We present a lattice quantum chromodynamics determination of the scalar and vector form factors for the $B_s\to D_s\ell \nu$ decay over the full physical range of momentum transfer. In conjunction with future experimental data, our results will provide a new method to extract $|V_{cb}|$, which may elucidate the current tension between exclusive and inclusive determinations of this parameter. Combining the form factor results at nonzero recoil with recent HPQCD results for the $B\to D\ell \nu$ form factors, we determine the ratios $f_0^{B_s\to D_s}(M_{\pi }^2)/f_0^{B\to D}(M_K^2)=1.000(62)$ and $f_0^{B_s\to D_s}(M_{\pi }^2)/f_0^{B\to D}(M_{\pi }^2)=1.006(62)$. These results give the fragmentation fraction ratios $f_s/f_d=0.310(30{)}_{\text{stat}}(21{)}_{\text{syst}}(6{)}_{\text{theor}}(38{)}_{\text{latt}}$ and $f_s/f_d=0.307(16{)}_{\text{stat}}(21{)}_{\text{syst}}(23{)}_{\text{theor}}(44{)}_{\text{latt}}$, respectively. The fragmentation fraction ratio is an important ingredient in experimental determinations of $B_s$ meson branching fractions at hadron colliders, in particular for the rare decay $\mathcal{B}(B_s\to {\mu }^{+}{\mu }^{-})$. In addition to the form factor results, we make the first prediction of the branching fraction ratio $R(D_s)=\mathcal{B}(B_s\to D_s\tau \nu )/\mathcal{B}(B_s\to D_s\ell \nu )=0.301(6)$, where $\ell$ is an electron or muon. Current experimental measurements of the corresponding ratio for the semileptonic decays of $B$ mesons disagree with Standard Model expectations at the level of nearly four standard deviations. Future experimental measurements of $R(D_s)$ may help understand this discrepancy.

• Received 5 April 2017

DOI:https://doi.org/10.1103/PhysRevD.95.114506