We report a study of the decay $B^0\to D{K}^{+}{\pi }^{-}$ followed by $D\to K^{-}{\pi }^{+}$, where $D$ indicates $D^0$ or ${\overline{D}}^0$. We reconstruct the $D{K}^{+}{\pi }^{-}$ state in a phase space corresponding to $D{K}^{*}(892{)}^0$. The $CP$-violating angle ${\varphi }_3$ affects its decay rate via the interference between $b\to u$ and $b\to c$ transitions. The result is obtained from a $711{\mathrm{fb}}^{-1}$ data sample that contains $772\times {10}^{6}B\overline{B}$ pairs collected at the ${\rm Y}(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. We measure the ratio ${\mathcal{R}}_{D{K}^{*0}}\equiv \Gamma (B^0\to [K^{-}{\pi }^{+}{]}_D{K}^{+}{\pi }^{-})/\Gamma (B^0\to [K^{+}{\pi }^{-}{]}_D{K}^{+}{\pi }^{-})$ to be $({4.5}_{-5.0-1.8}^{+5.6+2.8})\times {10}^{-2}$, and set an upper limit of ${\mathcal{R}}_{D{K}^{*0}}<0.16$ at the 95% credible interval.

• Received 2 May 2012

DOI:https://doi.org/10.1103/PhysRevD.86.011101