# Heavy-Light Mesons in the Nonrelativistic Quark Model Using Laplace Transformation Method

Authors: M. Abu-Shady; E. M. Khokha
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

An analytic solution of theN-dimensional radial Schrödinger equation with the combination of vector and scalar potentials via the Laplace transformation method (LTM) is derived. The current potential is extended to encompass the spin hyperfine, spin-orbit, and tensor interactions. The energy eigenvalues and the corresponding eigenfunctions have been obtained in theN-dimensional space. The present results are employed to study the different properties of the heavy-light mesons (HLM). The masses of the scalar, vector, pseudoscalar, and pseudovector forB,Bs,D, andDs mesons have been calculated in the three-dimensional space. The effect of the dimensional number space is discussed on the masses of the HLM. We observed 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 theB+,D+, and Bs+ mesons have been studied. Therefore, the used method with the current potential gives good results which are in good agreement with experimental data and are improved in comparison with recent theoretical studies.


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## Body

## 1. Introduction

One of the most important tasks in nonrelativistic quantum mechanics is to get the solution of the Schrödinger equation. The solution of the Schrödinger equation with spherically symmetric potentials plays a significant role in many fields of physics such as hadronic spectroscopy for understanding the quantum chromodynamics theory. Numerous works have been introduced to get the solution of Schrödinger equation using different methods like 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], supersymmetric quantum mechanics (SUSQM) [11], Hill determinant method (HDM) [12], and other numerical methods [13–15].Recently, the study of the different topics has received a great attention from theoretical physicists in the higher dimensional space. In addition, the study is more general and one can obtain the required results in the lower dimensions directly, such as the hydrogen atom [16–18], harmonic oscillator [19, 20], random walks [21], Casimir effects [22], and the quantization of angular momentum [23–27]. The N-dimensional Schrödinger equation has been studied with different forms of spherically symmetric potentials [28–33]. The N-dimensional Schrödinger equation has been investigated with the Cornell potential and extended Cornell potential [34–38] using different methods such as the Nikiforov-Uvarov (NU) method [32, 36, 39, 40], power series technique (PST) [41], the asymptotic iteration method (AIM) [34], Pekeris type approximation (PTA) [41, 42], and the analytical exact iteration method (AEIM) [43, 44].The LTM is one of the useful methods that contributed to finding the exact solution of Schrödinger equation in one-dimensional space for Morse potential [45, 46], the harmonic oscillator [47], and three-dimensional space with pseudoharmonic and Mie-type potentials [48] and with noncentral potential [49]. TheN-dimensional Schrödinger equation has been solved via the LTM in many studies for Coulomb potential [28], harmonic oscillator [50], Morse potential [51], pseudoharmonic potential [52], Mie-type potential [53], anharmonic oscillator [54], and generalized Cornell potential [38].The study of different properties of HLM is very vital for understanding the structure of hadrons and dynamics of heavy quarks. Thus, many theoretical and experimental efforts have been done for understanding distinct characteristics of HLM. In [4, 34, 55], the authors calculated the mass spectra of quarkonium systems as charmonium and bottomonium mesons with the quark-antiquark interaction potential using various methods in many works. 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 cc-, bb-, cs-, bs-, bu-, and cb- numerically using Jacobi method. The strong decays, spectroscopy, and radiative transition of heavy-light hadrons have been computed using the quark model predictions [58]. The decay constant of HLM has been calculated using the field correlation method [59]. Moreover, the spectroscopy of HLM has been investigated in the framework of the QCD relativistic quark model [60]. The spectroscopy and Regge trajectories of HLM have been obtained using quasi-potential approach [61]. The decay constants of heavy-light vector mesons [62] and heavy-light pseudoscalar mesons [63] have been calculated with QCD sum rules. A comparative study has been introduced for the mass spectrum and decay properties for theD meson with the quark-antiquark potential using hydrogeometric and Gaussian wave function [64]. In framework of Dirac formalism the mass spectra ofDs [65] andD [66] mesons have been obtained using Martin-light potential in which the hadronic and leptonic decays ofD andDs mesons have been evaluated [67]; besides the rare decays of B0 and Bs0 mesons into dimuon (μ+μ-) [68] and the decay constants ofB andBs have been calculated [69]. The mass spectra and decay constants for ground state of pseudoscalar and vector mesons have been obtained using the variational analysis in the light quark model [70]. The spectroscopy of bottomonium andB meson has been studied using the free-form smearing in [71]. The variational method has been employed to compute the masses and decay constants of HLM in [72]. In addition, the decay properties ofD andDs mesons have been investigated using the quark-antiquark potential in [73]. TheB and Bs mesons spectra and their decays have been studied with a Coulomb plus exponential type potential in [74]. The leptonic and semileptonic decays ofB meson into τ have been studied [75]. The degeneracy of HLM with the same orbital angular momentum has been broken with the spin-orbit interactions [76]. The relativistic quark model has been investigated to study the properties ofB and Bs mesons [77] and the excited charm and charm-strange mesons [78]. The perturbation method has been employed to determine the mass spectrum and decay properties of HLM with the mixture 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 HLM have been calculated within a relativistic quark model by using the Foldy-Wouthuysen transformation [81]. The isospin breaking of heavy meson decay constants had been compared with lattice QCD from QCD sum rules [82]. The decay constants of pseudoscalar and vectorB andD 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 DJ (3000) and DsJ (3040) with two 2P (1+) quantum number assignments. The leptonic (D→e+νe) and semileptonic (D→K(∗)l+νl,D→πl+νl) decays have been analyzed using the covariant quark model with infrared confinement within the standard model framework [85]. The weak decays ofB, Bs, and Bc into D-wave heavy-light mesons have been studied using Bethe-Salpeter equation [86]. In [87], the decay constant and distribution amplitude for the heavy-light pseudoscalar mesons have been evaluated using the light-front holographic wavefunction. By using the Gaussian wave function with quark-antiquark potential model, the Regge trajectories, spectroscopy, and decay properties have been studied for B and Bs mesons [88],D andDs mesons [89], and also the radiative transitions and the mixing parameters of the D-meson have been obtained [90]. The dimensional space dependence of the masses of heavy-light mesons has been investigated using the string inspired potential model [91].The goal of this work is to get the analytic solution of theN-dimensional Schrödinger equation for the mixture of vector and scalar potentials including the spin-spin, spin-orbit, and tensor interactions using LTM in order to obtain the energy eigenvalues in theN-dimensional space and the corresponding eigenfunctions. So far no attempt has been made to solve theN-dimensional Schrödinger equation using the 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 HLM in three-dimensional space and in the higher dimensional space. In addition, the decay constants, leptonic decay widths, and branching fractions of the HLM are calculated.The paper is systemized as follows: the contributions of previous works are displayed in Section1. In Section 2, a brief summary of Laplace transformation method is introduced. In Section 3, an analytic solution of theN-dimensional Schrödinger equation is derived. In Section 4, the obtained results are discussed. In Section 5, summary and conclusion are presented.


## 2. Overview of Laplace Transformation Method

The Laplace transformϕ(z) or L of a function f(t) is defined by [92](1)ϕz=Lft=∫0∞e-ztftdt.If there is some constant σ∈R such that e-σtf(t)≤M for sufficiently large t, the integral in (1) exists for Re z>σ for z>0. The Laplace transform may fail to exist because of a sufficiently strong singularity in the function f(t) as t→0. In particular(2)LtαΓα+1=1zα+1,α>-1,where Γ is the gamma function. The Laplace transform has the derivative properties(3)Lfnt=znLft-∑k=0n-1zn-1-kfk0,(4)Ltnft=-1nϕnz,where the superscript (n) stands for the n-th derivative with respect to t for f(n)(t) and with respect to z for ϕ(n)(z). If z0 is the singular point, the Laplace transform behaves near z→z0 as (5)ϕz=1z-z0υ,and then for t→∞(6)ft=1Γυtυ-1ez0t.On the other hand, if near origin f(t) behaves like tα with α>-1, then ϕ(z) behaves near z→∞ as(7)ϕz=Γα+1zα+1.


## 3. Analytic Solution of theN-Dimensional Radial Schrödinger Equation

TheN-dimensional radial Schrödinger equation that describes the interaction between quark-antiquark systems takes the form [41](8)d2dr2+N-1rddr-ll+N-2r2+2μE-Vqq-rΨr=0,where l,N represent the angular quantum number and the dimensional number greater than one, respectively, and μ=mqmq-/(mq+mq-) is the reduced mass of the quark-antiquark system.In the nonrelativistic quark model, the quark-antiquark potentialVqq-(r) consists of the spin independent potential V(r) and the spin dependent potential VSD(r), respectively:(9)Vqq-r=Vr+VSDr.The spin independent potential is taken as a combination of vector and scalar parts [93]: (10)Vr=VVr+VSr,(11)VVr=ηar2+br-cr,(12)VSr=1-ηar2+br,where VV(r) and VS(r) are the vector and scalar parts, respectively, and η stands for the mixing coefficient.ɑ,b, andc are arbitrary parameters wherea,b, andc > 0 which are fitted with experimental data. The harmonic and linear terms represent the confining part at long distance and the Coulomb term stands for the quark-antiquark interactions through one gluon exchange at short distances which gives better description of quark-antiquark interaction.The spin dependent potential is extended to three types of interaction terms as [94](13)VSDr=VLSrL.S+S12VTr+VSSrS1.S2,while the spin-orbit VLS(r) and tensor VT(r) terms give the fine structure of the states, the spin-spin VSS(r) interaction term describes the hyperfine splitting of the state, and L is an angular quantum operator, and S is a spin operator (for detail, see [94]).(14)VLSr=12mqmq-r3dVVdr-dVSdr,(15)VTr=112mqmq-1rdVVdr-d2VVdr2,(16)VSSr=23mqmq-∇2VV,where ∇2 is radial Laplace operator. (17)S1.S2=12SS+1-32,(18)L.S=12JJ+1-LL+1-SS+1,(19)S12=2S2-3S.r^S.r^.The diagonal elements of the S12 are defined. (20)S12=42L+32L-1S2L2-3L.S2-32L.S.Substituting (11)-(16) into (9) then the nonrelativistic quark-antiquark potential Vqq-(r) takes the form(21)Vqq-r=ar2+br+δ+gr+hr3,where(22)δ=2amqmq-2ηS1.S2+2η-12L.S,(23)g=bmqmq-η43S1.S2+112S12+2η-12L.S-c,(24)h=3c2mqmq-16S12+L.S.Substituting (21) into (8), then (25)d2dr2+N-1rddr-ll+N-2r2+ε-Ar2-Br-2μδ-Gr-Hr3Ψr=0,where(26)ε=2μE,A=2μa,B=2μb,G=2μg,H=2μh.The complete solution of (25) takes the form(27)Ψr=rke-αr2fr,k>0,withα=μa2,where the term rk confirms that the solution is bounded at r=0. The function f(r) is yet to be determined. From (27) we get(28)Ψ′r=rke-αr2f′r+kr-2αrfr.(29)Ψ′′r=rke-αr2f′′r+2kr-4αrf′r+kk-1r2+4α2r2-4αk-2αfr.Substituting (27), (28), and (29) into (25), then, (30)rf′′r+ω-4αr2f′r+λr-Br2+ζr-G-Hr2fr=0,where(31)ω=2k+N-1,(32)λ=kk+N-2-ll+N-2,(33)ζ=ε-4αk-2αN-2μδ.In order to apply the Laplace transform of the above differential equation, the parametric condition is taken as in [52, 54]. (34)kk+N-2-ll+N-2=0.Thus, (32) has a solution (35)k+=l,andk-=-l+N-2.We take the physical solution of (32) k=k+=l as in [52, 54].Substituting (34) into (30) yields(36)rf′′r+ω-4αr2f′r+ζr-Br2-G-Hr2fr=0.By expanding the term H/r2 around y=0, where y=r-υ and υ is a parameter as in [36, 56], we get (37)Hr2=Hy+υ2=Hυ43r2-8rυ+6υ2.Substituting (37) into (36) yields(38)rf′′r+ω-4αr2f′r+Qr-Pr2-C0fr=0,where(39)Q=ζ+8Hυ3,P=B+3Hυ4,andC0=G+6Hυ2.The Laplace transform is defined as ϕ(z)=Lf(r) and taking boundary condition f(0)=0 yields (40)z+τd2ϕzdz2+z24α+ρdϕzdz+γz+C04αϕz=0.Here(41)τ=P4α,ρ=Q4α+2,γ=2-ω4α.The singular point of (40) is z=-τ. By using the condition of (5), the solution of (40) takes the form (42)ϕz=Cz+τn+1,n=0,1,2,3,….From (42), (43)ϕ′z=-Cn+1z+τn+2,(44)ϕ′′z=Cn+1n+2z+τn+3.Substituting (42)-(44) into (40), we obtain the following relations:(45)γ=n+14α,(46)γτ+C04α=0,(47)n+1n+2-ρn+1+C0τ4α=0.Using (26), (39), and (41) and the set of (45)-(47), then, the energy eigenvalue of (8) in theN-dimensional space is given by the relation(48)EnlN=a2μ2n+2l+N-b24a+δ-8hυ3-ha9h4υ8+3b2υ4.Take the inverse Laplace transform such that f(r)=L-1ϕ(z). The function f(r) takes the following form:(49)fr=CΓn+1rne-τr.Using (11), (13), and (23), the eigenfunctions of (9) take the following form:(50)Ψr=CΓn+1rn+lexp⁡-μa2r2-μ2abr.From the condition ∫0∞Ψ(r)2rN-1dr=1, the normalization constant C can be computed. In addition, the wave equation Ψ(r) satisfies the boundary condition Ψ(r=0)=Ψ(r=∞)=0.


## 4. Discussion of Results

In Figure1, the current potential has been plotted in comparison to 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 to Cornell and Coulomb plus exponential potentials. The different states of B and D mesons have been shown in Figures 2 and 3, respectively, in which the principal number of states plays an important role in confining part of potential.Figure 1
The current potential and other potential models are plotted as functions of distancer.Figure 2
The current potential of B meson for different states.Figure 3
The current potential ofD meson for different states.In the following subsections, we employ the obtained results in the previous section to determine the mass spectra of scalar, vector, pseudoscalar, and pseudovector ofB, Bs, D, and Ds mesons in the N-dimensional space in comparison with the experimental data (PDG 2016) [95] and with other recent studies. In addition, the decay properties such as decay constants, leptonic decay width, and the branching ratio of HLM are calculated.


### 4.1. Mass Spectra of Heavy-Light Mesons

The masses of HLM in theN-dimensional space are defined [44]: (51)MB,D=mq+mq-+EnlN.Substituting (48) into (51), then the mass spectra of HLM in theN- dimensional space can be found from the relation(52)MB,D=mq+mq-+a2μ2n+2l+N-b24a+δ-8hυ3-ha9h4υ8+3b2υ4.In Tables 2–6, we have calculated the masses of the HLM in the three-dimensional space in comparison with the experimental data and 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 atN = 4 andN = 5 are calculated. In Tables 2 and 3, we observe that D and Bs meson masses close to experimental data and other meson masses are in good agreement with experimental data and become better in comparison to the results in recent studies [72–74, 81, 88, 89, 96]. In comparison with [72], they used the variational method for the Cornell potential to study the HLM with including the spin-spin and spin-orbit interactions. They ignored the tensor interactions in their calculations. The present results are good in comparison to the results in [72]. In addition, we used the LTM in the present calculations. Yazarloo and Mehiraban used the variational method to studyD and Ds mesons for the Cornell potential [73] and used the Nikiforov-Uvarov (NU) method to studyB and Bs mesons for the Coulomb plus exponential type potential [74]. The present results are in good agreement with the results of [73, 74]. Kher et al. [89] used a Gaussian wave function to calculate the mass spectra ofDand Ds in addition toB and Bs mesons [88] for the Cornell potential. Jing-Bin [81, 96] obtained the spectra of the HLM in the relativistic model from the Bethe-Salpeter equation using the Foldy-Wouthuysen transformation in his works.Table 1
Parameters for HLM.
m c m b m u , d m s η υ 1.45 (GeV) 4.87 (GeV) 0.38 (GeV) 0.48 (GeV) 0.25 1 (GeV−1)Table 2
Masses for pseudoscalar (LJ2S+1=S01 ) mesons in GeV. a=0.00085GeV3, b=0.01614GeV2, and c=0.7.
Meson Present Work Exp. [95] [72] [81] [96] [88, 89] [73, 74] N=4 N=5 D 1.864 1.864 1.895 1.871 1.859 1.884 [89] 1.864 [73] 1.902 1.939 D s 1.960 1.968 1.962 1.964 1.949 1.965 [89] 1.978 [73] 1.989 2.023 B 5.277 5.280 5.302 5.273 5.262 5.287 [88] 5.272 [74] 5.311 5.346 B s 5.366 5.366 5.340 5.363 5.337 5.367 [88] 5.385 [74] 5.397 5.428Table 3
Masses for vector(LJ2S+1=S13) mesons in GeV. a=0.026068GeV3, b=0.218058GeV2, and c=8×10-3.
Meson Present Work Exp. [95] [72] [81] [96] [88, 89] [73, 74] N=4 N=5 D 2.010 2.010 2.023 2.008 2.026 2.010 [89] 2.010 [73] 2.218 2.426 D s 2.100 2.112 2.057 2.107 2.110 2.120 [89] 2.102 [73] 2.244 2.434 B 5.374 5.325 5.356 5.329 5.330 5.323 [88] 5.327 [74] 5.567 5.759 B s 5.415 5.415 5.384 5.419 5.405 5.413 [88] 5.409 [74] 5.588 5.760We note that the present results forD and Bs meson masses become better in comparison to the results of [81, 88, 89, 96], where the values of pseudoscalarD and Bs mesons are close to the experimental data in Table 2. The values of vector D and Bs mesons close to the experimental data and the values of vectorDs and B mesons are good in comparison to the experimental results in Table 3.The masses of the scalar mesons are presented in Table4; the value ofD meson is close to the experimental value. The values ofDs andBare in agreement with the experimental values and the value of Bs meson is in good agreement with the theoretical studies [72–74, 81, 88, 89, 96]. In Table 5, we observe that all the values of pseudovector mesons are close to the experimental results except the value ofB meson which is in good agreement with the experimental value. The values of vectorDs andB mesons are in good agreement with the experimental results. In Table 6, the results of the p-wave state for the HLM are reported.Table 4
Masses for scalar(LJ2S+1=P03) mesons in GeV. a=0043GeV3, b=0.001GeV2, and c=10-3.
Meson Present Work Exp. [95] [72] [81] [96] [88, 89] [73, 74] N=4 N=5 D 2.289 2.318±0.029 2.316 2.364 2.357 2.357[89] 2.539[73] 2.374 2.459 D s 2.350 2.318 2.372 2.437 2.412 2.438[89] 2.311[73] 2.427 2.505 B 5.700 5.710 5.657 5.776 5.740 5.730[88] 5.745[74] 5.736 5.815 B s 5.720 - - - 5.719 5.811 5.776 5.812[88] 5.843[74] 5.785 5.856Table 5
Masses for pseudovector(LJ2S+1=P11) mesons in GeV. a=0.01359GeV3, b=0.08784GeV2, and c=0.008.
Meson Present Work Exp. [95] [72] [81] [96] [88, 89] [73, 74] N=4 N=5 D 2.421 2.421 2.362 2.507 2.434 2.425[89] 2.421[73] 2.571 2.722 D s 2.460 2.460 2.409 2.558 2.528 2.529[89] 2.429[73] 2.597 2.735 B 5.797 5.726 5.760 5.719 5.736 5.733[88] 5.744[74] 5.936 6.075 B s 5.828 5.829 5.775 5.819 5.824 5.828[88] 5.841[74] 5.952 6.077Table 6
Masses for mesons with p-wave state(LJ2S+1=P23) in GeV. a=0.0163GeV3, b=0.113GeV2, and c=6×10-5.
Meson Present work Exp. [95] [81] [96] [88, 89] [73, 74] N=4 N=5 D 2.463 2.463 2.460 2.482 2.461[89] 2.463[74] 2.628 2.792 D s 2.500 2.537 2.570 2.575 2.569[89] 2.528[74] 2.641 2.800 B 5.817 5.740 5.739 5.754 5.740[88] 5.743[73] 5.969 6.122 B s 5.840 5.840 5.838 5.843 5.840[88] 5.840[73] 5.976 6.113The present predictions ofD, Ds,B, andBs mesons are in agreement in comparison to the experimental data and the theoretical studies [73, 74, 81, 88, 89, 96].In addition, we have investigated the masses of the HLM in the higher dimensions atN=4 andN=5. In Tables 2–6, the effect of the dimensional number is investigated on the masses of the HLM. One can see that the masses increase with increasing dimensional number. The influence of the dimensional number is not considered on the masses of the HLM in the works [72–74, 81, 88, 89, 96]. Roy and Choudhury [91] have studied the masses of heavy flavor mesons in the higher dimensional space using string inspired potential. They found that an increase of the dimensional number leads to increase the meson masses. Therefore, the present results of the mass spectra of HLM are in good agreement in comparison with the results of [91].


### 4.2. Decay Constants

The study of the decay constants is one of the very significant characteristics of the HLM, as it provides a direct source of information on the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. Many theoretical studies have been done for determining the decay constants with different models as relativistic quark model [97–99], lattice QCD [100–102], QCD sum rules [62, 97, 103], and nonrelativistic model [72–74, 79, 97].The Van Royen-Weisskopf formula [104] can be used to calculate the decay constants of the pseudoscalar and vector mesonsfpandfv, respectively, in the nonrelativistic limit which is defined as(53)fp/v2=12Ψ02Mp/v.The Van Royen-Weisskopf formula with the QCD radiative corrections taken into account can be written as [105](54)f¯p/v2=12Ψ02Mp/vC2αs,where(55)Cαs=1-αsπΔp/v-mq-mq-mq+mq-ln⁡mqmq-and Δp=2 and Δv=8/3, for pseudoscalar and vector mesons, respectively.In Tables7 and 8, we have determined the decay constants of the pseudoscalar and vectorB andD mesons obtained from (53) and (54) in comparison with the results of other recent works. In [87], the authors evaluated the decay constant for the heavy-light pseudoscalar mesons using the helicity-improved light-front holographic wavefunction. In [83], the authors applied the variational method to study the decay constants of the pseudoscalar and vectorB andD mesons in the light-cone quark model for the relativistic Hamiltonian with the Gaussian-type function.Table 7
The decay constants of pseudoscalarB and D mesons in MeV.
Meson f p f - p [72] [83] [87] [97] D 220 235 228 200 ± 24 214.2 - 7.8 + 7.6 210 ± 11 D s 250 243 273 232 ± 17 253.5 - 7.1 + 6.6 259 ± 10 B 147 201 149 184 ± 32 191.7 - 6.5 + 7.9 192 ± 13 B s 174 213 187 215 ± 24 225.4 - 5.3 + 7.9 230 ± 13Table 8
The decay constants of vectorB and D mesons in MeV.
Meson f v f - v [83] [73, 74] [79] D 290 210 247 ± 35 307 [73] 353.8 D s 310 212 287 ± 29 344 [73] 382.1 B 196 182 210 ± 37 242.4 [74] 234.7 B s 216 191 239 ± 29 178.8 [74] 244.2In [72], the authors used the variational method to compute the decay constants of HLM from the radial Schrödinger equation with the Cornell potential. Zhi-Gang Wang [97] introduced an analysis of the decay constants of HLM with QCD sum rules. Yazarloo and Mehiraban [79] used the perturbation method to study the decay constants of D, Ds,B, and Bs mesons with the combination of harmonic and Yukawa-type potentials.In Table7, the obtained results are in good agreement in comparison to the results of [72, 83, 87, 97]. In Table 8, the present results are compatible with the results of [73, 74, 79, 83]. In addition, the ratio of decay constants forD mesons is (fDs/fD=1.140). This value is in good agreement with the experimental value fDs/fD=1.258±0.038 [95]. The present result is in agreement with the obtained values (fDs/fD=1.195) in [72] and (fDs/fD=1.160) in [83]. Also, we have (fDs∗/fD∗=1.070) which is in agreement with the calculated values (fDs∗/fD∗=1.183) in [87] and (fDs∗/fD∗=1.233) in [97]. The calculated ratio of decay constants forB mesons (fBs/fB=1.184) and (fBs∗/fB∗=1.102) are in good agreement in comparison with (fBs/fB=1.168) and (fBs∗/fB∗=1.138) in [83].


### 4.3. Leptonic Decay Widths and Branching Ratio

The charged HLM can decay to a charged lepton pairl+νl via a virtualW± boson. The leptonic decay widths of the HLM can be obtained from the relation [106](56)ΓB+,Dq→l+νl=GF2MB,Dq8πml21-ml2MB,Dq22fB,D2×Vub2forBmesonVcq2q∈d,s,forDmesonwhere GF=1.664×10-5 is the Fermi constant and the relevant CKM elements are taken from the PDG [95] as Vub=0.004, Vcd=0.227,  and  Vcs=0.974. The leptonic masses ml are taken as me=0.501×10-3GeV, mμ=0.105GeV, and mτ=1.776GeV. We obtain the decay constants of the HLM from Tables 7 and 8 into (56) to compute leptonic decay widths of the HLM. The obtained results of the leptonic decay width ofB+,D+, and Ds+ mesons are shown in Tables 9, 10, and 11, respectively. Vinodkumar et al. [107] calculated the leptonic decay widths ofB, Bs mesons besides,D andDs mesons [66, 67, 108] for the Martin-like potential with Dirac formalism. We have determined the leptonic decay widths ofB+ meson in Table 9 in comparison with the results of the [74, 79, 107], as well as the leptonic decay widths ofD+ meson in Table 10 in comparison with the results of [66, 79, 108] and the leptonic decay widths of Ds+ meson in Table 11 compared with the results of [66, 79, 108]. We note that the present results are in good agreement with the results of [66, 67, 74, 107, 108].Table 9
Leptonic decay width ofB+ meson in GeV.
Present Γ [74] [79] [107] B + → e + υ e 2.475 ×10−24 8.624 ×10−24 8.094 ×10−24 5.689 ×10−24 B + → μ + υ μ 1.086 ×10−19 3.685 ×10−19 3.459 ×10−19 2.439 ×10−19 B + → τ + υ τ 2.445 ×10−17 8.196 ×10−17 7.697 ×10−17 5.430 ×10−17Table 10
Leptonic decay width ofD+ meson in GeV.
Present Γ [79] [108] [66] D + → e + υ e 0.622 ×10−20 1.488 ×10−20 1.323 ×10−20 5.706 ×10−21 D + → μ + υ μ 2.715 ×10−16 6.322 ×10−16 5.641 ×10−16 2.433 ×10−16 D + → τ + υ τ 0.668 ×10−15 1.215 ×10−15 1.529 ×10−15 6.157 ×10−16Table 11
Leptonic decay width ofDs+ meson in GeV.
Present  Γ [79] [108] [67] D s + → e + υ e 1.529 ×10−19 2.962 ×10−19 3.157 ×10−19 1.792 ×10−19 D s + → μ + υ μ 0.668 ×10−14 1.259 ×10−14 1.347 ×10−14 7.648 ×10−15 D s + → τ + υ τ 0.586 ×10−13 1.296 ×10−13 1.326 ×10−13 7.508 ×10−14The branching ratio of the HLM is defined as(57)BrB+,Dq→l+υl=ΓB+,Dq→l+υl×τB+,Dqwhere the lifetime τ ofB+,D+, and Ds+ mesons is taken as τB+=1.638ps, τD+=1.040ps, and τDs+=0.5ps [95]. We have determined the branching ratio for the B+, D+, and Ds+ mesons compared with the experimental data and with the results of other recent studies [72–74, 88, 89].In Table12, we note that the present values of the branching ratio for theB+ meson are close to experimental data and are in agreement in comparison with the theoretical results [72, 74, 79, 88, 107]. In addition, in Tables 13 and 14, we note that the evaluated results of branching ratio for theD+ and Ds+ mesons are close to the experimental data and become better in comparison with works [72, 73, 79, 89, 108].Table 12
Leptonic branching ratio ofB+ meson.
PresentBr [88] [79] [107] [72] [74] Exp. [95] B + → e + υ e 6.162 ×10−12 8.640 ×10−12 2.015 ×10−11 1.419 ×10−11 6.220 ×10−12 2.147 ×10−11 <9.8 ×10−7 B + → μ + υ μ 2.705 ×10−7 0.370 ×10−7 8.611 ×10−7 6.085 ×10−7 2.630 ×10−7 9.174 ×10−7 <1.0 ×10−6 B + → τ + υ τ 6.088 ×10−5 0.822 ×10−4 1.916 ×10−4 1.354 ×10−4 1.140 ×10−4 2.040 ×10−4 (1.14±0.27) ×10−4Table 13
Leptonic branching ratio ofD+ meson.
PresentBr [89] [79] [73] [72] [108] Exp. [95] D + → e + υ e 0.984 ×10−8 0.580 ×10−8 2.351 ×10−8 1.77 ×10−8 1.130 ×10−8 2.105 ×10−8 <8.8 ×10−6 D + → μ + υ μ 4.293 ×10−4 2.470 ×10−4 9.991 ×10−4 7.54 ×10−4 4.770 ×10−4 8.977 ×10−4 (3.74±0.17) ×10−4 D + → τ + υ τ 1.055 ×10−3 0.860 ×10−3 1.920 ×10−3 1.79 ×10−3 2.030 ×10−3 2.933 ×10−3 <1.2×10−3Table 14
Leptonic branching ratio ofDs+ meson.
PresentBr [89] [79] [73] [72] [108] Exp. [95] D s + → e + υ e 1.163 ×10−7 0.940 ×10−7 2.251 ×10−7 1.82 ×10−7 1.630 ×10−7 1.391 ×10−7 <8.3 ×10−5 D s + → μ + υ μ 5.078 ×10−3 4.000 ×10−3 9.572 ×10−3 7.74 ×10−3 6.900 ×10−3 5.937 ×10−3 (5.56±0.25) ×10−3 D s + → τ + υ τ 4.451 ×10−3 3.780 ×10−3 9.864 ×10−2 8.2 ×10−2 6.490 ×10−2 5.844 ×10−3 (5.55±0.24)%


## 4.1. Mass Spectra of Heavy-Light Mesons

The masses of HLM in theN-dimensional space are defined [44]: (51)MB,D=mq+mq-+EnlN.Substituting (48) into (51), then the mass spectra of HLM in theN- dimensional space can be found from the relation(52)MB,D=mq+mq-+a2μ2n+2l+N-b24a+δ-8hυ3-ha9h4υ8+3b2υ4.In Tables 2–6, we have calculated the masses of the HLM in the three-dimensional space in comparison with the experimental data and 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 atN = 4 andN = 5 are calculated. In Tables 2 and 3, we observe that D and Bs meson masses close to experimental data and other meson masses are in good agreement with experimental data and become better in comparison to the results in recent studies [72–74, 81, 88, 89, 96]. In comparison with [72], they used the variational method for the Cornell potential to study the HLM with including the spin-spin and spin-orbit interactions. They ignored the tensor interactions in their calculations. The present results are good in comparison to the results in [72]. In addition, we used the LTM in the present calculations. Yazarloo and Mehiraban used the variational method to studyD and Ds mesons for the Cornell potential [73] and used the Nikiforov-Uvarov (NU) method to studyB and Bs mesons for the Coulomb plus exponential type potential [74]. The present results are in good agreement with the results of [73, 74]. Kher et al. [89] used a Gaussian wave function to calculate the mass spectra ofDand Ds in addition toB and Bs mesons [88] for the Cornell potential. Jing-Bin [81, 96] obtained the spectra of the HLM in the relativistic model from the Bethe-Salpeter equation using the Foldy-Wouthuysen transformation in his works.Table 1
Parameters for HLM.
m c m b m u , d m s η υ 1.45 (GeV) 4.87 (GeV) 0.38 (GeV) 0.48 (GeV) 0.25 1 (GeV−1)Table 2
Masses for pseudoscalar (LJ2S+1=S01 ) mesons in GeV. a=0.00085GeV3, b=0.01614GeV2, and c=0.7.
Meson Present Work Exp. [95] [72] [81] [96] [88, 89] [73, 74] N=4 N=5 D 1.864 1.864 1.895 1.871 1.859 1.884 [89] 1.864 [73] 1.902 1.939 D s 1.960 1.968 1.962 1.964 1.949 1.965 [89] 1.978 [73] 1.989 2.023 B 5.277 5.280 5.302 5.273 5.262 5.287 [88] 5.272 [74] 5.311 5.346 B s 5.366 5.366 5.340 5.363 5.337 5.367 [88] 5.385 [74] 5.397 5.428Table 3
Masses for vector(LJ2S+1=S13) mesons in GeV. a=0.026068GeV3, b=0.218058GeV2, and c=8×10-3.
Meson Present Work Exp. [95] [72] [81] [96] [88, 89] [73, 74] N=4 N=5 D 2.010 2.010 2.023 2.008 2.026 2.010 [89] 2.010 [73] 2.218 2.426 D s 2.100 2.112 2.057 2.107 2.110 2.120 [89] 2.102 [73] 2.244 2.434 B 5.374 5.325 5.356 5.329 5.330 5.323 [88] 5.327 [74] 5.567 5.759 B s 5.415 5.415 5.384 5.419 5.405 5.413 [88] 5.409 [74] 5.588 5.760We note that the present results forD and Bs meson masses become better in comparison to the results of [81, 88, 89, 96], where the values of pseudoscalarD and Bs mesons are close to the experimental data in Table 2. The values of vector D and Bs mesons close to the experimental data and the values of vectorDs and B mesons are good in comparison to the experimental results in Table 3.The masses of the scalar mesons are presented in Table4; the value ofD meson is close to the experimental value. The values ofDs andBare in agreement with the experimental values and the value of Bs meson is in good agreement with the theoretical studies [72–74, 81, 88, 89, 96]. In Table 5, we observe that all the values of pseudovector mesons are close to the experimental results except the value ofB meson which is in good agreement with the experimental value. The values of vectorDs andB mesons are in good agreement with the experimental results. In Table 6, the results of the p-wave state for the HLM are reported.Table 4
Masses for scalar(LJ2S+1=P03) mesons in GeV. a=0043GeV3, b=0.001GeV2, and c=10-3.
Meson Present Work Exp. [95] [72] [81] [96] [88, 89] [73, 74] N=4 N=5 D 2.289 2.318±0.029 2.316 2.364 2.357 2.357[89] 2.539[73] 2.374 2.459 D s 2.350 2.318 2.372 2.437 2.412 2.438[89] 2.311[73] 2.427 2.505 B 5.700 5.710 5.657 5.776 5.740 5.730[88] 5.745[74] 5.736 5.815 B s 5.720 - - - 5.719 5.811 5.776 5.812[88] 5.843[74] 5.785 5.856Table 5
Masses for pseudovector(LJ2S+1=P11) mesons in GeV. a=0.01359GeV3, b=0.08784GeV2, and c=0.008.
Meson Present Work Exp. [95] [72] [81] [96] [88, 89] [73, 74] N=4 N=5 D 2.421 2.421 2.362 2.507 2.434 2.425[89] 2.421[73] 2.571 2.722 D s 2.460 2.460 2.409 2.558 2.528 2.529[89] 2.429[73] 2.597 2.735 B 5.797 5.726 5.760 5.719 5.736 5.733[88] 5.744[74] 5.936 6.075 B s 5.828 5.829 5.775 5.819 5.824 5.828[88] 5.841[74] 5.952 6.077Table 6
Masses for mesons with p-wave state(LJ2S+1=P23) in GeV. a=0.0163GeV3, b=0.113GeV2, and c=6×10-5.
Meson Present work Exp. [95] [81] [96] [88, 89] [73, 74] N=4 N=5 D 2.463 2.463 2.460 2.482 2.461[89] 2.463[74] 2.628 2.792 D s 2.500 2.537 2.570 2.575 2.569[89] 2.528[74] 2.641 2.800 B 5.817 5.740 5.739 5.754 5.740[88] 5.743[73] 5.969 6.122 B s 5.840 5.840 5.838 5.843 5.840[88] 5.840[73] 5.976 6.113The present predictions ofD, Ds,B, andBs mesons are in agreement in comparison to the experimental data and the theoretical studies [73, 74, 81, 88, 89, 96].In addition, we have investigated the masses of the HLM in the higher dimensions atN=4 andN=5. In Tables 2–6, the effect of the dimensional number is investigated on the masses of the HLM. One can see that the masses increase with increasing dimensional number. The influence of the dimensional number is not considered on the masses of the HLM in the works [72–74, 81, 88, 89, 96]. Roy and Choudhury [91] have studied the masses of heavy flavor mesons in the higher dimensional space using string inspired potential. They found that an increase of the dimensional number leads to increase the meson masses. Therefore, the present results of the mass spectra of HLM are in good agreement in comparison with the results of [91].


## 4.2. Decay Constants

The study of the decay constants is one of the very significant characteristics of the HLM, as it provides a direct source of information on the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. Many theoretical studies have been done for determining the decay constants with different models as relativistic quark model [97–99], lattice QCD [100–102], QCD sum rules [62, 97, 103], and nonrelativistic model [72–74, 79, 97].The Van Royen-Weisskopf formula [104] can be used to calculate the decay constants of the pseudoscalar and vector mesonsfpandfv, respectively, in the nonrelativistic limit which is defined as(53)fp/v2=12Ψ02Mp/v.The Van Royen-Weisskopf formula with the QCD radiative corrections taken into account can be written as [105](54)f¯p/v2=12Ψ02Mp/vC2αs,where(55)Cαs=1-αsπΔp/v-mq-mq-mq+mq-ln⁡mqmq-and Δp=2 and Δv=8/3, for pseudoscalar and vector mesons, respectively.In Tables7 and 8, we have determined the decay constants of the pseudoscalar and vectorB andD mesons obtained from (53) and (54) in comparison with the results of other recent works. In [87], the authors evaluated the decay constant for the heavy-light pseudoscalar mesons using the helicity-improved light-front holographic wavefunction. In [83], the authors applied the variational method to study the decay constants of the pseudoscalar and vectorB andD mesons in the light-cone quark model for the relativistic Hamiltonian with the Gaussian-type function.Table 7
The decay constants of pseudoscalarB and D mesons in MeV.
Meson f p f - p [72] [83] [87] [97] D 220 235 228 200 ± 24 214.2 - 7.8 + 7.6 210 ± 11 D s 250 243 273 232 ± 17 253.5 - 7.1 + 6.6 259 ± 10 B 147 201 149 184 ± 32 191.7 - 6.5 + 7.9 192 ± 13 B s 174 213 187 215 ± 24 225.4 - 5.3 + 7.9 230 ± 13Table 8
The decay constants of vectorB and D mesons in MeV.
Meson f v f - v [83] [73, 74] [79] D 290 210 247 ± 35 307 [73] 353.8 D s 310 212 287 ± 29 344 [73] 382.1 B 196 182 210 ± 37 242.4 [74] 234.7 B s 216 191 239 ± 29 178.8 [74] 244.2In [72], the authors used the variational method to compute the decay constants of HLM from the radial Schrödinger equation with the Cornell potential. Zhi-Gang Wang [97] introduced an analysis of the decay constants of HLM with QCD sum rules. Yazarloo and Mehiraban [79] used the perturbation method to study the decay constants of D, Ds,B, and Bs mesons with the combination of harmonic and Yukawa-type potentials.In Table7, the obtained results are in good agreement in comparison to the results of [72, 83, 87, 97]. In Table 8, the present results are compatible with the results of [73, 74, 79, 83]. In addition, the ratio of decay constants forD mesons is (fDs/fD=1.140). This value is in good agreement with the experimental value fDs/fD=1.258±0.038 [95]. The present result is in agreement with the obtained values (fDs/fD=1.195) in [72] and (fDs/fD=1.160) in [83]. Also, we have (fDs∗/fD∗=1.070) which is in agreement with the calculated values (fDs∗/fD∗=1.183) in [87] and (fDs∗/fD∗=1.233) in [97]. The calculated ratio of decay constants forB mesons (fBs/fB=1.184) and (fBs∗/fB∗=1.102) are in good agreement in comparison with (fBs/fB=1.168) and (fBs∗/fB∗=1.138) in [83].


## 4.3. Leptonic Decay Widths and Branching Ratio

The charged HLM can decay to a charged lepton pairl+νl via a virtualW± boson. The leptonic decay widths of the HLM can be obtained from the relation [106](56)ΓB+,Dq→l+νl=GF2MB,Dq8πml21-ml2MB,Dq22fB,D2×Vub2forBmesonVcq2q∈d,s,forDmesonwhere GF=1.664×10-5 is the Fermi constant and the relevant CKM elements are taken from the PDG [95] as Vub=0.004, Vcd=0.227,  and  Vcs=0.974. The leptonic masses ml are taken as me=0.501×10-3GeV, mμ=0.105GeV, and mτ=1.776GeV. We obtain the decay constants of the HLM from Tables 7 and 8 into (56) to compute leptonic decay widths of the HLM. The obtained results of the leptonic decay width ofB+,D+, and Ds+ mesons are shown in Tables 9, 10, and 11, respectively. Vinodkumar et al. [107] calculated the leptonic decay widths ofB, Bs mesons besides,D andDs mesons [66, 67, 108] for the Martin-like potential with Dirac formalism. We have determined the leptonic decay widths ofB+ meson in Table 9 in comparison with the results of the [74, 79, 107], as well as the leptonic decay widths ofD+ meson in Table 10 in comparison with the results of [66, 79, 108] and the leptonic decay widths of Ds+ meson in Table 11 compared with the results of [66, 79, 108]. We note that the present results are in good agreement with the results of [66, 67, 74, 107, 108].Table 9
Leptonic decay width ofB+ meson in GeV.
Present Γ [74] [79] [107] B + → e + υ e 2.475 ×10−24 8.624 ×10−24 8.094 ×10−24 5.689 ×10−24 B + → μ + υ μ 1.086 ×10−19 3.685 ×10−19 3.459 ×10−19 2.439 ×10−19 B + → τ + υ τ 2.445 ×10−17 8.196 ×10−17 7.697 ×10−17 5.430 ×10−17Table 10
Leptonic decay width ofD+ meson in GeV.
Present Γ [79] [108] [66] D + → e + υ e 0.622 ×10−20 1.488 ×10−20 1.323 ×10−20 5.706 ×10−21 D + → μ + υ μ 2.715 ×10−16 6.322 ×10−16 5.641 ×10−16 2.433 ×10−16 D + → τ + υ τ 0.668 ×10−15 1.215 ×10−15 1.529 ×10−15 6.157 ×10−16Table 11
Leptonic decay width ofDs+ meson in GeV.
Present  Γ [79] [108] [67] D s + → e + υ e 1.529 ×10−19 2.962 ×10−19 3.157 ×10−19 1.792 ×10−19 D s + → μ + υ μ 0.668 ×10−14 1.259 ×10−14 1.347 ×10−14 7.648 ×10−15 D s + → τ + υ τ 0.586 ×10−13 1.296 ×10−13 1.326 ×10−13 7.508 ×10−14The branching ratio of the HLM is defined as(57)BrB+,Dq→l+υl=ΓB+,Dq→l+υl×τB+,Dqwhere the lifetime τ ofB+,D+, and Ds+ mesons is taken as τB+=1.638ps, τD+=1.040ps, and τDs+=0.5ps [95]. We have determined the branching ratio for the B+, D+, and Ds+ mesons compared with the experimental data and with the results of other recent studies [72–74, 88, 89].In Table12, we note that the present values of the branching ratio for theB+ meson are close to experimental data and are in agreement in comparison with the theoretical results [72, 74, 79, 88, 107]. In addition, in Tables 13 and 14, we note that the evaluated results of branching ratio for theD+ and Ds+ mesons are close to the experimental data and become better in comparison with works [72, 73, 79, 89, 108].Table 12
Leptonic branching ratio ofB+ meson.
PresentBr [88] [79] [107] [72] [74] Exp. [95] B + → e + υ e 6.162 ×10−12 8.640 ×10−12 2.015 ×10−11 1.419 ×10−11 6.220 ×10−12 2.147 ×10−11 <9.8 ×10−7 B + → μ + υ μ 2.705 ×10−7 0.370 ×10−7 8.611 ×10−7 6.085 ×10−7 2.630 ×10−7 9.174 ×10−7 <1.0 ×10−6 B + → τ + υ τ 6.088 ×10−5 0.822 ×10−4 1.916 ×10−4 1.354 ×10−4 1.140 ×10−4 2.040 ×10−4 (1.14±0.27) ×10−4Table 13
Leptonic branching ratio ofD+ meson.
PresentBr [89] [79] [73] [72] [108] Exp. [95] D + → e + υ e 0.984 ×10−8 0.580 ×10−8 2.351 ×10−8 1.77 ×10−8 1.130 ×10−8 2.105 ×10−8 <8.8 ×10−6 D + → μ + υ μ 4.293 ×10−4 2.470 ×10−4 9.991 ×10−4 7.54 ×10−4 4.770 ×10−4 8.977 ×10−4 (3.74±0.17) ×10−4 D + → τ + υ τ 1.055 ×10−3 0.860 ×10−3 1.920 ×10−3 1.79 ×10−3 2.030 ×10−3 2.933 ×10−3 <1.2×10−3Table 14
Leptonic branching ratio ofDs+ meson.
PresentBr [89] [79] [73] [72] [108] Exp. [95] D s + → e + υ e 1.163 ×10−7 0.940 ×10−7 2.251 ×10−7 1.82 ×10−7 1.630 ×10−7 1.391 ×10−7 <8.3 ×10−5 D s + → μ + υ μ 5.078 ×10−3 4.000 ×10−3 9.572 ×10−3 7.74 ×10−3 6.900 ×10−3 5.937 ×10−3 (5.56±0.25) ×10−3 D s + → τ + υ τ 4.451 ×10−3 3.780 ×10−3 9.864 ×10−2 8.2 ×10−2 6.490 ×10−2 5.844 ×10−3 (5.55±0.24)%


## 5. Summary and Conclusion

In this work, we have presented an approximate-analytic solution of theN-dimensional radial Schrödinger equation for the mixture of vector and scalar potentials via the LTM. 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 theN-dimensional space. In three-dimensional space, we have employed the obtained results to study the different properties of the HLM that are not considered in many recent studies. The masses of the scalar, vector, pseudoscalar, and pseudovector forB,Bs,D, andDs mesons have been calculated in the three-dimensional space and in the higher dimensional space in Tables 2–6. Most of the present calculations are close to the experimental data and are improved in comparison with the recent calculations [72–74, 81, 88, 89, 96]. As well, we have computed the masses of the HLM in the higher dimensional space atN=4 andN=5. The dependence of the masses of HLM on the dimensional number is discussed. We found that the masses increase with increasing dimensional number. This result is obtained in [91]. In Tables 7 and 8, the decay constants of the pseudoscalar and vector mesons have been determined in comparison with the results of [72–74, 79, 83, 87, 97]. The calculated ratios of the decay constants ofD mesons (fDs/fD=1.140) and (fDs∗/fD∗=1.070) are close to the experimental ratio (fDs/fD=1.258±0.038).The present results of the decay ratio ofB mesons are in good agreement with the results of [72, 83]. The leptonic decay widths ofB+ meson have been studied in comparison with the results of [74, 79, 107] and the leptonic decay widths ofD+ meson in comparison with the results of [66, 79, 108]. In addition, the leptonic decay widths of Ds+ meson have been studied in comparison with the results of [66, 79, 108].The obtained results of the leptonic decay widths are compared with the results of [66, 67, 74, 107, 108]. We have determined the branching ratio for theB+,D+, and Ds+ mesons that are in good agreement with the experimental data and with the recent studies [72–74, 88, 89]. Therefore, the current potential with used method gives very good predictions for the heavy-light meson properties. We hope to extend this work to include external force as a future work.



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