THERMAL PROPERTIES AND MASS SPECTRA OF HEAVY MESONS IN THE PRESENCE OF A POINT-LIKE DEFECT 1

In this research, the radial Schrödinger equation is solved analytically using the Nikiforov-Uvarov method with the Cornell potential. The energy spectrum and the corresponding wave function are obtained in close form. The effect of Topological Defect on the thermal properties and mass spectra of heavy mesons such as charmonium and bottomonium are studied with the obtained energy spectrum. It is found that the presence of the Topological Defect increases the mass spectra and moves the values close to the experimental data. Our results agreed with the experimental data and are seen to be improved when compared with other works.

157 Thermal Properties and Mass Spectra of Heavy Mesons in the Presence...

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thermal properties of the heavy mesons by employing the Nikiforov-Uvarov approach to solve the Schrodinger equation with the Cornell potential.For convenience, we have assumed that our heavy mesons are spinless particles [42,57,[68][69][70].This is because most potentials with spin addition cannot be solved analytically, necessitating the employment of numerical methods like the Runge-Kutte approximation [71], Numerov matrix approach [72], Fourier grid Hamiltonian method [73], and so on [74].
II.THE MODEL For space time with a point-like global monopole (PGM), the line element that explains it, takes the form [75] Where α is the parameter related to the PGM which depends on the energy scale . Furthermore, Eq. ( 1) portrays a space time with scalar curvature.On inserting the potential under consideration, we have ( ) In this way, the Schrodinger equation takes the form 2 2 ( , ) ( , ) ( , ) ( , ) 2 LB r t r t V r t r t i t where μ is the particle's mass, ( ) , is the Laplace-Beltrami operator and ( , ) ( ) V r t V r = is GMP (1).Thereby, the Schrodinger equation for the GMP in a medium with the presence of the PGM (1) Where the 1 2 ( ) / V r w r w r = − is the Cornell potential employed for modeling the quarkonium interaction [70].This model has been greatly utilized for this purpose in recent past by numerous researchers.
Here, let us consider a particular solution to Eq. ( 5) given in terms of the eigenvalues of the angular momentum operator 2  L as , ( ) ( , , , ) ( , ) where , ( , ) Y θ ϕ are spherical harmonics and ( ) R r is the radial wave function.On substitution of Eq. ( 6) into Eq.( 4), the radial part of the Schrodinger equation for the Cornell potential in the presence of TD is obtained as follows Eq. ( 6) is not solvable in its present form, Eq. ( 6) needs to be transformed from r x → coordinate using the following , on application, Eq. ( 6) is rewritten as follows: The approximation scheme on the term 1 w x is introduced by assuming that there is a characteristic radius 0 r of the meson.The approximation scheme is achieved by the expansion of 1 w x in a power series around 0 r ; i.e. around 0 ~1 / r δ ≡ ,up to the second order [56].By setting y x δ = − and around 0 y = we have; EEJP. 1 (2024) Etido P. Inyang, et al. where The equation above is expressed in the form solvable by the Nikiforov-Uvarov formalism.The major equation closely related with this method is given in the following form; 2 ( ) ( ) ( ) ( ) ( ) 0 ( ) ( ) The following is obtained;  which shows explicitly that our Eq.( 8) satisfies the requirement of the Nikiforov-Uvarov approach.It is worthy to point out also that ( ), ( ) x and x σ σ  are polynomials of at most second degree, and ( ), x τ is at most a polynomial of first degree.The Nikiforov-Uvarov method is a really popular method amongst mathematical scientist and related discipline.Several authors have used this method to solve similar problems of interest [10][11][12][13][14][15][16][17][18][19][20].Even through the method is quite popular, it will be useful to highlight some details, so as to make our paper self-contained.For this reason, this will be detailed in the appendix.Following the steps outlined in the appendix (Eqs.(A1-A7), the energy equation and radial wave function are obtained as follows and

THERMAL PROPERTIES OF THE CORNELL POTENTIAL IN THE PRESENCE OF DEFECT
We introduce the partition function Z(β), which provides a measure of thermally accessible states, to explore the thermal properties of the heavy mesons.It can be determined by adding together all possible energy states.Following the Boltzmann-Gibbs distribution, Z(β) is given by the relation [76]; max 0 ( ) Where 1 kT and with k is the Boltzmann constant.Substituting Eq. ( 10) in Eq. ( 12), summing over all accessible energy levels, we obtain the partition function as follows: ( , , ) Where the following non-dimensional parameters have been defined for simplicity; On successful evaluation of the partition function, several other thermodynamic variables can be obtained by using the following;

DISCUSSION AND RESULTS
The prediction the mass spectra (MS) of HQS such as charmonium and bottomonium is carried out using the following relation [77,78].
where µ is quarkonium mass and E nℓ is energy eigenvalues.Substituting Eq. ( 10) into Eq.( 14) gives, The numerical values of bottomonium and charmonium masses are µ b = 4.823GeV and µ c = 1.209GeV, and the corresponding reduced mass are µ˜b = 2.4115GeV and µ˜c = 0.6045GeV respectively [79].The potential parameters were fitted with experimental data (ED) [80].This was achieved by solving a simultaneous equation for α equals to 0.1,0.2 and 1 respectively.The mass spectra of the heavy mesons were predicted in the absent and present of the topological defect for different quantum states.In the case of charmonium predictions for 1S and 2S states we noticed that the prediction were accurate with the experimental data in the present and absent of the topological defect.In 3S and 4S, it was noticed that as the topological defect increased to 0.2 the value of the mass spectra was very close to the experimental data.A similar trend was noticed in 1P,2P,1D and 2D states when the topological defect was introduced and the predicted values were close to the experimental data and was seen to be improved from works reported by [27,30,42] as shown in Table 1.In the case of bottomonium, it was observed that for 1S, and 2S quantum states the mass spectra were all equal to the experimental data and works reported by [27,30,42] as shown in Table 2.It was noticed that for 3S and 4S states, a significant change in the mass spectra was observed when the topological defect was set to 0.1 and 0.2.A similar trend was observed with other predicted states when the topological defect was increased as shown in Table 2.We observed that the results obtained from the prediction of the mass spectra of charmonium and bottomonium for different quantum states are in agreement with the experimental data and are improved with the reports of [27,30,42].The thermal properties for charmonium are plotted as shown in Fig. 1(a-e).In Fig. 1 (a), the partition function for topological defect = 0.1 shows a linear increase when the temperature is increased.When topological defect is equal to 0.2 and 1.0, the partition function is seen to decrease with an increase in temperature; same behavior is reported by Abu Shady et al., [55] and Kumar et al., [58].In Fig. 1 (b), the free energy (FE) is plotted against temperature, we noticed that as the topological defect increases from 0.1 to 1.0, the free energy increases, which is in agreement with the experimental data.In Fig. 1 (c), the entropy of the system for charmonium is plotted.It is observed that when topological defect is equal to 0.1 and 0.2, the entropy is seen to decrease as the temperature increases, but when topological defect = 1.0, we noticed a steady entropy as the temperature is increased.In Fig. 1 (d), the internal energy (IE) is plotted as a function of temperature.When topological defect = 1.0, a steady internal energy is noticed, but for topological defect = 0.1 and 0.2 an exponential decrease is observed followed by the internal energy of the system being steady when the temperature increases.Abu-Shady et al., [55] reported a decrease with increasing of temperature and maximum quantum number, our trend is on the expected line.

Table I. Mass spectra of Charmonium in (GeV). The following fitting parameters has been employed
In Fig. 2 (a-e), the thermal properties of bottomonium is plotted as shown.In Fig. 2 (a), the partition function is plotted as a function of temperature.When the topological defect = 0.1 a linear increase is noticed.For topological defect = 0.2 and 1.0, a slight increase and no increase on the partition function is seen respectively.In Fig. 2 (b), the free energy of the bottomonium is plotted against the temperature.We observed as the topological Thermal Properties and Mass Spectra of Heavy Mesons in the Presence...

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defect increases, the free energy is seen to increase.A similar observation was reported by Abu-Shady et al [55] and Kumar et al [58].In Fig. 2 (c), the entropy is plotted against temperature for different values of topological defect, it was noticed that the entropy decreases with an increase in temperature.In [55,58] the authors found the entropy decreases with increasing temperature.In Fig. 2 (d), the inter energy plots shows that when topological defect = 1.0, no increase in the internal energy is noticed, but for topological defect = 0.1 and 0.2 a slight decrease is seen at the beginning then followed a constant value in the internal energy.In Fig. 2 (e), the specific heat capacity is plotted against temperature.A sharp increase in specific heat capacity is noticed for topological defect = 0.1 and 0.2 and later converges at a point when the m specific heat capacity = 1.An exponential increase is noticed when topological defect = 1.CONCLUSION In this study, the effect of the topological defect on the mass spectra of heavy mesons is studied with the Cornell potential.The Schrodinger equation was solved analytically using the Nikiforov-Uvarov method.The approximate solutions of the energy spectrum and wave function in terms of Laguerre polynomials were obtained.We apply the present results to predict the mass spectra of heavy mesons such as charmonium and bottomonium in the present and absent of the topological defect for different quantum states and its thermal properties.We noticed that when the topological defect increases the mass spectra and moves closer to the experimental data.However, the results obtained showed an improvement when compared with the work of other researchers.EEJP. 1 (2024) Etido P. Inyang, et al.
The discriminant of the quadratic expression under the square root above is given as; Recalling the expression for ( ), x τ we obtain the expression for ( ), x τ and its derivative respectively as follows; From Eq. (A7) and (A6), we have the following; Eq.(B3) yields the energy equation of the Cornell potential presented in Eq. (10).