Heavy-Light Mesons in the Non-Relativistic Quark Model Using Laplace Transformation Method

An analytic solution of the N-dimensional radial Schr\"odinger equation with the mixture of vector and scalar potentials via the Laplace transformation method (LTM) is studied. The present potential is extended to include the spin hyperfine, spin-orbit and tensor interactions. The energy eigenvalues and the corresponding eigenfunctions have been determined in the N-dimensional space. The present results are employed to study the different properties of the heavy-light mesons. The masses of the scalar, vector, pseudoscalar and pseudovector of B, Bs, D and Ds mesons have been calculated in the three dimensional space. The effect of the dimensional number space is studied on the masses of the heavy-light mesons. We find that the meson mass increases with increasing dimensional space. The decay constants of the pseudoscalar and vector mesons have been computed. In addition, the leptonic decay widths and branching ratio for the B+, D+ and Ds+ mesons have been studied. Therefore, the present method with the present potential gives good results which are in good agreement with experimental data and are improved in comparison with recently theoretical works.


1-Introduction
One of the most important tasks in non-relativistic quantum mechanics is to find the solution of the Schrödinger equation.The solution of the Schrödinger equation with spherically symmetric potentials plays an important role in many fields of physics such as hadronic spectroscopy for understanding the quantum chromodynamics theory.Numerous studies have been presented to find the solution of Schrodinger equation using different methods as the operator algebraic method [1], path integral method [2], the conventional series solution method [3][4], Fourier transform [5][6], shifted (1/N) expansion [7][8], point canonical transformation [9], Quasi-linearization method [10], super-symmetric quantum mechanics (SUSQM) [11], Hill determinant method (HDM) [12], and other numerical methods [13][14][15].
The study of different properties of heavy-light mesons is very important for understanding the structure of hadrons and dynamics of heavy quarks.Thus, many of theoretical and experimental efforts have been done for understanding the different properties of heavy-light mesons.In Refs.[4, 34, 55], the authors calculated the mass spectra of quarkonium systems as charmonium and bottomonium mesons with the quark-antiquark interaction potential using different methods in many studies.Al-Jamel and Widyan [56] studied the spin-averaged mass spectra of heavy quarkonia with Coulomb plus quadratic potential using (NU) method.Abou-Salem [57] has computed the masses and leptonic decay widths of , , , , and cc bb cs bs bu cb numerically using Jacobi method.The strong decays, spectroscopy and radiative transition of heavy-light hadrons have been computed with quark model predictions [58].The decay constant of heavy-light mesons have been calculated using the field correlation method [59].The Quasipotential approach the spectroscopy of heavy-light mesons have been investigated with the QCD motivated relativist quark model [60].The spectroscopy and Regge trajectories of heavy-light mesons have been obtained with Quasi-potential approach [61].The decay constants of heavy-light vector mesons [62] and heavylight pseudoscalar mesons [63] have been calculated with QCD sum rules.A comparative study has been presented of the mass spectrum and decay properties for the D-meson with the quark-antiquark potential using hydrogeometric and Gaussian wave function [64].In framework of Dirac formalism the mass spectra of D s [65] and D [66] mesons have been obtained using Martin-light potential in which the hadronic and leptonic decays of D and D s mesons have been determined [67], besides the rare decays of B 0 and B 0 s mesons into dimun (μ + μ -) [68] and the decay constants of B and B s have been calculated [69].The mass spectra and decay constants for ground state of pseudoscalar and vector mesons have been computed using the variation analysis in the light quark model [70].The spectroscopy of bottomonium and B-meson have been studied using the free-form smeary in [71].The variational method has been employed to calculate the masses and decay constants of heavy-light mesons in [72], and has been investigated the decay properties of D and D s mesons with quark-antiquark potential in [73].The B and B s mesons spectra and their decays have been studied with a Coulomb plus exponential type potential in [74].The leptonic and semileptonic decays of B meson into τ have been studied [75].The degeneracy of heavy-light mesons with the same orbital angular momentum has been broken using the spin orbit interactions [76].The relativized quark model has been investigated to study the properties of B and B s mesons [77] and the excited charm and charm-strange mesons [78].The perturbation method has been employed to calculate the mass spectrum and decay properties of heavy-light mesons with the combination of harmonic and Yukawa-type potentials [79].In [80], the authors have investigated the leptonic decays of seven types of heavy vector and pseudoscalar mesons.The spectra and wave functions of heavy-light mesons have been calculated within a relativistic quark model by applying the Foldy-Wouthuysen transformation [81].
The isospin breaking of heavy meson decay constants have been compared with lattice QCD from QCD sum rules [82].The decay constants of pseudoscalar and vector B and D mesons have been studied in the light-cone quark model with the variational method [83].In [84], the authors have calculated the strong decays of newly observed D J (3000) and D sJ (3040) with two 2P(1 + ) quantum number assignments.The leptonic (D + → e + ν e ) and semileptonic (D → K ( * ) ℓ + ν ℓ , D → πℓ + ν ℓ ) decays have been analyzed using covariant quark model with infrared confinement within the standard model framework [85].The weak decays of B, B s and B c into D-wave heavy light-mesons have been studied using Bethe-Salpeter equation [86].In Ref. [87], the decay constant and distribution amplitude for the heavy-light pseudoscalar mesons have been evaluated by using the light-front holographic wavefunction.By using the Gaussian wave function with quarkantiquark potential model the Regge trajectories, spectroscopy and decay properties have been studied for B and B s mesons [88], D and D s mesons [89], and also the radiative transitions and the mixing parameters of the D-meson have been obtained [90].The dimensional dependence of the masses of heavy-light mesons has been investigated using the string inspired potential model [91].
The aim of this work is to find the analytic solution of the N-dimensional Schrödinger equation for the mixture of vector and scalar potentials including the spin-spin, spin-orbit and tensor interactions via (LTM) in order to obtain the energy eigenvalues in the N-dimensional space and the corresponding eigenfunctions.So far no attempt has been made to solve the N-dimensional Schrodinger equation using (LTM) when the spin hyperfine, spin-orbit and tensor interactions are included.To show the importance of present results, the present results are employed to calculate the mass spectra of the heavy-light mesons in three dimensional space and in the higher dimensional space.In addition, the decay constants, leptonic decay widths and branching fractions of the heavy-light mesons are calculated.
The paper is organized as follows: the contributions of previous works are displayed in Section 1, In Section 2, a brief summary of Laplace Transformation method is introduced.In Section 3, An analytic solution of the N-dimensional Schrödinger equation is derived.In Section 4, the obtained results are discussed.In Section 5, summary and conclusion are presented.

Overview of Laplace Transform Method
The Laplace transform () z  or L of a function () ftis defined by [92].
If there is some constant  for sufficiently large t , the integral in equation ( 2) exist for Re z   .The Laplace transform may fail to exist because of a sufficiently strong singularity in the function () The Laplace transform has the derivative properties where the superscript () n stands for the n -th derivative with respect to t for () () n ft , and with respect z to for () () Then for where ()   is the gamma function.On the other hand, if near origin ()

Analytic Solution of the N-dimensional Radial Schrödinger Equation
The N-dimensional radial Schrödinger equation for the interaction between quarkantiquark systems takes the form where , N represent the angular quantum number and the dimensional number, respectively, and is the reduced mass of the quark-antiquark system.
In the non-relativistic quark-antiquark potential () qq Vr consists of the spin independent potential () Vr and spin dependent potential () The spin independent potential is taken as a combination of vector and scalar parts where () Vr and ()

S
Vrare the vector and scalar parts, respectively and  stands the mixing coefficient.The harmonic and linear terms represent the confining part and the Coulomb term represents one gluon exchange The spin dependent potential is extended to three types of interaction terms as .
while the spin-orbit    The diagonal elements of the 12 S is defined as [94]   
Thus, Eq. ( 32) has a solution We take the physical solution of Eq. ( 32)  .

83
,, The Laplace transform defined as and taking the boundary condition where, The singular point of Eq. ( 40) is .z   By using the condition of Eq. ( 5), the solution of Eq. ( 40) takes the form From Eq. ( 42), 2 Substituting from Eqs. (42)(43)(44) into Eq.( 40), we obtain the following relations Using Eqs. ( 26), ( 39), ( 41) and the set of Eqs.(45)(46)(47), then, the energy eigenvalue of Eq. ( 8) in the N-dimensional is given by the relation By taking the inverse Laplace transform such that The function () fr takes the following form Using Eqs. ( 11), ( 13) and ( 23) the eigenfunctions of Eq. ( 9) are take the following form   In the following subsections, we employ the obtained results in the previous section to calculate the mass spectra of scalar, vector, pseudoscalar and pseudovector of B, B s , D and D s mesons in the N-dimensional space in comparison with the experimental data (PDG 2016) [95] and with other recent studies.Addition, the decay properties such as decay constants, leptonic decay width and the branching ratio of heavy-light mesons are calculated.

Mass spectra of heavy-light mesons
The masses of heavy-light mesons in the N-dimensional space are defined [44] , .
Substituting from Eq. ( 48) into Eq.( 51), then the mass spectra of heavy-light mesons in the N-dimensional space can be found from the relation  [72], they used the variational method for the Cornell potential to study the heavy-light mesons with including the spin-spin and spin-orbit interactions.They ignored the tensor interactions in their calculations.The present results are improved in comparison with the results in Ref. [72].In addition, we used the Laplace transform method in the present calculations.Yazarloo and Mehiraban used the variational method to study D and D s mesons for the Cornell potential [73], and used the Nikiforov-Uvarov (NU) method to study B and B s mesons for the Coulomb plus exponential type potential [74].The present results are a good agreement with the results of Refs [73,74].Kher et al. [89] used a Gaussian wave function to calculate the mass spectra of D and D s mesons, and B and B s mesons [88] for the Cornell potential.Jing-Bin [81,96] obtained the spectra of the heavy-light mesons in a relativistic model from the Bethe-Salpeter equation using the Foldy-Wouthuysen transformation in his works.We note that present results for D and B s meson masses are improved with the results of Refs [81,[88][89]96], where the values of pseudoscalar D and B s mesons close the experimental results in Table ( Addition, we have investigated the masses of the heavy-light mesons in the higher dimensions at N=4 and N=5.The effect of the dimensional number on the masses of the heavy-light mesons is investigated in Tables (2)(3)(4)(5)(6).One can note that the masses increase with increasing dimensional number.The influence of the dimensional number on the masses of the heavy-light mesons is not considered in the works [72-74, 81, 88-89, 96].Roy and Choudhury [91] have presented a study of masses of heavy flavor mesons in the higher dimensional space using string inspired potential.They found that the meson mass increases with the dimensional number.Therefore, the present results of the mass spectra of heavy-light mesons are in a good agreement in comparison with the results of the Ref. [91].

Meson Present work
Exp. [

Decay Constants
The study of the decay constants is one of the very important characteristics of the heavy-light mesons, as it provides a direct source of information on the Cabbio-Kobayashi-Maskawa (CKM) matrix elements.Many theoretical studies have been done for determining the decay constants with different models as relativistic quark model [97][98][99], lattice QCD [100][101][102], QCD sum rules [62,97,103], and non-relativistic model [72-74, 79, 97].
The Van Royen-Weisskopf formula [104] can be used to calculate the decay constants of the pseudoscalar and vector mesons in the non-relativistic limit which defined as: The Van Royen-Weisskopf formula with the QCD radiative corrections taken into account can be written as [105]: where, / ( ) 1 ln v  for pseudoscalar and vector mesons respectively.In Table (6) and Table (7), we have determined the decay constants of the pseudoscalar and vector B and D mesons obtained from Eq. ( 53) and Eq. ( 54) in comparison with the results of other recent works.In Ref. [87], the authors evaluated the decay constant for the heavy-light pseudoscalar mesons by using the helicity-improved light-front holographic wavefunction.In Ref. [83], the authors applied the variational method to study the decay constants of the pseudoscalar and vector B and D mesons in the light-cone quark model for the relativistic Hamiltonian with the Gaussian-type function.In Ref. [72], the authors used the variational method to compute the decay constants of heavy-light mesons from the radial Schrodinger equation with the Cornell potential.Zhi-Gang Wang [97] presented an analysis of the decay constants of heavy-light mesons with QCD sum rules.Yazarloo and Mehiraban [79] used the perturbation method to study the decay constants of D, D s , B and B s mesons with the combination of harmonic and Yukawa-type potentials.In Table (7) and Table (8), the present results are a good agreement with the results of Refs [72,83], and in a good agreement with the results of Refs [87,97].The present result is (

Summery and Conclusion
In this work, we have calculated an analytic solution of the N-dimensional Schrodinger equation for the mixture of vector and scalar potentials via the Laplace transformation method.The spin-spin, spin-orbit, and tensor interactions have been included in the extended Cornell potential model.The energy eigenvalues and the corresponding eigenfunctions have been determined in the Ndimensional space.In 3-dimensional space, we have employed the obtained results to study the different properties of the heavy-light mesons which are not considered in many recent works.The masses of the scalar, vector, pseudoscalar and pseudovector of B, B s , D and D s mesons have been calculated in the three dimensional space and in the higher dimensional space in Tables (2-6), Most of present calculations close with the experimental data and are improved in comparison with the recent calculations [72-74, 81, 88-89, 96].Addition, we have calculated the masses of the heavy-light mesons in the higher dimensional space at N=4 and N=5.The effect of the dimensional number is studied on the masses of the heavy-light mesons.We note that the masses increase with increasing dimensional number.This behavior is agreement with Ref. [91].The decay constants of the pseudoscalar and vector mesons have been computed in Tables (7-

T
Vr terms give the fine structure of the states and the spin-spin () SS Vr interaction term describes the hyper-fine splitting of the state[94]

4 .
Discussion of ResultsIn Fig. (1), the present potential have been plotted in comparison with other potential models, we see that the present potential is in a qualitative agreement with other potential models [72, 74, 79], in which the confining part is clearly obtained in comparison with Cornell and Coulomb plus exponential potentials.The different states of B and D meson have been shown in Fig. (2) and Fig. (3), respectively, in which the principal number of state plays an important role in confining part of potential.

Fig. ( 1 ).
Fig. (1).The present potential and other potential models are plotted as functions of distance r.

Fig. ( 2
Fig. (2).The present potential of B meson for different states Fig. (3).The present potential of D meson for different states.

1
decay constants of D meson.This value is good agreement with experimental value 1[97].Present result of the decay ratio of B mesons are ( 8), in comparison with the results of Refs [72-74, 79, 83, 87, 97].The calculated ratio of the decay constants of D mesons ( the results of Refs.[72, 83].The leptonic decay widths of B + meson have been studied in

Table ( 1
). Parameters for heavy-light mesons.Table (6), we have calculated the masses of the heavy-light mesons in the three dimensional space in comparison with the experimental data and with other recent studies [72-74, 81, 88-89, 96].The parameters used in the present calculations are shown in Table (1).In addition, the masses at N = 4 and N In comparison with the Ref.

2), the
values of vector D and B s mesons close to the experimental data, and the values of vector D s and B mesons are a good in comparison with the experimental results in Table(3).The masses of the scalar mesons are presented in Table (4), the value of D meson closes with the experimental value.The values of D s and B are in agreement with the experimental values and the value of B s meson is a good agreement with the theoretical studies[72-74,

81, 88-89, 96].
(5)note that all the values of pseudovector mesons in Table(5)close to the experimental results except the value of B meson is a good agreement with the experimental value.The values of vector D s and B mesons are a good agreement with the experimental results.In

Table 6 ,
the results of the p-wave state for the heavy-light mesons are reported.The present predictions of D s and B mesons close to the experimental data.The predictions of B s and D mesons are agreement with the experimental data and improved in comparison with the theoretical studies[73-74,

Table ( 7
).The Decay constants of pseudoscalar B and D mesons in (MeV)

Table ( 8
).The Decay constants of vector B and D mesons in (MeV)

74, 79,107], as
(7)stants of the heavy-light mesons from Table(7)and Table (8) into Eq.(56) to compute leptonic decay widths of the heavy-light mesons.The obtained results of the leptonic decay width of B + , D + and D s + mesons are shown in Tables (9, 10) and Table (11), respectively.Vinodkumar et al. [107] calculated the leptonic decay widths of B and B s mesons and D and D s mesons [66-67,108] for the Martin-like potential with Dirac formalism.We have determined the leptonic decay widths of B + meson in Table ( 9) in comparison with the results of the Refs.[well as the leptonic decay widths of D + meson in Table (10) in comparison with the results of the Refs.[66, 79, 108], and the leptonic decay widths of D s + meson in Table (11) compared with the results of the Refs.[

66, 79, 108].
We note that the present results are in a good agreement with the results of the Refs.[74,

72-74, 88-89]. In Table (12), we
note the present values of the branching ratio for the B + meson close to experimental and with the theoretical results[72,

74, 79, 88, 107].
In addition, we note that the evaluated results of branching ratio for the D + and D s Leptonic branching ratio of B + meson.

Table ( 9
[74,79,107]ison with the results of the Refs.[74,79,107]and the leptonic decay widths of D + meson in Table (10) in comparison with the results of the Refs.[66, 79, 108].In addition, the leptonic decay widths of D s + meson have been studied in Table (11) compared with the results of the Refs.[66, 79, 108].All the obtained results of the leptonic decay widths are agreement with the results of the Refs.[74, 66-67, 107-108].We have determined the branching ratio for the B + , D + and D s + mesons that are good agreement with the experimental data and with the recent results [72-74, 88-89].Therefore, the present method with the present potential gives good results for heavy-light meson which are in good agreement with experimental data and are improved in comparison with recently theoretical works.