${\bf B_s\to K^{(*)} \ell\bar \nu}$, Angular Analysis, S-wave Contributions and ${\bf |V_{ub}|}$

Abstract

We analyse the $\overline B_s^0\to K^+l^-\bar \nu$ and $\overline B_s^0\to K^{*+}(\to K\pi) \ell^-\bar \nu$ decays that are valuable for extracting the CKM matrix element $|V_{ub}|$. We calculate the differential and integrated partial widths in units of $|V_{ub}|^2$ based on various calculations of hadronic form factors and in particular the latest Lattice QCD calculation of the $B_s\to K^*$ form factors. For the decay $\overline B_s^0 \to K\pi \ell\bar \nu$, we formulate the general angular distributions with the inclusion of the various partial-wave $K\pi$ contributions. Using the results for the $K\pi$ scalar form factor calculated from unitarized chiral perturbation theory, we explore the S-wave effects on angular distribution variables and demonstrate that they may not be negligible, considering the high precision expected in future measurements. We also briefly discuss the impact of the S-wave $\pi\pi$ contributions in the $B^-\to \pi^+\pi^-\ell \bar \nu$ decay and provide estimates for the mode $B^-\to K^+K^-\ell \bar \nu$. The studies of these channels in future can not only be used to determine $|V_{ub}|$, but may also provide valuable information on the $K\pi$ and $\pi\pi$ phase shifts.