Faddeev equations for the $J/\psi\,NN$ and $\phi\,NN$ three-body systems in momentum space

Abstract

We construct Faddeev equations for the $J/\psi\,NN$ and $\phi\,NN$ three-body systems in momentum space. The two-body subsystem $t$-matrix is obtained by solving the Lippmann-Schwinger equations. The $NN$ interactions are constructed using the Malfliet-Tjon potential. The $J/\psi\,N$ interaction potentials in the ${}^4S_{3/2}$ and ${}^2S_{1/2}$ channels are taken from HAL QCD. The $\phi\,N$ interaction potential in the ${}^4S_{3/2}$ channel is also taken from HAL QCD. The ${}^2S_{1/2}$ $\phi N$ interaction is obtained from the combination of the $\phi p$ correlation function analysis and the HAL QCD results, as well as from the assumption that the spin-spin parts of the $\phi N$ and $J/\psi N$ interactions are inversely proportional to the respective hadron masses. The Faddeev equations are solved to find bound states in the $J/\psi\,NN$ and $\phi\,NN$ three-body systems. The numerical results suggest that there exist bound states in the $\phi\,NN$ three-body system, while there is no bound state in the $J/\psi\,NN$ system.