Bound states of $_{c\bar{c}}^{9}$Be within $c\bar{c}+\alpha+\alpha$ cluster models based on state-of-the-art HAL QCD charmonium-nucleon interactions

The possible bound state of the $ _{c\bar{c}}^{9}$Be, a charmonium-nucleus system, is investigated. The analysis is carried out within a three-cluster model, where its binary subsystems are represented as $ c\bar{c}\textrm{+}\alpha $ and $\alpha+\alpha$. The hyperspherical harmonics method is employed to facilitate a convenient description of this three-cluster configuration. The calculations are done by employing the effective $ c\bar{c}\textrm{-}\alpha $ potentials. These potentials were derived recently based on state-of-the-art lattice QCD calculations, 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 $J/\psi N$ interactions, all obtained with nearly physical pion masses. The Coulomb interaction was also incorporated into the current calculations. It is determined that, despite neither the $ _{c\bar{c}}^{5}$He nor the $^{8}$Be binary subsystems being bound, a bound state of the $ c\bar{c}\textrm{-} \alpha\alpha$ nuclear system could potentially exist. The maximum central binding energy is found to be approximately 1.71 MeV, based on the spin 1/2 $J/\psi N $ interaction, while a minimum value of about 0.56 MeV is obtained from calculations involving the spin 1/2 $\eta_{c}N$ interaction.