Probing $ \phi $N interaction through bound states of $ \phi\textrm{N-}\alpha $ system

The possible bound state of the $\phi N \alpha$ system is explored within the framework of the three-body cluster model. The calculations are done by employing the state-of-the-art $\phi$N interactions obtained from the analysis of the pure elastic scattering and the coupled-channel in the $\phi p$ correlation functions. The $\phi\alpha$ potentials are constructed by two methods: the single-folding potential (SFP) method for the given spin-averaged $\phi$N potentials in coordinate space, and the optical model potential (OMP) approximation within the multiple-scattering framework for the given scattering length of the $\phi$N interaction. It is found that, when only the single-channel $\phi N$ interactions are employed, the $\phi N \alpha$ system could be bound with a binding energy in the interval [3-26] MeV. However, the coupled-channel $\phi p$ interaction, which is most consistent with experimental measurements, does not yield any bound state, even when the spin-averaged $\phi N$ potential is employed, and this potential is found to be more attractive than the corresponding coupled-channel counterpart. It is essential to consider the contributions from the dynamics of the vector-baryon coupled channels $\phi p$ interaction. This effect is capable of playing a decisive role in the existence of mesic nuclei.