Examination of the $c\bar{c}+n+^{10}$Be bound-state problem within three cluster models based on QCD charmonium-nucleon interactions

The possible bound state of the $c\bar{c}+n+^{10}$Be system, representing a hypothetical charmonium-nucleus configuration, is investigated. The analysis is conducted within a three-cluster framework, in which the binary subsystems are treated as $n+^{10}\textrm{Be}$, $^{10}\textrm{Be}+c\bar{c}$, and $c\bar{c}+n$. The hyperspherical harmonics method is employed to provide a convenient description of this three-cluster configuration. The calculations are performed using effective $^{10}\textrm{Be}\textrm{-}c\bar{c}$ potentials constructed via the single-folding procedure. These potentials have been derived recently on the basis of state-of-the-art lattice QCD results from the HAL QCD Collaboration, which provided interactions for the spin-$3/2$ $J/\psi N$, spin-$1/2$ $J/\psi N$, spin-$1/2$ $\eta_{c}N$, and spin-averaged $J/\psi N$ channels, all obtained at nearly physical pion masses. The numerical results indicate that the central binding energies of the spin-$3/2$ $J/\psi+n+^{10}$Be, spin-$1/2$ $J/\psi+n+^{10}$Be, and spin-$1/2$ $\eta_{c}+n+^{10}$Be systems are 3.47, 3.55, and 1.91 MeV, respectively. The corresponding root-mean-square nuclear matter radii are predicted to be approximately 2.49, 2.48, and 2.60 fm.