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

We investigate the existence of the three-body molecular state composed of $DD^\bar{K}^$ 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 $I\left(J^P\right)=1 / 2\left(0^{-}\right)$ three-body molecular bound state exists when $Z_c(3900)$ is a virtual state located within approximately $-10~\text{MeV}$ of the $D\bar{D}^*$ 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 $BB^K^$ 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.

💡 Research Summary

The authors investigate whether three‑body hadronic molecules composed of two heavy mesons and a strange vector meson—specifically $DD^\bar K^$ and its bottom counterpart $BB^K^$—can form bound states. They work within the one‑boson‑exchange (OBE) framework, which includes exchanges of $\pi$, $\eta$, $\rho$, $\omega$, and $\sigma$ mesons. A well‑known difficulty of OBE models is the large uncertainty in the scalar ($\sigma$) and vector ($\rho$, $\omega$) coupling constants. To reduce this model dependence, the authors adopt a calibration strategy: the scalar coupling is tied to the pole position of $Z_c(3900)$, treated as a virtual state near the $D\bar D^$ threshold, while the vector couplings are fixed by reproducing the binding energies of the well‑established $X(3872)$ (binding $-4.09$ MeV) and $T_{cc}(3875)$ (binding $-1.6$ MeV). With these parameters, the same model also describes the $T_{cs0}(2870)$, interpreted as a $D^\bar K^*$ molecule, without further adjustment.

The two‑body potentials derived from the OBE Lagrangians are regularized with a monopole form factor, and a cutoff $\Lambda$ is varied between 1.10 and 1.35 GeV to assess sensitivity. The authors introduce three scaling factors $R_s$, $R_\beta$, and $R_\lambda$ to explore variations in scalar and vector couplings; $R_s$ is directly linked to the $Z_c(3900)$ pole, while $R_\beta$ and $R_\lambda$ stay close to unity after fitting the $X(3872)$ and $T_{cc}(3875)$ data.

For the three‑body problem they employ a non‑relativistic Hamiltonian with the OBE two‑body potentials and solve the Schrödinger equation using the Gaussian Expansion Method (GEM). All three Jacobi coordinate sets are included, but only S‑wave Gaussian basis functions are retained, which effectively captures the dominant dynamics while keeping the calculation tractable. To search for possible resonant states, they also apply the Complex Scaling Method (CSM), which rotates the Hamiltonian into the complex plane and allows bound and resonant solutions to be identified on the same footing.

The numerical results show that an $I(J^P)=\tfrac12(0^-)$ three‑body bound state appears in the $DD^\bar K^$ system provided the $Z_c(3900)$ virtual pole lies within roughly $-10$ MeV of the $D\bar D^*$ threshold. If the pole is deeper (e.g., $-20$ MeV) the binding disappears. In the bottom sector, the larger masses increase the attraction, and a bound state survives even if the $Z_c(3900)$ pole is as low as $-25$ MeV. The binding energies are modest (a few MeV), and the spatial size is comparable to that of typical hadronic molecules. No three‑body resonances are found in any of the investigated channels; the CSM spectra contain only the bound‑state poles and the continuum.

The authors emphasize that the existence of the three‑body molecule is highly sensitive to the precise nature of $Z_c(3900)$. An accurate determination of its pole position—whether it is a virtual state or a narrow resonance—would directly test the predictions. They propose experimental searches in the $DD\bar K\pi\pi$ and $DD\bar K$ final states, where a narrow peak with $J^P=0^-$ could be identified. For the bottom analog, future facilities such as upgraded LHCb or a high‑luminosity $Z$‑factory would be needed.

In summary, by anchoring the OBE model to well‑measured two‑body exotic states, the paper provides a coherent and relatively model‑independent prediction that $DD^\bar K^$ (and $BB^K^$) can form a shallow three‑body molecular bound state under realistic assumptions. The work highlights the interplay between two‑body dynamics and three‑body binding, clarifies the role of scalar σ exchange, and offers concrete experimental signatures to validate or refute the existence of these exotic three‑body hadrons.