Structure of multi-$Λ$ hypernuclei with a Skyrme-type $ΛΛ$ interaction constrained by data on double-$Λ$ hypernuclei and neutron stars

We investigate multi-$\Lambda$ hypernuclear systems with Skyrme-type $\Lambda\Lambda$ interactions constrained by the data on double-$\Lambda$ hypernuclei and neutron stars. The roles of the repulsive $p$-wave and density-dependent terms in the $\Lambda\Lambda$ interaction are examined by considering the homogeneous hyperonic matter around the normal density and finite multi-$\Lambda$ hypernuclei within the spherical Hartree-Fock approach. In homogeneous matter, the $\Lambda$ chemical potential and corresponding $\Lambda$ drip point depend strongly on the repulsive $p$-wave term, while the effect of density-dependent term is relatively weak in the density range relevant to finite nuclei. In the multi-$\Lambda$ hypernuclei built on doubly closed stable cores from light to heavy systems, $\Lambda$ radius, separation energy and single-particle structure show a clear dependence on the repulsive $p$-wave interaction, and this dependence becomes stronger as the number of $\Lambda$ hyperons increases. A second and distinct effect appears near the $\Lambda$ drip line: when the last occupied $\Lambda$ orbit approaches the continuum, the repulsive $p$-wave term shifts the state upward and can produce a weakly bound state with an extended radial distribution. As a result, $\Lambda$ radius can increase rapidly near the threshold. This threshold effect should be distinguished from the moderate enhancement of the dependence on $p$-wave interaction with increasing number of $\Lambda$ hyperons. These results indicate that the multi-$\Lambda$ hypernuclei are particularly useful for isolating the role of $p$-wave $\Lambda\Lambda$ interacion around the normal density, whereas the density-dependent term is expected to be more important interaction in the high-density domain relevant to neutron stars.