Estimating $r_B^{D\pi }$ as an input to the determination of the CKM angle $\gamma$

The interference between Cabibbo-favored and Cabibbo-suppressed $B\to D\pi$ decay amplitudes provides sensitivity to the CKM angle $\gamma$. The relative size of the interfering amplitudes is an important ingredient in the determination of $\gamma$. Using branching fractions from various $B\to Dh$ decays, and the measured value for $r_B^{DK}$, the magnitude of the amplitude ratio of $B^{+}\to D^0{\pi }^{+}$ and $B^{+}\to {\overline{D}}^0{\pi }^{+}$ decays is estimated to be $r_B^{D\pi }=0.0053\pm 0.0007$.

Figure 1

(a), (b) The color-allowed (tree) ($T$) and color-suppressed ($C$) topologies contributing to the $B^{+}\to {\overline{D}}^0{\pi }^{+}$ amplitude proceeding through $V_{cb}$, and (c), (d) the color-suppressed ($C^{ub}$) and annihilation topologies contributing to the $B^{+}\to D^0{\pi }^{+}$ amplitude proceeding through $V_{ub}$.