We search for $CP$ violation by measuring a $T$-odd asymmetry in the Cabibbo-suppressed $D^{+}\to K^{+}{K}_S^0{\pi }^{+}{\pi }^{-}$ decay, and in the Cabibbo-favored $D_s^{+}\to K^{+}{K}_S^0{\pi }^{+}{\pi }^{-}$ and $D^{+}\to K^{+}{K}^{-}{K}_S^0{\pi }^{+}$ 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 $a_{CP}^{T\text{−}\mathrm{odd}}(D^{+}\to K^{+}{K}_S^0{\pi }^{+}{\pi }^{-})=(0.34\pm 0.87\pm 0.32)\%$, $a_{CP}^{T\text{−}\mathrm{odd}}(D_s^{+}\to K^{+}{K}_S^0{\pi }^{+}{\pi }^{-})=(-0.46\pm 0.63\pm 0.38)\%$, and $a_{CP}^{T\text{−}\mathrm{odd}}(D^{+}\to K^{+}{K}^{-}{K}_S^0{\pi }^{+})=(-3.34\pm 2.66\pm 0.35)\%$, where the first uncertainty is statistical and the second is systematic. We also report the first observation of the Cabibbo-suppressed decay $D_s^{+}\to K^{+}{K}^{-}{K}_S^0{\pi }^{+}$. The branching fraction is measured relative to that of the analogous Cabibbo-favored decay: $\mathcal{B}(D_s^{+}\to K^{+}{K}^{-}{K}_S^0{\pi }^{+})/\mathcal{B}(D_s^{+}\to K^{+}{K}_S^0{\pi }^{+}{\pi }^{-})=(1.36\pm 0.15\pm 0.04)\%$.

• Received 18 May 2023
• Accepted 21 November 2023

DOI:https://doi.org/10.1103/PhysRevD.108.L111102

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Published by the American Physical Society