$B^+$ decay to $K^+ηη$ with ($ηη$) from the $D\bar{D}(3720)$ bound state

Abstract: We search for a $B$ decay mode where one can find a peak for a $D \bar{D}$ bound state predicted in effective theories and in Lattice QCD calculations, which has also been claimed from some reactions that show an accumulated strength in $D \bar{D}$ production at threshold. We find a good candidate in the $B^+\to K⁺ \eta\eta$ reaction, by looking at the $\eta\eta$ mass distribution. The reaction proceeds via a first step in which one has the $B^+\to D_s^{*+} \bar{D}<sup>0$ reaction followed by $D_s<sup>{*+}$ decay to $D⁰ K<sup>+$ and a posterior fusion of $D⁰ \bar{D}<sup>0$ to $\eta \eta$, implemented trough a triangle diagram that allows the $D⁰ \bar{D}<sup>0$ to be virtual and produce the bound state. The choice of $\eta\eta$ to see the peak is based on results of calculations that find the $\eta\eta$ among the light pseudoscalar channels with stronger coupling to the $D \bar{D}$ bound state. We find a neat peak around the predicted mass of that state in the $\eta\eta$ mass distribution, with an integrated branching ratio for $B^+\to K<sup>+$ ($D\bar{D}$, bound) ; ($D\bar{D}$, bound) $\to \eta \eta$ of the order of $1.5 \times 10<sup>{-4}$, a large number for hadronic $B$ decays, which should motivate its experimental search.