Measurements of the relative and absolute branching fractions of the decays $B^0_s\rightarrow D^\mp_s K^\pm$ and $B^0_s\rightarrow D^-_s \pi^+$

This report documents the observation of the decay $B^0_s\rightarrow D^\mp_s K^\pm$ at LHCb and a measurement of its branching fraction relative to the Cabibbo-favoured mode $B^0_s\rightarrow D^-_s \pi^+$. The analysis is based on a data sample of 336 pb$^{-1}$ $pp$ collisions at a centre-of-mass energy $\sqrt{s} = 7$ TeV collected in 2011. The measured relative branching fraction is found to be $\frac{\mathcal{B}\left(B^0_s\rightarrow D^\mp_s K^\pm\right)}{\mathcal{B}\left(B^0_s\rightarrow D^-_s \pi^+ \right)} = 0.0647 \pm 0.0044 \; {\rm(stat.)} \; {}^{+0.0039}_{-0.0043} \; {\rm(syst.)}$ . The LHCb measurement of the ratio of $b$-fragmentation fractions $f_s/f_d$ using semileptonic decays, combined with the yields obtained with 2010 data, is used to obtain a measurement of the branching fraction of the decay $B^0_s\rightarrow D^-_s \pi^+$ : $\mathcal{B}(B^0_s\rightarrow D^-_s \pi^+ ) = (3.04 \pm 0.19 \; {\rm(stat.)} \pm 0.23 \; {\rm(syst.)} \; {}^{+0.18}_{-0.16} \; (f_s/f_d) ) \times 10^{-3}$. Finally, the two measurements are combined to obtain $\mathcal{B} (B^0_s\rightarrow D^\mp_s K^\pm) = 1.97 \pm 0.18^{+0.19 + 0.11}_{-0.20 - 0.10} \times 10^{-4}$. Both branching fraction measurements are significantly more precise than the existing world average values.