Search for $CP$ violation in $D_{(s)}^{+}\to K^{+}{K}_S^0{h}^{+}{h}^{-}$ $(h=K,\ddot{I}€)$ decays and observation of the Cabibbo-suppressed decay $D_s^{+}\to K^{+}{K}^{-}{K}_S^0\ddot{I}{€}^{+}$

Abstract

We search for $CP$ violation by measuring a $T$-odd asymmetry in the Cabibbo-suppressed $D^{+}\to K^{+}{K}_S^0\ddot{I}{€}^{+}\ddot{I}{€}^{-}$ decay, and in the Cabibbo-favored $D_s^{+}\to K^{+}{K}_S^0\ddot{I}{€}^{+}\ddot{I}{€}^{-}$ and $D^{+}\to K^{+}{K}^{-}{K}_S^0\ddot{I}{€}^{+}$ decays. We use 980 ${\mathrm{fb}}^{-1}$ of data collected by the Belle detector running at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. The $CP$-violating $T$-odd parameter $a_{CP}^{T\text{-}\mathrm{odd}}$ is measured to be $a_{CP}^{T\text{-}\mathrm{odd}}(D^{+}\to K^{+}{K}_S^0\ddot{I}{€}^{+}\ddot{I}{€}^{-})=(0.34\pm 0.87\pm 0.32)\%,$ $a_{CP}^{T\text{-}\mathrm{odd}}(D_s^{+}\to K^{+}{K}_S^0\ddot{I}{€}^{+}\ddot{I}{€}^{-})=(-0.46\pm 0.63\pm 0.38)\%,$ and $a_{CP}^{T\text{-}\mathrm{odd}}(D^{+}\to K^{+}{K}^{-}{K}_S^0\ddot{I}{€}^{+})=(-3.34\pm 2.66\pm 0.35)\%,$ where the first uncertainty is statistical and the second is systematic. We also report the first observation of the Cabibbo-suppressed decay $D_s^{+}\to K^{+}{K}^{-}{K}_S^0\ddot{I}{€}^{+}$. The branching fraction is measured relative to that of the analogous Cabibbo-favored decay : $B(D_s^{+}\to K^{+}{K}^{-}{K}_S^0\ddot{I}{€}^{+})/B(D_s^{+}\to K^{+}{K}_S^0\ddot{I}{€}^{+}\ddot{I}{€}^{-})=(1.36\pm 0.15\pm 0.04)\%$.