Existence of the $DD^*\bar{K}^*$ and $BB^*K^*$ three-body molecular states

Abstract: We investigate the existence of the three-body molecular state composed of $DD<sup><em>\bar{K}^</em>$ within the one-boson-exchange (OBE) model. A major challenge is that while the pseudoscalar-meson couplings are well-determined, the couplings for scalar- and vector-meson exchanges render significant model dependence. To ensure the reliability of our predictions and reduce model dependence, we recalibrate the coupling constants of the OBE model. We treat the pole position of $Z_c(3900)$, or equivalently the scalar $σ$-exchange coupling constant, as the only unknown parameter. The coupling constants for the vector $ρ$- and $ω$-exchanges are determined by the pole positions of the well established states $X(3872)$ and $T_{cc}(3875)$. We demonstrate that these parameter sets also successfully describe the $T_{cs0}(2870)$ without further tuning. For the three-body system, our results indicate that an $$I\left(J^P\right)=1 / 2\left(0<sup>{-}\right)$$ three-body molecular bound state exists when $Z_c(3900)$ is a virtual state located within approximately $-10~\text{MeV}$ of the $D\bar{D}<sup>*$ threshold. Furthermore, we extend our analysis to the complex energy plane using the complex scaling method to search for molecular resonances, though no evidence of resonances is found in considered channels. We also apply this formalism to the bottom analog $BB<sup><em>K^</em>$ system. In this sector, the conditions for the existence of a three-body bound state are more relaxed, as a $Z_c(3900)$ virtual state located within $-25~\text{MeV}$ below the threshold suffices, although three-body molecular resonances remain absent. We suggest that future experiments precisely measure the pole position of $Z_c(3900)$ or search for the three-body bound state in $DD\bar{K}ππ$ and $DD\bar{K}$ channels, as these efforts would mutually illuminate the nature of the associated states.