  We present a lattice quantum chromodynamics determination of the scalar and
vector form factors for the $B_s \rightarrow D_s \ell \nu$ 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 non-zero recoil with
recent HPQCD results for the $B \rightarrow D \ell \nu$ form factors, we
determine the ratios $f^{B_s \rightarrow D_s}_0(M_\pi^2) / f^{B \rightarrow
D}_0(M_K^2) = 1.000(62)$ and $f^{B_s \rightarrow D_s}_0(M_\pi^2) / f^{B
\rightarrow D}_0(M_\pi^2) = 1.006(62)$. These results give the fragmentation
fraction ratios $f_s/f_d =
0.310(30)_{\mathrm{stat.}}(21)_{\mathrm{syst.}}(6)_{\mathrm{theor.}}(38)_{\mathrm{latt.}}
$ and $f_s/f_d =
0.307(16)_{\mathrm{stat.}}(21)_{\mathrm{syst.}}(23)_{\mathrm{theor.}}(44)_{\mathrm{latt.}}$,
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 ${\cal B}(B_s \rightarrow \mu^+
\mu^-)$. In addition to the form factor results, we make the first prediction
of the branching fraction ratio $R(D_s) = {\cal B}(B_s\to D_s\tau\nu)/{\cal
B}(B_s\to D_s\ell\nu) = 0.301(6)$, 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.
