We fit the $^1 S_0 \Lambda\Lambda$ interaction in the nuclear medium to the masses of the experimentally known double-$\Lambda$ hypernuclei: $^{ \phantom{ 6}6}_{ \Lambda\Lambda}$He, $^{ 10}_{ \Lambda\Lambda}$Be and $^{ 13}_{ \Lambda\Lambda}$B. We derive this effective interaction from OBE J\"ulich $\Lambda\Lambda$-type potentials and using both Hartree-Fock and variational approaches. We find that the inclusion of $\Lambda\Lambda$ correlations in the variational scheme leads to significant differences and a better understanding of the dynamical features of the system. We investigate the sensitivity of the binding energies and the mesonic decay widths of the above double-$\Lambda$ hypernuclei to the $\omega \Lambda\Lambda $ coupling and the form factor at the $\sigma \Lambda\Lambda$ vertex. We also use this effective interaction to predict binding energies and pionic decay widths of heavier double-$\Lambda$ hypernuclei, not discovered yet. Finally, we discard the existence of $^1 S_0$ $\Lambda\Lambda$ bound states provided the $\Lambda\Lambda-\Xi N$ coupling can be neglected.
Categories: nucl-th