Precision measurement of CP violation and branching fractions in $B^{\pm} \to K^0_{\mathrm{S}} h^{\pm}$ $(h = π, K)$ decays and search for the rare decay $B_c^{\pm} \to K^0_{\mathrm{S}} K^{\pm}$

The decay $B^{\pm} \to K^0_{\mathrm{S}} π^{\pm}$, with a $CP$ asymmetry expected to be close to zero in the Standard Model, is theoretically clean and sensitive to potential new physics. An analysis of the decays $B^{\pm} \to K^0_{\mathrm{S}} π^{\pm}$ and $B^{\pm} \to K^0_{\mathrm{S}} K^{\pm}$ is performed using proton-proton collision data collected by the LHCb experiment at a center-of-mass energy of $13\,\mathrm{TeV}$, corresponding to an integrated luminosity of $5.4\,\mathrm{fb}^{-1}$. The \CP asymmetries are determined to be ${\cal A}^{CP}(B^{\pm} \to K^0_{\mathrm{S}} π^{\pm})=-0.028\pm 0.009\pm 0.009$ and ${\cal A}^{CP}(B^{\pm} \to K^0_{\mathrm{S}} K^{\pm})=0.118\pm 0.062 \pm 0.031$, and the branching fraction ratio is measured to be ${\cal B}(B^{\pm} \to K^0_{\mathrm{S}} K^{\pm})/{\cal B}(B^{\pm} \to K^0_{\mathrm{S}} π^{\pm})=0.055\pm 0.004 \pm 0.002$, where the first uncertainties are statistical and the second are systematic. These results are the most precise measurements of these quantities to date. A search for the rare decay $B_c^{\pm} \to K^0_{\mathrm{S}} K^{\pm}$ is also performed. No significant signal is observed, and the upper limit on the product of the branching fraction ratio ${\cal B}(B_c^{\pm} \to K^0_{\mathrm{S}} K^{\pm})/{\cal B}(B^{\pm} \to K^0_{\mathrm{S}} π^{\pm})$ and the fragmentation-fraction ratio $f_c/f_u$ is set to be 0.015 (0.016) at the 90\% (95\%) confidence level.