Probing the existence of $\eta$d and $\eta^3$He with a few-body approach

We investigate the possible $\eta$-deuteron and $\eta-^3$He bound states with the $\eta NN$ and $\eta NNN$ few body method. In order to solve the three body and four body Schr\"odinger equations, we apply the Gaussian expansion method. The realistic AV8' $NN$ potential together with an extra 3N three body force which reproduce the binding energies of deuteron, $^3$H and $^3$He, are used. We construct the relations between the complex Gaussian-type energy-independent $\eta N$ interactions and the $\eta N$ scattering lengths $a$. The relations between the binding energies in $\eta$d and $\eta^3$He, and the scattering lengths are then obtained. We find that to have a bound $\eta$d nucleus, the real $\eta N$ scattering length should exceed at least 1.35 fm. As for the $\eta^3$He system, the bound state exists when the real $a>0.75\sim0.8$ fm if we neglect the imaginary part of the $\eta N$ interaction. After solving the full four body complex Schr\"odinger equation, we find the imaginary $\eta N$ potential causes the system more unbound. Thus, we give the relations between the bound or unbound scenarios of the $\eta^3$He system and the complex $\eta N$ scattering lengths.Comment: 7 pages, 11 figure