Onset of $\eta$-nuclear binding in a pionless EFT approach

$\eta NNN$ and $\eta NNNN$ bound states are explored in stochastic variational method (SVM) calculations within a pionless effective field theory (EFT) approach at leading order. The theoretical input consists of regulated $NN$ and $NNN$ contact terms, and a regulated energy dependent $\eta N$ contact term derived from coupled-channel models of the $N^{\ast}(1535)$ nucleon resonance plus a regulated $\eta NN$ contact term. A self consistency procedure is applied to deal with the energy dependence of the $\eta N$ subthreshold input, resulting in a weak dependence of the calculated $\eta$-nuclear binding energies on the EFT regulator. It is found, in terms of the $\eta N$ scattering length $a_{\eta N}$, that the onset of binding $\eta\,^3$He requires a minimal value of Re$\,a_{\eta N}$ close to 1 fm, yielding then a few MeV $\eta$ binding in $\eta\,^4$He. The onset of binding $\eta\,^4$He requires a lower value of Re$\,a_{\eta N}$, but exceeding 0.7 fm.