# The semileptonic \(\bar{B}\rightarrow D\ell \bar{\nu }\) and \(\bar{B}_s \rightarrow D_s \ell \bar{\nu }\) decays in Isgur–Wise approach

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## Abstract

We consider a combination of linear confining and Hulthén potentials in the Hamiltonian and, via the perturbation approach, report the corresponding Isgur–Wise function parameters. Next, we investigate the Isgur–Wise function for \(\bar{B}\rightarrow D\ell \bar{\nu }\) and \(\bar{B}_s \rightarrow D_s \ell \bar{\nu }\) semileptonic decays and report the decay width, branching ratio, and \(\vert V_{cb} \vert \)CKM matrix element. A comparison with other models and experimental values is included.

## 1 Introduction

The semileptonic \(B\) to \(D\) mesonic decay is the focus of many current studies in the annals of particle physics. Although a plethora of approaches have been applied to the field, the relatively old but powerful Isgur–Wise function (IWF) approach is a good candidate for use to analyze the problem. Isgur and Wise obtained simple and appealing relations of the form factors for weak pseudoscalar to pseudoscalar and pseudoscalar to vector transitions for various hadronic matrix elements [1, 2]. If \(m_Q \gg \Lambda _\mathrm{QCD}\,(m_Q \) is the heavy-quark mass and \(\Lambda _\mathrm{QCD} \) is QCD scale parameter), the number of form factors in semileptonic decay reduces. Next, all form factors of semileptonic decays in the heavy-quark limit can be defined in terms of a single universal function, i.e. the IWF [3]. 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. The calculation of the IWF is the essential step in all calculations of the branching ratios, \(V_\mathrm{cb} \) element of the CKM matrix, decay rates, and branching ratios [4]. To get the \(V_\mathrm{cb}\) element of the CKM matrix we can use the IWF by the experimental data on \(V_\mathrm{cb}\). There have been many attempts to obtain the IWF in several models and different non-perturbative methods [5, 6, 7, 8]. As the kinematic dependence of the IWF is unknown, there exist different parameterizations of the IWF and different non-perturbative methods to calculate the value at zero recoil as well as the slope. In fact, in the heavy-quark symmetry the form factor can only depend on \(\omega =v.v{^\prime } \), which connects the rest frames of the initial and final state mesons. Because of current conservation, this form factor is normalized to unity at zero recoil. The hadronic form factors of \(B\) to \(D\) meson transitions are among the important applications of the heavy-quark symmetry. In this limit, all form factors of \(B\) to \(D\) meson transitions can be analyzed via the IWF \(\xi (\omega )\) of the velocity transfer \(\omega \) [9]. Until now, valuable papers have been published and various aspects of formalism have been discussed. Bouzas and Gupta discussed the constraints on the IWF using sum rules for the \(B\) meson decays [10]. Charm and bottom baryons and mesons have been studied within the framework of the Bethe–Salpeter equation by Ivanov et al. They also reported the decay rates of charm and bottom baryons and mesons [11]. Kiselev determined the slope of the IWF and the \(\vert V_\mathrm{cb} \vert \) matrix element for semileptonic \(B\rightarrow D\ell \nu \) decay [12]. The theory and phenomenology of weak decays of \(B\) mesons were reviewed by Neubert [13]. Leptonic decays of heavy pseudoscalar mesons and semileptonic decays of mesons were studied in Refs. [14, 15]. Ebert et al. studied the exclusive semileptonic decays of \(B\) mesons to orbitally excited \(D\) mesons in the framework of the relativistic quark model [16]. A lattice study of semileptonic \(B\) decays was presented by Bowler et al. (UKQCD collaboration) [17]. Measurements of the semileptonic Decays \(B\rightarrow D\ell \nu \) were studied by Ref. [18] in 2008. Bernlochner et al. explored the rates of semileptonic \(B\) decays [19]. Atoui investigated the results of lattice QCD study of the exclusive semileptonic \(B_s \rightarrow D_s \ell \nu \) decay form factors in the region near zero recoil [20]. The hadronic form factors of \(B\) semileptonic decays at both zero and nonzero recoil were computed by Qiu et al. [21].

The main aim of this manuscript is the study of the IWF for the \(B\) 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 \(\vert V_\mathrm{cb} \vert \) element of the CKM matrix in Sect. 3. Section 4 includes the numerical results and comparison with other models. The relevant conclusions are given in Sect. 5.

## 2 Mesonic wave function

## 3 Isgur–Wise function, decay width, and branching ratio of \(\bar{B}\rightarrow D\ell \bar{\nu }\) decay

Slope, curvature of IWF for \(B\), \(D, B_{s}\), \(D_{s}\) mesons (\(V_0 =-\)1.61 GeV, \(\alpha =0.1\) GeV, \(b=0.76\,\mathrm{GeV}^2\) for \(B,D\) and \(V_0 =-\)1.61 GeV, \(\alpha =0.1\) GeV, \(b=0.6\,\mathrm{GeV}^2\) for \(B_s ,D_s \))

Decay width, branching ratios, and \(\vert V_\mathrm{cb} \vert \) for \(\bar{B}\rightarrow D\ell \bar{\nu }\)

Quantity | Our model | Other models | Uncertainty (%) |
---|---|---|---|

\(\Gamma \) (in \(10^{10}\mathrm{s}^{-1})\) | 1.51 | 1.413 [12] | 6.8 |

\(1.43\pm 0.08\) [18] | 5.5 | ||

\(Br\,(\mathrm{in}\,\,\% )\) | 2.48 | 2.23 \(\pm \) 0.12 [29] | 11.2 |

\(2.31\pm 0.09\) [19] | 7.3 | ||

\(2.34 \,\pm \, 0.03 \,\pm \, 0.13\) [18] | 5.9 | ||

\(\vert V_\mathrm{cb} \vert \) | \(0.037\le \left| {\,V_\mathrm{cb} } \right| \le 0.039\) | \(0.042\pm 0.001\) [29] | 7.1–11.9 |

Decay width, branching ratios, and \(\vert V_\mathrm{cb} \vert \) for \(\bar{B}_s \rightarrow D_s \ell \bar{\nu }\)

Quantity | Our model | Uncertainty (%) |
---|---|---|

\(\Gamma (\mathrm{in}\,\,10^{10}\,\,\mathrm{s}^{-1})\) | 0.89 | 4.3 |

\(Br\,(\mathrm{in}\,\,\% )\) | 1.35 | 2.7 |

\(\vert V_\mathrm{cb} \vert \) | 0.038 | 7.3 |

## 4 Results and discussion

## 5 Conclusions

We considered a mesonic system influenced by linear and Hulthén interactions. Next, using the perturbation technique, and the Isgur–Wise formalism, we obtained the corresponding decay width and branching rations for some \(B\) to \(D\) decays. The results are in good agreement with experimental values and other models. Our model is also simple, with no intricacies of a mathematical nature.

## Notes

### Acknowledgments

The authors would like to thank the referee for giving valuable suggestions and criticisms improving the paper.

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