The Semileptonic $\overline{B}\longrightarrow Dl\overline{\nu}$ and $\overline{B_{s}}\longrightarrow D_{s}l\overline{\nu}$ Decays in Isgur-Wise Approach

We consider a combination of linear confining and Hulth\'en potentials in the Hamiltonian and via the perturbation approach, report the corresponding Isgure-Wise function parameters. Next, we investigate the Isgur-Wise Function for $\overline{B}\longrightarrow Dl\overline{\nu}$ and $\overline{B_{s}}\longrightarrow D_{s}l\overline{\nu}$ semileptonic decays and report the decay width, branching ratio and $|V_{cb}|$ CKM matrix element. A comparison with other models and experimental values is included.


Introduction
The semi-leptonic B to D mesonic decay is the focus of many current studies in the annals of particle physics. Although a verity of approaches have been applied to field, the relatively old but powerful Isgur-Wise function (IWF) approach can be a good candidate to analyze the problem. All form factors of semi-leptonic decays in heavy quark limit can be defined in terms of a single universal function, i.e. the IWF. The main part of the IWF includes the wave function of the meson and some kinematic factors which depend on the four velocities of heavy-light mesons before and after recoil. Decay rates, elements of CKM matrix and branching ratios can be derived from IWF [1]. There are many attempts to obtain IWF in several models [2][3][4][5]. Although different versions of the IWF exist the literature, they all assume the normalization at zero recoil, i.e. the 4-velocities (v and v′) of mesons before and after transitions are identical. Till now, valuable papers have been released and various aspects of formalism have been discussed. Bouzas and Gupta discussed the constraints on the IWF using sum rules for B meson decays [6]. Charm and bottom baryons and mesons have studied within the framework of the Bethe-Salpeter equation by Ivanov et al. and they have reported decay rates of charm and bottom baryons and mesons [7]. Kiselev [11].
The mail aim of this manuscript is the study of IWF for B to D transition. In the next section we will obtain the mesonic wave function using the perturbation method. We then investigate the IWF for semileptonic B to D decay and present the slope, curvature, decay-width, branching ratio and || cb V element of CKM matrix in section (3). Section (4) includes numerical results and comparison with other models. The relevant conclusions are given in section (5).

Mesonic wave function
Our starting square is the three-dimensional radial Schrödinger equation possessing the form where  is the reduced meson mass and , n E denotes the energy of the system. We choose the potential as which is a combination of a linear confinement term and the Hulthén potential. The latter behaves like a Coulomb potential for small values of r and decreases exponentially for large r values [12]. This behavior of the interaction in particular is interest in particle physics. Moreover, the potential has been used in other areas such as nuclear, atomic, solid-state, and chemical physics. As an example, it has been shown that the potential in the form respectively represent the strength and screening range of the potential, can acceptably account for description of interactions between the nucleon and heavy nucleus [13]. In our calculations, we consider the linear term as the parent; and the Hulthén interaction therefore plays the role of the perturbation and the perturbed Hamiltonian is As already mentioned, our parent Hamiltonian is ( 1  ) 2 , , , Now let us limit the study to the ground-state with 1, 0 n  . The corresponding equation is [14] which possesses the wave function where  defines as and Ai denotes the Airy function. The corresponding energy of the system is with 0  being the zero of Airy function which is -2.3194 in the case of ground-state (1s) [14]. Now, we calculate the perturbed wave function by using the first-order perturbation and w  is the perturbed energy. Replacing Eqs. (3), (4), (11) into Eq. (10), we can write We now propose introduce the transformation r ze    and use the approximation to bring Eq. (12) into the form By simplifying Eq. (15), we can write Equating the corresponding powers on both sides of Eq. (16), we get As a result, we have where tot N is the normalization constant of the total wave function. We now go through the semileptonic decay BD   within the IWF approach.

Isgur-Wise Function, Decay-width and Branching Ratio of BD   decay
The IWF is often written as which is well supported by the experimental data [15]. On the other hand, the kinematic accessible region in the semileptonic decays is limited to 1 which depends on the momentum transfer ( 22 2 ( 1) p   ). Because of the implication of the current conservation the form factor, the IWF is normalized to unity at 2 0 p  which corresponds to 1   demonstrating the zero recoil limit [9]. Extending cos(pr) and comparing Eqs. (20) and (21) , gives the so-called slope and curvature parameters as where  is referred to the mesonic zero recoil point. In Fig. (1), the behavior of IWF for some B and D mesons is plotted. The differential semileptonic decay width of BD   in the heavyquark limit has the form [9] where cb V is the element of CKM matrix. Eq. (23) indicates the dependence of the differential semileptonic decay width on the  parameter and the product of 4-velocities of two mesons in Fig. (2).

Conclusions
We considered a mesonic system influenced by linear and Hulthén interactions. Next, using the perturbation technique, and the Isgure-Wise formalism, we obtained the corresponding decay width and branching rations for some B to D decays. Results, when compared with the exsiting data, are motivating and acceptable.