COMPARATIVE STUDY OF THE MASS SPECTRA OF HEAVY QUARKONIUM SYSTEM WITH AN INTERACTING POTENTIAL MODEL †

,


INTRODUCTION
The study of the fundamental or constituent blocks of matter has been for long time a fascinating field in Physics.In the nineteenth century, the atom was considered to be the fundamental particles from which all matters were composed.This idea was used to explain the basic structure of all elements.Experiments performed at the end of the nineteenth century and beginning of the twentieth century provided evidence for the structure of an atom [1].
The conclusions were that all atoms have a nucleus containing protons which is surrounded by elements and that the nucleus was very small compared with the size of the atom.The neutron was introduced to explain the discrepancy between the mass of the atom and the mass from the number of protons.In 1932, Chadwick discovered the neutron and the fundamental particles were considered to be proton, the neutron and the electron.The discovery of antimatter in cosmic radiation supported the theory developed from the special theory of relativity and quantum theory that all fundamental particles have corresponding antimatter particles.The matter and antimatter particles have the same mass but opposite charge.The problem of what were considered to be fundamental particles was resolved by the quarks.Quarks are the basic building blocks of hadrons, particles interacting with each other through strong interaction [2,3].In nuclear physics, we are mostly concerned with the lightest members of the hadron's family; nucleons, which make up all the nuclei and pions which constitute the main carriers of nuclear force.Since their discoveries, investigation of heavy quarkonium system (HQS) provides us with great tools for quantitative tests of quantum chromodynamics (QCD) [4].Because of the heavy masses of the constituent quarks, a good description of many features of these systems can be obtained using non-relativistic models, where one assumes that the motion of constituent quarks is non-relativistic, so that the quark-antiquark strong interaction is described by a phenomenological potential [5,6].Heavy quarkonium system have turned out to provide extremely useful probes for the deconfined state of matter because the force between a heavy quark and anti-quark is weakened due to the presence of gluons which lead to the dissociation of quarkonium bound states [7].The quarkonia with heavy quark and antiquark and their interaction are well described by the Schrodinger equation (SE).The solution of the Schrodinger equation with spherically symmetric potential is of major concern in describing the mass spectra (MS) of quarkonium system [8,9].In simulating the interaction potentials for these systems, confining-type potentials are generally used.The holding potentials can be of any form.For instance, a variety of this type of potential is the Cornell potential (CP) with two terms one of which is responsible for the Coulomb interaction of the quarks and the other correspond to a confining term [10,11].Researchers have studied the MS of heavy and heavylight quarkonia using the CP and its extended form [12][13][14].For studying the behavior of several physical problems in Physics, we require to solve the Schrodinger equation.The solutions to the Schrodinger equation can be established with analytical methods such as the Nikiforov-Uvarov (NU) method [15][16][17][18][19][20][21][22][23][24], the asymptotic iterative method (AIM) [25], the extended NU method [26], the Nikiforov-Uvarov functional analysis (NUFA) method [27][28][29][30], the series expansion EEJP. 3 (2023) method [30][31][32][33][34], the WKB approximation [35][36][37], and so on [38].Recently, the study of MS of HQS with exponentialtype potentials has attracted the attention of most researchers.For example, Inyang et al. [39] studied the MS of HQS with Yukawa potential using the NU method.Also, Akpan et al. [40], presented the mass spectra of HQS using Hulthen and Hellmann potential model through the solutions of the Schrodinger equation.Furthermore, Ibekwe et al. [41] studied the mass spectra of HQS using the NU with the combination of screened Coulomb and Kratzer potential.Abu-Shady and Inyang [42], suggested trigonometric Rosen-Morse potential as the quark-antiquark interaction potential for studying the masses of heavy and heavy-light mesons.In the present research, our interest is to compare the mass spectra of HQS with the class of Yukawa potential (CYP) using the Nikiforov-Uvarov and the series expansion methods.The CYP is a combination of Yukawa potential [43], Hellmann potential [44] and inverse quadratic Yukawa potential [45].The CYP applications cut across other fields of physics such has atomic, nuclear and condensed matter physics, among others.The CYP takes the form [46], r ) where , a b and c are potential strengths, I α is the screening parameter.
The exponential terms in Eq. ( 1) are expanded with Taylor series up to order three, so that the potential can interact in the quark-antiquark system, and Eq. ( 2) is obtained. where

The Nikiforov-Uvarov method
The Nikiforov-Uvarov (NU) method is based on solving the hypergeometric-type second-order differential equations by means of the special orthogonal functions [47].For a given potential, the Schrodinger-like equations in spherical coordinates are reduced to a generalized equation of hypergeometric-type with an appropriate coordinate transformation r x → and then they can be solved systematically to find the exact solutions.The main equation which is closely associated with the method is given in the following form [48].
The most useful demonstration of Eq. ( 8) is The new parameter π(x) is a polynomial of degree at most one.In addition, the term which appears in the coefficient of ( ) y x in Eq. ( 5) is arranged as follows, In this case, the coefficient of ( ) y x is transformed into a more suitable form by taking the equality given in Eq.( 31); where Substituting the right-hand sides of Eq. ( 6) and Eq. ( 11) into Eq.( 5), an equation of hypergeometric-type is obtained as follows; 2 As a consequence of the algebraic transformations mentioned above, the functional form of Eq. ( 4) is protected in a systematic way.If the polynomial ( ) x σ in Eq. ( 13) is divisible by ( ) x where λ is a constant, Eq. ( 13) is reduced to an equation of hypergeometric-type And so its solution is given as a function of hypergeometric-type.To determine the polynomial ( ) x π , Eq. ( 12) is compared with Eq. ( 14) and then a quadratic equation for ( ) where The solution of this quadratic equation for π(x) yields the following equality In order to obtain the possible solutions according to plus and minus of Eq. ( 18), the parameter k within the square root sign must be known explicitly.To provide this requirement, the expression under the square root sign has to be the square of a polynomial, since ( ) x π is a polynomial of degree at most one.In this case, an equation of the quadratic form is available for the constant k .Setting the discriminant of this quadratic equal to zero, the constant  is determined clearly.After determining k , the polynomial ( ) x π is obtained from Eq. ( 18), and then ( ) x τ and λ are also obtained by using Eq.( 8) and Eq.( 17), respectively.A common trend that has been followed to generalize the solutions of Eq. ( 15) is to show that all the derivatives of hypergeometric-type functions are also of the hypergeometric-type.Equation ( 15) is differentiated by using the representation 1 ( ) where 1 ( ) ( ) ( ) τ is a polynomial of degree at most one and  is a parameter that is independent of the variables.It is clear that Eq. ( 19) is an equation of hypergeometric-type.By taking 2 ( ) ( ) v x y x ′ ′ = as a new representation, the second derivative of Eq. ( 15) becomes In a similar way, an equation of hypergeometric-type can be constructed as a family of particular solutions of Eq. ( 15) by taking ( ) ( ) And here the general recurrence relations for ( ) n x τ and n μ are found as follows, respectively, ( ) When 0 n μ = , Eq. ( 25) becomes as follows And then Eq. ( 23) has a particular solution of the form ( ) ( ) n y x y x = which is a polynomial of degree n.To obtain an eigenvalue solution through the NU method, the relationship between λ and n λ must be set up by means of Eq.( 17) and Eq.( 26).( ) n y x is the hypergeometric -type function whose polynomial solutions are given by the Rodrigues relation where  is a normalization constant and the weight function ( ) x ρ must satisfy the condition below ( )

The series expansion method
The series expansion method is based on solving the hypergeometric-type second-order differential equations.For a given potential the wave function of SE is chosen in the form.
where     parameters whose values are to be determined in terms of potential strength parameters.The functional series for () is taken to be 2 0 (r) where  is an expansion coefficient [48].
By substituting ( ), ( ) into the SE, rearranging and equating coefficients of the corresponding powers of  to zero.The eigen-values are subsequently obtained.

APPROXIMATE SOLUTIONS OF THE SCHRODINGER EQUATION WITH CLASS OF YUKAWA
POTENTIAL USING THE NU METHOD The Schrodinger equation takes the form [49] ( ) ( )
where , l is the angular momentum quantum number, , μ is the reduced mass for the quark-antiquark particle, r is the inter-particle distance and  is reduced plank constant respectively.Substituting Eq.( 2) into Eq.(31) gives, Transforming the coordinate of Eq.( 32) we set Using Eqs. ( 32) and (33) we have ( ) Next, we propose the following approximation scheme on the term 1 x α and 2 2 x α .
Let us assume that there is a characteristic radius  of the meson.Then the scheme is based on the expansion of 1 x α and 2 2 x α . in a power series around 0 r ; i.e. around 0 1 r δ ≡ , in the x-space up to the second order.This is similar to Pekeris approximation, which helps to deform the centrifugal term such that the modified potential can be solved by the NU method [12].Setting y x δ = − and around 0 y = it can be expanded into a series of powers we obtain; and Putting Eqs. ( 35) and (36) into Eq.( 34) and simplifying gives ( 1) Comparing Eq. (37) and Eq. ( 4) we obtain ( ) 2 , ( ) , ( ) Substituting Eq. (39) into Eq.( 18) gives ( ) To determine k , we take the discriminant of the function under the square root.

EXACT SOLUTIONS OF THE SCHRODINGER EQUATION WITH CLASS OF YUKAWA POTENTIAL USING THE SEM
We consider the radial Schrodinger equation of the form [50] ( ) where l is angular quantum number taking the values 0,1,2,3,4…, μ is the reduced mass for the quarkonium particle, and r is the internuclear separation.Putting Eq. ( 2) into Eq.( 51) gives where
( ) From Eq. ( 54) we have ( ) From Eq. ( 29), Eqs.( 56) and (57) are obtained ( ) Substituting Eqs. ( 29), ( 56) and (57) into Eq.( 51) and divide through by Also, from Eq. ( 30), we obtain the following We substitute Eqs. ( 30),( 59) and (60) into Eq.(58) and obtain ) ( ) By collecting powers of r in Eq. ( [ ] Equation ( 62) is linearly independent implying that each of the terms is separately equal to Zero, noting that r is a non- zero function; therefore, it is the coefficient of r that is zero.With this in mind, we obtain the relation for each of the terms.

Determination of the potential strength parameters
The reduced mass μ is defined in the standard way as 2 respectively.In the same vain, the parameters for charmonium and bottomonium of the Eq. ( 73

Discussion of results
The mass spectra of charmonium and bottomonium for class of Yukawa potential for the NU and the SEM were calculated as shown in Tables 1and 2 respectively using Eqs.( 72) and (73).The free parameters are fitted with experimental data.In addition, quark masses are obtained from Ref. [55].We note the spectra masses of charmonium from states 1s,2s, 3s and 2p from both the NU and the series expansion methods agree with experimental data and 1s,2s,3s and 4s states for bottomonium agree with experimental data for both methods as shown in Tables 1 and 2. Other states appear to be close with experimental data, but the SEM solutions appear to be very close to experimental data for charmonium and bottomonium compared to the NU method.It was noticed that in the 1f state for charmonium and 2d and 1f states for bottomonium the values of the experimental data are not available.The mass spectra obtained agree with Ref. [12].Our results are improved in comparison with works of other researcher like Ref. [12] as shown in the Tables in which the author investigated the N-radial SE analytically.The Cornell potential was extended to finite temperature.The energy eigenvalue and the wave functions were calculated in the N-dimensional form using the NU method.Also, the mass spectra obtained using Eqs.(72) and (73) are improved in comparison with the works of Ref. [13] in which they studied the N-dimensional radial Schrodinger equation using the analytical exact iteration method, in which the Cornell potential is generalized to finite temperature and chemical potential.

CONCLUSION
In this work, the Schrodinger equation is analytically solved using the Nikiforov-Uvarov and series expansion methods with the class of Yukawa potential.The approximate solutions of the eigen energy equation and corresponding eigenfunction in terms of Laguerre polynomials were obtained using the NU method.The solutions of the eigen energy equation were also obtained with the SEM.The mass spectra for heavy quarkonium system for the potential under study were obtained for bottomonium bb and charmonium cc .We adopted the numerical values of these masses as b m = 4.823 GeV for bottomonium and c m = 1.209GeV for charmonium.We compared the results obtained between the Nikiforov-Uvarov and series expansion methods.It was noticed that SEM solutions yield mass spectra very close to experimental data compared to solutions with the NU method.The obtained results were also compared with works by some other authors [12,13] with different analytical methods.The values obtained are improved in comparison with their works.This work can be extended by using other exponential-type potential models with other analytical approach and a different approximation scheme to obtain the mass spectra of heavy quarkonium system.The relativistic properties using Klein-Gordon or Dirac equations can be explored to obtain the mass spectra of light quarkonia.Finally, the information entailed in the normalized wave-functions can also be studied.

2
We calculate mass spectra of the heavy quarkonium system such as charmonium and bottomonium that have the quark and antiquark flavor, and apply the following relation[51][52][53] Mass spectra of the heavy quarkonium, m = Quarkonium bare mass, nl E = Energy eigenvalue.By substituting Eq. (48) into Eq.(71) we obtain the mass spectra for class of Yukawa potential using the NU method as,

mμ=
,where m = mass of the constituent quarks and antiquarks.For bottomonium bb and charmonium cc systems we adopt the numerical values of these masses as b m = 4.823 GeV for bottomonium and c m = 1.209GeV for charmonium [54].Then, the corresponding reduced mass are b μ = 2.4115 GeV and c μ = 0.6045 GeV .The potential parameters of Eqs.(72) and (73) are fitted with experimental data.Experimental data are taken from [55].The parameters for charmonium and bottomonium of the Eq.(72) are 1Joseph A. Obu, et al.

Table 1 .
Comparison of mass spectra of charmonium in (GeV) for the class of Yukawa potential  , , between the NU, SEM, some authors and experimental data

Table 2 .
Comparison of mass spectra of bottomonium in (GeV) for the class of Yukawa potential  , , between the NU, SEM, some authors and experimental dataState , with the NU  , with the SEM