We investigate the three-body $DDK$ system with quantum numbers $I(J^P)=\frac{1}{2}(0^{-})$ within a coupled-channel framework that incorporates the $DDK$, $D^{*}{D}^{*}K$, $D{D}_s\eta$ and $D{D}_s\pi$ channels. The $D^{(*)}{D}^{(*)}$ interactions are described using the one-boson-exchange model constrained by heavy-quark symmetry and fitted to the pole positions of $X(3872)$, $T_{cc}^{+}$, and $Z_c(3900)$. The $D^{(*)}K$ interaction is from the chiral effective field theory, motivated by the molecular interpretation of $D_{s0}^{*}(2317)$, and is further constrained by lattice-QCD results for the $DK$ scattering lengths. The resulting three-body problem is solved using the Gaussian expansion method, while the complex scaling method is employed to search for possible resonant states. We find that coupled-channel effects from $D^{*}{D}^{*}K$ are negligible, and the $DDK$ system supports a deeply bound state across a wide range of parameters. Depending on the long-range behavior of the $DK$ interaction, an additional shallow state may emerge near the particle-dimer ($D\text{−}DK$) threshold. The deeply bound state exhibits a compact three-body structure, whereas the shallow state displays characteristic features of a three-body halo configuration. Similar results are obtained for the $D^{*}{D}^{*}K$ system. We also observe that the coupled-channel effects originating from the $D{D}_s\eta$ channel produce only minor corrections to the $DDK$ bound state and lead to a modest reduction in its binding energy. By contrast, opening the $D{D}_s\pi$ channel substantially alters the nature of the three-body $DDK$ state and transforms the bound state into a resonance. These findings may provide new insight into few-body dynamics in systems involving charmed mesons and kaons.

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• Phenomenology
• Hadrons