Branching fraction and CP asymmetry of the decays $B^{+}\to K_{\mathrm{S}}^0{\pi }^{+}$ and $B^{+}\to K_{\mathrm{S}}^0{K}^{+}$

An analysis of $B^{+}\to K_{\mathrm{S}}^0{\pi }^{+}$ and $B^+ \to K_{\rm \scriptscriptstyle S}^0 K^+$ decays is performed with the LHCb experiment. The $pp$ collision data used correspond to integrated luminosities of $1{fb}^{-1}$ and $2{fb}^{-1}$ collected at centre-of-mass energies of $\sqrt{s}=7\mathrm{T}\mathrm{e}\mathrm{V}$ and $\sqrt{s}=8\mathrm{,Te\kern -0.1em V}$, respectively. The ratio of branching fractions and the direct {\it CP} asymmetries are measured to be $$\mathcal{B}$(B^+ \to K_{\rm \scriptscriptstyle S}^0 K^+)/$\mathcal{B}$(B^+ \to K_{\rm \scriptscriptstyle S}^0 \pi^+) = 0.064 \pm 0.009$\text{(stat.)}$ \pm 0.004$\text{(syst.)}$$,$$\mathcal{A}$^{\it CP}(B^+ \to K_{\rm \scriptscriptstyle S}^0 \pi^+) = -0.022 \pm 0.025$\text{(stat.)}$ \pm 0.010$\text{(syst.)}$$and$$\mathcal{A}$^{\it CP}(B^+ \to K_{\rm \scriptscriptstyle S}^0 K^+) = -0.21 \pm 0.14$\text{(stat.)}$ \pm 0.01$\text{(syst.)}$$\mathrm{.}Thedatasampletakenat$\sqrt{s}=7\mathrm{,Te\kern -0.1em V}$isusedtosearchfor$B_c^+ \to K_{\rm \scriptscriptstyle S}^0 K^+$decaysandresultsintheupperlimit$(f_c\cdot$\mathcal{B}$(B_c^+ \to K_{\rm \scriptscriptstyle S}^0 K^+))/(f_u\cdot$\mathcal{B}$(B^+ \to K_{\rm \scriptscriptstyle S}^0 \pi^+)) < 5.8\times10^{-2}\textrm{at 90% confidence level}$,where$f_c$and$f_u$denotethehadronisationfractionsofa$\bar{b}$quarkintoa$B_c^+$ora$B^+$ meson, respectively.