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Study of light $ϕ$-mesic nuclei with HAL QCD $ϕN$ interactions

We explore the possible existence of light $ϕ$-mesic nuclei using HAL QCD $ϕN$ interactions for the $^2S_{1/2}$ and $^4S_{3/2}$ channels. Particularly, using the Faddeev formalism in configuration space, the $ϕNN$ system, and $^{9}_ϕ$Be and $^{6}_{ϕϕ}$He nuclei within the framework of the three-body cluster model, are investigated. The $ϕα$ effective potential, obtained through a folding procedure, involves the HAL QCD $ϕN$ interaction in the $^4S_{3/2}$ channel which does not lead to a bound state of the $ϕN$ pair while the $ϕN$ $^2S_{1/2}$ channel yields the bound state as the $^3_ϕ$H nucleus. The $^4S_{3/2}$ potential ensures that the folding procedure is appropriate because there are no open channels like $ϕ+N$ and $ϕ+2N $ near or below the $ϕ+ 4N$ threshold, and it utilizes different matter distributions of $^4$He proposed in the literature. The folding potential is approximated by the Woods-Saxon formula. The mirror systems $ϕ$+$α$+$α$ and $ϕ$+$ϕ$+$α$ have energy ranges from 1-11 MeV and 3-10~MeV, respectively. The predicted binding energies represent the minimal values for the hypothetical $ϕ$ mesic nuclei $^{5}_ϕ$He, $^{9}_ϕ$Be and $^{6}_{ϕϕ}$He. The phenomenological $αα$ and $ϕϕ$ potentials are adopted from the literature.