Prediction of the $CP$ asymmetry $C_{00}$ in $B^0 \to D^0\overline{D^0}$ decay

Of all $B \to D \overline{D}$ decays, the $B^0 \to D^0 \overline{D^0}$ decay has the smallest observed branching ratio as it takes place primarily via the suppressed $W$-exchange diagram. The $CP$ asymmetry for this mode is yet to be measured experimentally. By exploiting the relationship among the decay amplitudes of $B \to D\overline{D}$ decays (using isospin and topological amplitudes) we are able to relate the $CP$ asymmetries and branching ratios by a simple expression. This enables us to predict the $CP$ asymmetry $C_{00}$ in $B^0 \to D^0 \overline{D^0}$. While the predicted central values of $C_{00}$ are outside the physically allowed region, they are currently associated with large uncertainties owing to the large errors in the measurements of the $B^0 \to D^0 \overline{D^0}$ branching ratio ($B_{00}$), the other $CP$ asymmetries $C_{+-}$ (of $B^0 \to D^+ D^-$) and $A_{\text{CP}}$ (of $B^+ \to D^+ \overline{D^0}$). With a precise determination of $B_{00}$, $C_{+-}$ and $A_{\text{CP}}$, one can use our analytical result to predict $C_{00}$ with a reduced error and compare it with the experimental measurement when it becomes available. The correlation between $B_{00}$ and $C_{00}$ is an interesting aspect that can be probed in ongoing and future particle physics experiments such as LHCb and Belle II.