized wave function in two cases (first case (, )). The result in this case is acceptable with other works. In the second case (, ), the generalized fractional parameter plays an important role where the results are in agreement with other works [44–47].

In Table 4, the radial mean value (GeV) for various levels of charmonium and topological defect causes to split nP and nD states. The mean value was calculated with the normalized wave function in two models: the classical model (, ) and the fractional model (, ). The present results are in agreement with other works [44–47]. In Table 5, the radial mean value ((GeV)^{-1}) for various levels of bottomonium and topological defect causes to split nP and nD states. The mean value was calculated using the normalized wave function in two models (, ); we note that the effect of the topological defect on S-states does not appear like P-states and D-states. In the fractional model (, ), the generalized fractional parameter plays an important role especially in S-states and P-states and D-states where the results are in agreement with other works [44–46]. Also, we can see that the average value ((GeV)^{-1}) decreases for bottomonium with an increase in the reduced mass compared with charmonium. In Table 6, the radial mean value (GeV) for various levels of bottomonium and topological defect causes to split nP and nD states. The mean values were calculated with the normalized wave function in the classical and fractional models. In the classical model (, ), we note that the effect of topological defect on S-states does not appear like P-states and D-states; the results in this model are in good agreement with other works [4, 44, 45]. In the fractional model (, ), the generalized fractional parameter plays an important role especially in S-states and P-states and D-states. In addition, the results increase for bottomonium compared with charmonium.

##### 4.4. Thermodynamic Properties of Heavy Quarkonia

The partition function serves as the foundation for thinking about the thermodynamic properties of heavy quarkonia inside the cosmic string framework [48, 49]. According to statistical mechanics, the partition function can be built as follows.

###### 4.4.1. Partition Function

where , the Boltzmann constant is , and the system’s absolute temperature is .

In Figures 7 and 8, we plot the partition function of P-states of charmonium and bottomonium as a function of at different values from under the effect of topological defect. We note that by increasing the values of , the curve becomes higher. In Figures 7(a) and 8(a), we take the topological defect . We note that the curves are not connected to the standard Minkowski curve, and we also note that the splitting increases as decreases which is in agreement with [50–52] and decreases with increasing which is in agreement with [50]. The authors of [51] within the framework of the NU approach have used DFDEP to solve the Klein-Gordon equation, and they obtained the energy eigenvalues and associated wave function in dimensions in great detail and noticed that the for the Minkowskian case monotonically lowers with increasing *β*. In Ref. [52], the partition function decreases monotonically with increasing for the two diatomic molecules considered and reaches a constant value for some typical values of . In Ref. [53], the author plotted the partition function of P-states of charmonium and bottomonium as a function of at different values of topological defect, and they show that by increasing the topological defect, decreases with increasing . In Ref. [50], based on the generalized Dunkl derivative in quantum mechanics, we study the one-dimensional Schrödinger equation with a harmonic oscillator potential and obtain the energy eigenvalues. For the principal thermodynamic properties, they obtain that the decreases with . In Figures 9 and 10, we plot the free energy of P-states of charmonium and bottomonium as a function of at different values from under the effect of topological defect. We note that with increasing the values of , the curve becomes higher. In Figures 9(a) and 10(a), we took the topological defect , and we note that the curves are not connected to the standard Minkowski curve. We note also that the splitting increases as decreases and free energy increases with increasing , and we note that free energy decreases by increasing the temperature. In Ref. [54], they show that decreases by increasing the temperature of neutral particles. This is in agreement with [48, 49, 53, 54]. In Figures 11 and 12, we plot the mean energy of P-states of charmonium and bottomonium as a function of at different values of under the effect of topological defect. We note that by increasing the values of , the curve becomes lower values. In Figures 11(a) and 12(a), we took the topological defect and note that the curves are separated from the classical Minkowski curve. We note also that the splitting increases as decreases and mean energy () decreases with increasing , and with increasing the parameter of the topological defect, its values shift to lower values. In Ref. [54], they calculated the thermodynamic properties of neutral particles under the effect of topological defect from the nonrelativistic Schrödinger-Pauli equation. The authors noted that by increasing the temperature, increases. In Ref. [55], the authors applied their proposed proper quantization rule to obtain the energy spectrum of the modified Rosen-Morse potential. The beauty and symmetry of this proper rule come from its meaning whenever the number of the nodes of or the number of the nodes of the wave function increases by one; the momentum integral will increase by . Based on this new approach, they present a vibrational high-temperature partition function to study thermodynamic functions and they found that mean energy () decreases with increasing . In Ref. [50], the authors found that the mean energy () decreases with increasing . This is in agreement with [50–55]. In Figures 13 and 14, we plot the specific heat () of P-states of charmonium and bottomonium as a function of at different values from topological defect under effect . We note that with increasing the topological defect, the specific heat decreases. This is in agreement with [56]. In Figures 13(b) and 14(b), we take . We note that by increasing , the curves became lower as in Figures 13(a) and 14(a), and we note that the specific heat increases with temperature such that in Ref. [55]. In Figures 15 and 16, we plot the entropy () of P-states of charmonium and bottomonium as a function of at different values from topological defect under effect . We note that there is no influence by topological defect parameters like the solution of Minkowski space-time. This is in agreement with [53, 54]. In Figures 15(b) and 16(b), we take . We note that by increasing , the curves became lower as in Figures 15(a) and 16(a). We note that the entropy decreases with . This is in agreement with [50, 55].

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###### 4.4.2. Free Energy

###### 4.4.3. Mean Energy

###### 4.4.4. Specific Heat

###### 4.4.5. The Entropy

#### 5. Conclusion

In this paper, we solved the radial Schrödinger equation in the cosmic string with the extended Cornell potential by using a generalized fractional derivative of the extended Nikiforov-Uvarov (GFD-ENU) method under the effect of topological defect. We obtained eigenvalues of energy and wave functions of heavy quarkonia in the fractional model. Firstly, we obtained the special cases from the general model, and then, we calculated the mass of charmonium and bottomonium in the two models (the first case is the classical model and the second is the fractional model) under the effect of topological defect; it has been noted that the splitting between nP and nD states is caused by the presence of the topological defect. The excited states are split into components, suggesting that a topological defect’s gravity field interacts with energy levels in a way that is comparable to the Zeeman effect brought on by the magnetic field where the results are close with experimental data and in good agreement with other works such as Refs. [36–40]. We note that generalized fractional derivative plays an important role in this work since we obtain lower total error in calculating the mass of heavy quarkonia. As we noted from Tables 1 and 2, the states of S are in very good agreement with experimental data; in particular, 1S state is close with the data. Also, the states 1P, 2P, and 1D are in very good agreement with the data. We also obtained the wave function; then, we calculated the root mean of charmonium and bottomonium in two models under the effect of topological defect, and the results are in agreement with Refs. [44–47]. The mass and thermodynamic properties were analyzed graphically. Concerning the classical limit, the thermodynamic quantities exhibit a shift. When the cosmic string parameter is low, this fluctuation becomes more important. Free energy increases with increasing and we note that free energy decreases by increasing temperature; we note also that the splitting increases as decreases, and this is in agreement with Refs. [48, 49, 53, 54]. By increasing the values of , the curve becomes higher. The shifts to lower values as decreases; this is in agreement with Refs. [51–53], and by increasing the values of , the curve becomes higher values. The Minkowski curve is separated from the curves with ; then, by raising the , the mean energy values are changed to lower values. This is in agreement with Refs. [50–55]. By increasing , the curves became lower. We note that by increasing the topological defect, the specific heat decreases; this is in agreement with [56]. Also, by increasing , the curves became lower. We note that there is no influence by topological parameters on the entropy, and this is in agreement with [53, 54]. By increasing , the curves became higher values. We hope to extend this work to include spin-spin, spin-orbital, and tensor interactions in future work.

#### Data Availability

All data are included in the manuscript.

#### Ethical Approval

The authors declare that they comply with ethical standards regarding the content of this paper.

#### Disclosure

The version of the work has been submitted [57].

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the Sponsoring Consortium for Open Access Publishing in Particle Physics.