Density Functional approach for multi-strange hypernuclei: competition between $\Lambda$ and $\Xi^{0,-}$ hyperons

The question of the competition between $\Lambda$ and $\Xi^{0,-}$ in the ground-state of multi-strange hypernuclei is addressed within a non-relativistic density functional approach, partially constrained by ab-initio calculations and experimental data. The exploration of the nuclear chart for $10<Z<120$ as a function of the strangeness number is performed by adding hyperons to a nuclear core imposing either conserved total charge $Q$ or conserved proton number $Z$. We find that almost all $\Lambda$ hypernuclei present an instability with respect to the strong interaction decay of $\Lambda$ towards $\Xi^{0,-}$ and that most of the instabilities generates $\Xi^-$ (resp. $\Xi^0$) in the case of conserved total charge $Q$ (resp. proton number $Z$). The strangeness number at which the first $\Xi^{0,-}$ appear is generally lower for configurations explored in the case of conserved $Q$ compared to the case of conserved $Z$, and corresponds to the crossing between the $\Lambda$ and the neutron or proton chemical potentials. About two to three hundred thousands pure $\Lambda$ hypernuclei may exist before the onset of $\Xi^{0,-}$. The largest uncertainty comes from the unknown $\Lambda\Xi$ interaction, since the $N\Lambda$ and the $N\Xi$ ones can be constrained by a few experimental data. The uncertainty on the $\Lambda\Xi$ interaction can still modify the previous estimation by 30-40\%, while the impact of the unknown $\Xi\Xi$ interaction is very weak.