Past exam of the particle and nuclear physics course of the University of Cambridge / 2021 / Section B / 4 / d / Solution

The antiquark $\bar{d}$ in the specified ${\overline{B}}_d^0=b\bar{d}$ is a spectator. For the charmed final state the leading external-W boson tree iswith $c\bar{d}=D^{+}$ and $d\bar{u}={\pi }^{-}$. For the charmless final state replace the active transition by $b\to u+W^{-}$, giving $u\bar{d}={\pi }^{+}$ and the same $d\bar{u}={\pi }^{-}$. Their weak amplitudes have factors $V_{cb}{V}_{ud}^{\ast }$ and $V_{ub}{V}_{ud}^{\ast }$, respectively, up to an irrelevant overall conjugation convention.

The diagrams determine the leading weak flavour factors. Strong-interaction matrix elements and, particularly for the charmless channel, additional loop diagrams can also contribute to physical branching fractions.