  We present a (2+1)-flavor lattice QCD calculation of the form factor ratio
between the semileptonic decays $\bar{B}^0_s \to D^+_sl^-\bar{\nu} $ and
$\bar{B}^0 \to D^+l^-\bar{\nu} $. This ratio is an important theoretical input
to the hadronic determination of the $B$ meson fragmentation fraction ratio
$f_s/f_d$ which enters in the measurement of $\mathrm{BR}(B^0_s\to
\mu^+\mu^-)$. Small lattice spacings and high statistics enable us to simulate
the decays with a dynamic final $D$ meson of small momentum and reliably
extract the hadronic matrix elements at nonzero recoil. We report our
preliminary result for the form factor ratio at the corresponding momentum
transfer of the two decays $f_0^{(s)}(M^2_\pi)/f_0^{(d)}(M^2_K)$.

High Energy Physics - Lattice