Measurement of the branching fractions of the decays $B_s^0 \rightarrow D_s^\mp K^\pm$ and $B_s^0 \rightarrow D_s^- \pi^+$

The decay mode $B^{0}_{s} \to D^{\mp}_{s} K^{\pm} $ allows for one of the theoretically cleanest measurements of the CKM angle $\gamma$ through the study of time-dependent $CP$ violation. This paper reports a measurement of its branching fraction relative to the Cabibbo-favoured mode $B^{0}_{s} \to D^{-}_{s} \pi^{+}$ based on a data sample of 0.37 fb$^{-1}$ proton-proton collisions at $\sqrt{s} = 7$ TeV collected in 2011 with the LHCb detector. In addition, the ratio of $B$ meson production fractions $f_s/f_d$, determined from semileptonic decays, together with the known branching fraction of the control channel $B^{0} \to D^{-} \pi^{+}$, is used to perform an absolute measurement of the branching fractions: $B (B^0_s \to D^-_s \pi^+) \;= (2.95 \pm 0.05 \pm 0.17^{\,+\,0.18}_{\,-\,0.22}) \times 10^{-3}$, $B (B^0_s \to D^\mp_s K^\pm) = (1.90 \pm 0.12 \pm 0.13^{\,+\,0.12}_{\,-\,0.14}) \times 10^{-4}\,$, where the first uncertainty is statistical, the second the experimental systematic uncertainty, and the third the uncertainty due to $f_s/f_d$.