Production of $D_s\bar{D}_s$ and $D\bar{D}$ bound states in the $B$ decays within the Bethe-Salpeter framework

Abstract

Within the Bethe--Salpeter framework, we investigate the production of possible $D_s\bar{D}_s$ $(X_{s\bar{s}})$ and $D\bar{D}$ $(X_{q\bar{q}})$ bound states in $B$ decays. The bound state properties of the two heavy meson systems are studied in the one-boson-exchange model, and the resulting normalized Bethe--Salpeter wave functions are used to calculate the branching fractions of $B^+\to X_{s\bar s}K^+$ and $B^+\to X_{q\bar q}K^+$. We find that bound state solutions for the $D\bar{D}$ system exist for all the three coupling sets considered, whereas the $D_s\bar{D}_s$ system supports a bound-state solution only in a restricted parameter region. The predicted branching fractions are in the ranges of $1.09\times10^{-5}$--$20.06\times10^{-4}$ for the $D_s\bar{D}_s$ bound state and $1.56\times10^{-6}$--$4.14\times10^{-4}$ for the $D\bar{D}$ bound state. In particular, if $X(3915)$ is interpreted as a predominantly $D_s\bar{D}_s$ bound state, its production