Bound states of \(^{9}_{\phi}\)Be and \(^{6}_{\phi\phi}\)He nuclei with \(\phi\)+\(\alpha\)+\(\alpha\) and \(\phi\)+\(\phi\)+\(\alpha\) cluster models

We investigate the \(^{9}_{\phi}\)Be and \(^{6}_{\phi\phi}\)He \(\phi\) mesic nuclei within the framework of the three-body cluster model as the \(\phi\)+\(\alpha\)+\(\alpha\) and \(\phi\)+\(\phi\)+\(\alpha\) systems, using the Faddeev formalism in configuration space. The \(\phi\)-\(\alpha\) potential is determined through a folding procedure of the HAL QCD \(\phi\)-\(N\) interaction in the \(^4S_{3/2}\) channel with the matter distribution of \(^4\)He. The phenomenological \(\alpha\)-\(\alpha\) and \(\phi\)-\(\phi\) potentials are taken from the literature. Additionally, we construct a Wood-Saxon (WS) type interaction to simulate the \(\phi\)-\(\alpha\) potential, also taken from the literature, based on an effective Lagrangian approach that includes \(K\bar{K}\) meson loops in the \(\phi\)-meson self-energy. A comparison of binding energies obtained for both types of the \(\phi\)-\(\alpha\) interactions reveals qualitative agreement. %between the obtained approaches. We predict the binding energy for the \(^{9}_{\phi}\)Be and \(^{6}_{\phi\phi}\)He \(\phi\) mesic nuclei as the mirror \(\phi\)+\(\alpha\)+\(\alpha\) and \(\phi\)+\(\phi\)+\(\alpha\) systems in the range of 1-11 MeV and 3-10 MeV, respectively. The range of values of the binding energies relies on the choice of the WS \(\phi\)-\(\alpha\) interaction parameters.