Three-body molecular states composed of $D^{(*)}$ and two nucleons

Abstract

We study the three-body systems $DNN$ and $D^{*}NN$ within a hadronic molecular framework by combining a realistic nucleon-nucleon interaction with a $D^{(*)}N$ potential constrained by heavy-quark symmetry. The three-body Schrödinger equation is solved with the Gaussian Expansion Method, and the analytic structure of the spectrum is investigated using the Complex Scaling Method. We find that the $DNN$ system supports a robust and compact bound state in the $I(J^{P})=\tfrac{1}{2}(1^-)$ channel over a broad range of cutoff values, even when the corresponding $DN$ subsystem is weakly bound or unbound. For $D^{*}NN$, the spin-$1$ nature of the heavy meson and the associated spin-dependent forces generate a clear spin hierarchy: deeply bound states appear in both $0^-$ and $2^-$ channels, while the $1^-$ channel exhibits a characteristic two-branch pattern with a strongly bound compact branch and a more weakly bound, spatially extended branch. The root-mean-square radii indicate pronounced spatial compression compared with the deuteron scale, highlighting the cooperative roles of realistic $NN$ correlations, the $D^{(*)}N$ interactions, and heavy-quark symmetry in forming compact heavy-flavor few-body bound states. No three-body resonances under complex scaling are found in the explored parameter space. Our results provide quantitative benchmarks for future experimental searches for such charmed-meson-nuclear bound states.

Figures (5)

• Figure 1: Feynman diagrams used to derive OBE potentials for the $D^{(*)}N$ and $NN$ systems.
• Figure 2: Two sets of Jacobi coordinates corresponding to different spatial configurations, where blue disks indicate the nucleons $N$ and the red disk denotes the $D^{(\ast)}$ meson. The third set of Jacobi coordinates, obtained by exchanging the two nucleons in panel (b), is not shown explicitly, but is included in the calculation.
• Figure 3: Binding energies of the three-body $D^{(*)}NN$ and the corresponding two-body subsystems, calculated using coupling constants fitted to the $Z_c(3900)$ pole, are shown as functions of the $Z_c(3900)$ virtual state pole position (from $-5$ to $-35$ MeV). The panels are organized in columns of fixed $\Lambda$ (1.1, 1.2, 1.3 GeV) and rows corresponding to specific physical systems: three spin configurations of $D^*NN$ and one $DNN$ system. Each panel displays both three-body $D^{(*)}NN$ and two-body $D^{(*)}N$ binding energies, enabling a direct comparison of cutoff dependence, sensitivity to the $Z_c(3900)$ pole, and the interplay between two- and three-body dynamics.
• Figure 4: RMS radii of the $D^{(*)}NN$ systems and the corresponding two-body subsystems, calculated with the same coupling constants $(\lambda,\beta, g_s)$ as in Fig. \ref{['fig:binding']}, which are determined by fitting the $Z_c(3900)$ pole. The RMS radii are shown as functions of the $Z_c(3900)$ pole position from $-5$ to $-35~\mathrm{MeV}$ for three cutoff values $\Lambda=1.1$, $1.2$, and $1.3~\mathrm{GeV}$. The panel layout is identical to that of Fig. \ref{['fig:binding']}: each column corresponds to a fixed cutoff $\Lambda$, and each row to a specific hadronic system. In each panel, the RMS radii of the three-body $D^{(*)}NN$ states are shown together with those of the corresponding two-body $D^{(*)}N$ subsystems, allowing for a direct comparison of their spatial structures and their dependence on the cutoff and the $Z_c(3900)$ pole.
• Figure 5: Pole trajectories of the $D^{(*)}NN$ systems in the complex energy plane for (a) $DNN$, (b) $D^{*}NN$ with $J^P=0^-$, (c) $D^{*}NN$ with $J^P=1^-$,and (d) $D^{*}NN$ with $J^P=2^-$.The calculations are performed with $R_s=R_\lambda=R_\beta=1$ and a fixed cutoff $\Lambda=1.4~\mathrm{GeV}$. The real and imaginary parts of the three-body energies are shown as the interaction strength is varied. In each panel, the trajectories originate from the three-body threshold, the $D^{(*)}N$ two-body threshold, and the $NN$ two-body threshold, respectively. The straight-line trajectories along the real-energy axis indicate genuine bound-state poles, while the absence of poles with sizable imaginary parts confirms the non-resonant nature of the states. In the $D^{*}NN$$J^P=1^-$ channel, two distinct bound-state poles are observed, corresponding to the two branches identified in the binding energy and rms analyses.