Search for $B^{\pm }\to [K^{\mp }{\pi }^{\pm }{]}_D{K}^{\pm }$ and Upper Limit on the $b\to u$ Amplitude in $B^{\pm }\to D{K}^{\pm }$

We search for $B^{\pm }\to [K^{\mp }{\pi }^{\pm }{]}_D{K}^{\pm }$ decays, where $[K^{\mp }{\pi }^{\pm }{]}_D$ indicates that the $K^{\mp }{\pi }^{\pm }$ pair originates from the decay of a $D^0$ or ${\overline{D}}^0$. Results are based on $120\times {10}^6$ ${\rm Y}(4S)\to B\overline{B}$ decays collected with the BABAR detector at SLAC. We set an upper limit on the ratio ${\mathcal{R}}_{K\pi }\equiv \frac{[\Gamma (B^{+}\to [K^{-}{\pi }^{+}{]}_D{K}^{+})+\Gamma (B^{-}\to [K^{+}{\pi }^{-}{]}_D{K}^{-})]}{[\Gamma (B^{+}\to [K^{+}{\pi }^{-}{]}_D{K}^{+})+\Gamma (B^{-}\to [K^{-}{\pi }^{+}{]}_D{K}^{-})]}<0.026$ ($90\%\mathrm{C}.\mathrm{L}.$). This constrains the amplitude ratio $r_B\equiv |A(B^{-}\to {\overline{D}}^0{K}^{-})/A(B^{-}\to D^0{K}^{-})|<0.22$ (90% C.L.), consistent with expectations. The small value of $r_B$ favored by our analysis suggests that the determination of the Cabibbo-Kobayashi-Maskawa phase $\gamma$ from $B\to DK$ will be difficult.