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

The interference between Cabibbo-favoured 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 $B^+\to \bar{D}^0\pi^+$ decays is estimated to be $r_{B}^{D\pi} = 0.0053 \pm 0.0007$.