Three-body final state interactions in $B^+\to D\bar{D}K^+$ decays

This paper presents a detailed analysis of the three-body final state interactions in the $B^+\to D\bar{D}K^+$ process, whose phase space is sufficient small. To precisely extract the resonance parameters, for instance the $\chi_{c0/2}(3930)$ in the $D\bar{D}$ invariant mass distributions, in this process, one has to take into account final state interaction, especially the three-body final state interaction. We employ the dispersive Khuri-Treiman formalism, combined with a parameterization of the $D\bar{D}$ interaction based on Heavy Quark Spin Symmetry. By performing a simultaneous fit to the experimental data from LHCb, BaBar, and Belle collaborations, the scheme with three-body interaction successfully describes the invariant mass distributions of the three two-body subsystems. We precisely extract the pole structures of $\chi_{c0}(3930)$ and $\psi(3770)$ as $3.913-0.018i~\mathrm{GeV}$ and $3.764-0.002i~\mathrm{GeV}$ in $B^+$ decay. By performing the pole trajectory analysis on a uniformized complex plane, we find that both of them stem from the input bare state.