Branching fraction measurements of $B_{d,s}^0$ decays to $K_{\rm \scriptscriptstyle S}^0h^{\pm}h^{'\mp}$ final states, including first observation of $B_s^0 \to K_{\rm \scriptscriptstyle S}^0K^{\pm}\pi^{\mp}$

The data recorded in 2011 by the LHCb experiment, corresponding to an integrated luminosity of 1.0 fb$^{-1}$, have been analysed to search for charmless three-body decays of both $B^0$ and $B_{s}^0$ mesons with a $K_{\rm \scriptscriptstyle S}^0$ meson in the final state. Branching fractions of these $B_{d,s}^0 \to K_{\rm \scriptscriptstyle S}^0h^{\pm}h^{'\mp}$ decay modes, relative to the well measured $B^0 \to K_{\rm \scriptscriptstyle S}^0\pi^{+}\pi^{-}$ decay, are derived. The first observation of the decay mode $B_s^0 \to K_{\rm \scriptscriptstyle S}^0K^{\pm}\pi^{\mp}$ and confirmation of the recent observation by the \babar experiment of the decay $B^0 \to K_{\rm \scriptscriptstyle S}^0K^{\pm}\pi^{\mp}$ are reported in this document, with relative branching fractions \begin{eqnarray*} \nonumber \frac{{\cal B}(B_s^0 \to K_{\rm \scriptscriptstyle S}^0K^{\pm}\pi^{\mp})}{{\cal B}(B^0 \to K_{\rm \scriptscriptstyle S}^0\pi^{+}\pi^{-})} &=& 1.96 \pm 0.15 \; {\rm(stat.)} \, \pm 0.20 \; ({\rm syst.}), \\ \nonumber \frac{{\cal B}(B^0 \to K_{\rm \scriptscriptstyle S}^0K^{\pm}\pi^{\mp})}{{\cal B}(B^0 \to K_{\rm \scriptscriptstyle S}^0\pi^{+}\pi^{-})} &=& 0.117 \pm 0.018 \; {\rm(stat.)} \, \pm 0.018 \; ({\rm syst.}). \end{eqnarray*} Additionally, statistical evidence for the CKM-suppressed decay modes $B_s^0 \to K_{\rm \scriptscriptstyle S}^0\pi^{+}\pi^{-}$ and $B_s^0 \to K_{\rm \scriptscriptstyle S}^0K^{+}K^{-}$ is obtained.