Measurement of the Wrong-Sign Decays $D^0\to K^{+}{\pi }^{-}{\pi }^0$ and $D^0\to K^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$, and Search for $CP$ Violation

Using $281{\mathrm{fb}}^{-1}$ of data from the Belle experiment recorded at or near the ${\rm Y}(4S)$ resonance, we have measured the rates of the “wrong-sign” decays $D^0\to K^{+}{\pi }^{-}{\pi }^0$ and $D^0\to K^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ relative to those of the Cabibbo-favored decays $D^0\to K^{-}{\pi }^{+}{\pi }^0$ and $D^0\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$. These wrong-sign decays proceed via a doubly Cabibbo-suppressed amplitude or via $D^0\mathrm{\text{−}}{\overline{D}}^0$ mixing; the latter has not yet been observed. We obtain $R_{\mathrm{WS}}(K\pi {\pi }^0)=[0.229\pm 0.015(\mathrm{stat}{)}_{-0.009}^{+0.013}(\mathrm{syst})]\%$ and $R_{\mathrm{WS}}(K3\pi )=[0.320\pm 0.018(\mathrm{stat}{)}_{-0.013}^{+0.018}(\mathrm{syst})]\%$. The $CP$ asymmetries are measured to be $-0.006\pm 0.053$ and $-0.018\pm 0.044$ for the $K^{+}{\pi }^{-}{\pi }^0$ and $K^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ final states, respectively.