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.