Advances in High Energy Physics

Volume 2018, Article ID 7032041, 12 pages

https://doi.org/10.1155/2018/7032041

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

^{1}Department of Applied Mathematics, Faculty of Science, Menoufia University, Shebeen El-Kom, Egypt^{2}Department of Basic Science, Modern Academy of Engineering and Technology, Cairo, Egypt

Correspondence should be addressed to M. Abu-Shady; moc.liamg@ydahsuba.rd

Received 8 March 2018; Revised 15 May 2018; Accepted 3 June 2018; Published 12 July 2018

Academic Editor: Chun-Sheng Jia

Copyright © 2018 M. Abu-Shady and E. M. Khokha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

An analytic solution of the* N*-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 the* N*-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 for* B*,* B*_{s},* D*, and* D*_{s} 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 the* B*^{+},* D*^{+}, and 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.

#### 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 () 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 -dimensional Schrödinger equation has been studied with different forms of spherically symmetric potentials [28–33]. The -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]. The* N*-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 , , , , , and 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 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 evaluated [67]; besides the rare decays of and mesons into dimuon (*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 obtained using the variational analysis in the light quark model [70]. The spectroscopy of bottomonium and* B* 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 of* D* and* D*_{s} mesons have been investigated using the quark-antiquark potential in [73]. The* B* and 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 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 of* B* and 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 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 (3000) and (3040) with two 2P (1^{+}) quantum number assignments. The leptonic and semileptonic decays have been analyzed using the covariant quark model with infrared confinement within the standard model framework [85]. The weak decays of* B*, , and into -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 and mesons [88],* D* and* D*_{s} mesons [89], and also the radiative transitions and the mixing parameters of the -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 the* N*-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 the* N*-dimensional space and the corresponding eigenfunctions. So far no attempt has been made to solve the* N*-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 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.

#### 2. Overview of Laplace Transformation Method

The Laplace transform or of a function is defined by [92]If there is some constant such that for sufficiently large , the integral in (1) exists for Re for . The Laplace transform may fail to exist because of a sufficiently strong singularity in the function as . In particularwhere is the gamma function. The Laplace transform has the derivative propertieswhere the superscript stands for the -th derivative with respect to for and with respect to for . If is the singular point, the Laplace transform behaves near as and then for On the other hand, if near origin behaves like with , then behaves near as

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

The* N*-dimensional radial Schrödinger equation that describes the interaction between quark-antiquark systems takes the form [41]where represent the angular quantum number and the dimensional number greater than one, respectively, and is the reduced mass of the quark-antiquark system.

In the nonrelativistic quark model, the quark-antiquark potential consists of the spin independent potential and the spin dependent potential , respectively:The spin independent potential is taken as a combination of vector and scalar parts [93]: where and are the vector and scalar parts, respectively, and stands for the mixing coefficient.* ɑ*,

*b*, and

*c*are arbitrary parameters where

*a*,

*b*, and

*c*> 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]while the spin-orbit and tensor terms give the fine structure of the states, the spin-spin interaction term describes the hyperfine splitting of the state, and is an angular quantum operator, and is a spin operator (for detail, see [94]).where is radial Laplace operator. The diagonal elements of the are defined. Substituting (11)-(16) into (9) then the nonrelativistic quark-antiquark potential takes the formwhereSubstituting (21) into (8), then whereThe complete solution of (25) takes the formwhere the term confirms that the solution is bounded at . The function is yet to be determined. From (27) we getSubstituting (27), (28), and (29) into (25), then, whereIn order to apply the Laplace transform of the above differential equation, the parametric condition is taken as in [52, 54]. Thus, (32) has a solution We take the physical solution of (32) as in [52, 54].

Substituting (34) into (30) yieldsBy expanding the term around , where and is a parameter as in [36, 56], we get Substituting (37) into (36) yieldswhereThe Laplace transform is defined as and taking boundary condition yields HereThe singular point of (40) is By using the condition of (5), the solution of (40) takes the form

From (42), Substituting (42)-(44) into (40), we obtain the following relations:Using (26), (39), and (41) and the set of (45)-(47), then, the energy eigenvalue of (8) in the* N*-dimensional space is given by the relationTake the inverse Laplace transform such that The function takes the following form:Using (11), (13), and (23), the eigenfunctions of (9) take the following form:From the condition , the normalization constant can be computed. In addition, the wave equation satisfies the boundary condition .

#### 4. Discussion of Results

In Figure 1, 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 and 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.