Anatomy of $B\rightarrow D\bar{D}$ Decays

The decays $B_d^0\rightarrow D_d^-D_d^+$ and $B_s^0\rightarrow D_s^-D_s^+$ probe the CP-violating mixing phases $\phi_d$ and $\phi_s$, respectively. The theoretical uncertainty of the corresponding determinations is limited by contributions from penguin topologies, which can be included with the help of the $U$-spin symmetry of the strong interaction. We analyse the currently available data for $B^0_{d,s}\rightarrow D_{d,s}^-D_{d,s}^+$ decays and those with similar dynamics to constrain the involved non-perturbative parameters. Using further information from semileptonic $B_d^0\rightarrow D_d^-\ell^+\nu_{\ell}$ decays, we perform a test of the factorisation approximation and take non-factorisable $SU(3)$-breaking corrections into account. The branching ratios of the $B_d^0\rightarrow D_d^-D_d^+$, $B_s^0\rightarrow D_s^-D_d^+$ and $B_s^0\rightarrow D_s^-D_s^+$, $B_d^0\rightarrow D_d^-D_s^+$ decays show an interesting pattern which can be accommodated through significantly enhanced exchange and penguin annihilation topologies. This feature is also supported by data for the $B_s^0\rightarrow D_d^-D_d^+$ channel. Moreover, there are indications of potentially enhanced penguin contributions in the $B_d^0\rightarrow D_d^-D_d^+$ and $B_s^0\rightarrow D_s^-D_s^+$ decays, which would make it mandatory to control these effects in the future measurements of $\phi_d$ and $\phi_s$. We discuss scenarios for high-precision measurements in the era of Belle II and the LHCb upgrade.