In search for signals of the $D\bar{D}$ bound state $X(3700)$ from study of the $B^+ \to D^+ D^- K^+$, $B^0 \to D^+ D^- K^0$ and $Λ_b \to D^+ D^- Λ$ reactions
We perform a theoretical study of the $B^+ \to D^+ D^- K^+$, $B^0 \to D^+ D^- K^0$ and $Λ_b \to D^+ D^- Λ$ reactions by looking at the production mechanisms, with special emphasis in the final state interaction of the charmed mesons, which gives rise to the $X(3930)$, coupling strongly to $D_s\bar{D}_s$, and another state that we call $X(3700)$, coupling strongly to $D \bar D$ that qualifies as a $D \bar D$ bound state. The combined study of all these reactions shows that the final state interaction responsible for the production of these resonances is more important in the $B^+ \to D^+ D^- K^+$ reaction than in the $Λ_b \to D^+ D^- Λ$ one. We have taken this into account and shown that normalizing the $D^+ D^-$ mass distributions to the same value at the peak of the $ψ(3770)$ production, the two mass distributions are quite different in a range of 10 MeV above the $D^+ D^-$ threshold, with a value about an order of magnitude bigger for the $B^+ \to D^+ D^- K^+$ reaction than for the $Λ_b \to D^+ D^- Λ$ one. The $D^+D^-$ mass distribution by itself also shows an enhancement of about one order of magnitude in the presence of the $X(3700)$ with respect to a calculation where this resonance is absent. We make a call to measure these magnitudes in the coming LHCb upgrades, which would bring great support to the existence of the bound $D^+ D^-$ state.