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We present a (2+1)-flavor lattice QCD calculation of the form factor ratiobetween 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 inputto 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 simulatethe decays with a dynamic final $D$ meson of small momentum and reliablyextract the hadronic matrix elements at nonzero recoil. We report ourpreliminary result for the form factor ratio at the corresponding momentumtransfer of the two decays $f_0^{(s)}(M^2_\pi)/f_0^{(d)}(M^2_K)$.

Semileptonic form factor ratio B_s->D_s/B->D and its application to BR(B^0_s->\mu^+\mu^-)