We report a study of the suppressed $B$ meson decay $B^{-}\to D{K}^{-}$ followed by $D\to K^{+}{\pi }^{-}$, where $D$ indicates a $D^0$ or ${\overline{D}}^0$ state. The two decay paths interfere and provide information on the $CP$-violating angle ${\varphi }_3$. We use a data sample containing $657\times {10}^6$ $B\overline{B}$ pairs recorded at the ${\rm Y}(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ storage ring. We do not find significant evidence for the mode $B^{-}\to D{K}^{-}$, $D\to K^{+}{\pi }^{-}$, and set an upper limit of $r_B<0.19$, where $r_B$ is the magnitude of the ratio of amplitudes $|A(B^{-}\to {\overline{D}}^0{K}^{-})/A(B^{-}\to D^0{K}^{-})|$. The decay $B^{-}\to D{\pi }^{-}$, $D\to K^{+}{\pi }^{-}$ is also analyzed as a reference, for which we observe a signal with $6.6\sigma$ significance, and measure the charge asymmetry ${\mathcal{A}}_{D\pi }$ to be $-{0.02}_{-0.16}^{+0.15}(\mathrm{stat})\pm 0.04(\mathrm{syst})$. In addition, the ratio $\mathcal{B}(B^{-}\to D^0{K}^{-})/\mathcal{B}(B^{-}\to D^0{\pi }^{-})$ is measured to be $[6.77\pm 0.23(\mathrm{stat})\pm 0.30(\mathrm{syst})]\times {10}^{-2}$.

• Received 13 April 2008

DOI:https://doi.org/10.1103/PhysRevD.78.071901