Density Functional approach for multistrange hypernuclei: Competition between $\mathrm{\Lambda }$ and ${\mathrm{\Xi }}^{0,-}$ hyperons

The question of the competition between $\mathrm{\Lambda }$ and ${\mathrm{\Xi }}^{0,-}$ in the ground state of multistrange hypernuclei is addressed within a nonrelativistic 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 $\mathrm{\Lambda }$ hypernuclei present an instability with respect to the strong interaction decay of $\mathrm{\Lambda }$ towards ${\mathrm{\Xi }}^{0,-}$ and that most of the instabilities generates ${\mathrm{\Xi }}^{-}$ (${\mathrm{\Xi }}^0$) in the case of conserved total charge $Q$ (proton number $Z$). The strangeness number at which the first ${\mathrm{\Xi }}^{0,-}$ 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 $\mathrm{\Lambda }$ and the neutron or proton chemical potentials. About two to three hundred thousand pure $\mathrm{\Lambda }$ hypernuclei may exist before the onset of ${\mathrm{\Xi }}^{0,-}$. The largest uncertainty comes from the unknown $\mathrm{\Lambda }\mathrm{\Xi }$ interaction, since the $N\mathrm{\Lambda }$ and the $N\mathrm{\Xi }$ ones can be constrained by a few experimental data. The uncertainty on the $\mathrm{\Lambda }\mathrm{\Xi }$ interaction can still modify the previous estimation by 30–40%, while the impact of the unknown $\mathrm{\Xi }\mathrm{\Xi }$ interaction is very weak.

Figure 1

Potentials $v_{N\mathrm{\Lambda }}({\rho }_N,{\rho }_{\mathrm{\Lambda }})$ (a) and $v_{\mathrm{\Lambda }\mathrm{\Lambda }}\left({\rho }_{\mathrm{\Lambda }}\right)$ (b) as a function of the nucleon density ${\rho }_N$ in units of the saturation density ${\rho }_0$, for the functionals DF-NSC89, DF-NSC97a, and DF-NSC97f. In panel (a) two different $\mathrm{\Lambda }$ densities are considered: ${\rho }_{\mathrm{\Lambda }}=0.0{\mathrm{fm}}^{-3}$ (solid lines) and ${\rho }_{\mathrm{\Lambda }}=0.1{\mathrm{fm}}^{-3}$ (empty squares). In panel (b) the simplified prescription EmpC (solid lines) is compared to the prescription EmpB2 from Ref. [10] (empty squares) in the case when 30% of the baryons are taken as $\mathrm{\Lambda }$.

Figure 2

Binding energy $E/A$ at conserved $Q$ for different multi-$\mathrm{\Lambda }$ hypernuclei as a function of the strangeness number $-S$: ${}^{40\text{–}S\mathrm{\Lambda }}\mathrm{Ca}$, ${}^{56\text{–}S\mathrm{\Lambda }}\mathrm{Ni}$, ${}^{120\text{–}S\mathrm{\Lambda }}\mathrm{Sn}$, ${}^{208\text{–}S\mathrm{\Lambda }}\mathrm{Pb}$. We compare predictions from two functionals for the $\mathrm{\Lambda }\mathrm{\Lambda }$ interaction, DF-NSC89+EmpC (lines) and DF-NSC89+EmpB2 (symbols), using SLy4 for the $N$ interaction and VY0 for the $\mathrm{\Xi }$ interaction. The calculations are stopped at the $\mathrm{\Xi }$-instability threshold.

Figure 3

Hyperon potentials in uniform matter $v_{\mathrm{\Lambda }}^{\mathrm{unif}}({\rho }_N,{\rho }_{\mathrm{\Lambda }},{\rho }_{\mathrm{\Xi }})$ (a) and $v_{\mathrm{\Xi }}^{\mathrm{unif}}({\rho }_N,{\rho }_{\mathrm{\Lambda }})$ (b) as a function of the nucleon density ${\rho }_N$ (in units of the saturation density ${\rho }_0$) for the functional DF-NSC89+EmpC+VY0 for various choices of the densities ${\rho }_{\mathrm{\Lambda }}$ and ${\rho }_{\mathrm{\Xi }}$ (in ${\mathrm{fm}}^{-3}$) (panel a), and for various strengths of the $\mathrm{\Lambda }\mathrm{\Xi }$ interaction (panel b). See text for more details.

Figure 4

Representation of the energy difference $\mathrm{\Delta }{E}_{\mathrm{tot}}(S,N_{\mathrm{\Xi }})=E_{\mathrm{tot}}(N_{\mathrm{\Lambda }}=-S-N_{\mathrm{\Xi }}/2,N_{\mathrm{\Xi }})-E_{\mathrm{tot}}(N_{\mathrm{\Lambda }}=-S,N_{\mathrm{\Xi }}=0)$ for various multistrange hypernuclei conserving the mass $A=132$ and the electric charge $Q=50$, and for which the strangeness charge $-S$ is varied from 0 to 30, for the functionals SLy4+DF-NSC89+EmpC+VY0 (a), VY1 (b), and VY2 (c).

Figure 5

Chemical potential for the functional SLY4+DF-NSC89+EmpC+VY0.

Figure 6

Comparison of the strangeness number $-S$ at the S-drip line for $\mathrm{\Lambda }$ hypernuclei (black lines) with $S_{\mathrm{inst}.}$ and strangeness number $-S$ at the ${\mathrm{\Xi }}^{0,-}$ instability (red, blue, and pink lines/points). A sample of various total charges $Q$ are considered, $Q=20$, 50, and 82, and $A_{\mathrm{core}}$ runs from the proton to the neutron drip line. Results are from the functional SLy4+DF-NSC89+EmpC+VY0.

Figure 7

Prediction for $S_{\mathrm{inst}.}$ through the nuclear chart. Calculations are performed with the density functional DF-NSC89+EmpC for the $\mathrm{\Lambda }$ interaction and SLy4 and VY0 for the nucleon and other hyperon interactions. Each point represents a calculation performed at constant $A$ and $Q$ (total charge) and they are represented as function of the baryonic number $A_{\mathrm{core}}$ and charge $Z_{\mathrm{core}}$ associated with neutrons and protons. The value of the total charge $Q$ is written at the end of each isocharge line. See text for more details.

Figure 8

Same as Fig. 7 using the functional DF-NSC97a+EmpC.

Figure 9

Same as Fig. 7 using the functional DF-NSC97f+EmpC.

Figure 10

Distribution of hypernuclei function of the strangeness number $-S$. The functionals that we consider are SLY4+${\mathrm{V}}_{N\mathrm{\Lambda }}$+EmpC+VY0 where ${\mathrm{V}}_{N\mathrm{\Lambda }}=\mathrm{DF}\text{−}\mathrm{NSC}89$ (solid lines), DF-NSC97a (dashed lines), and DF-NSC97f (dotted lines). The black lines show the distributions of hypernuclei below the ${\mathrm{\Xi }}^{0,-}$ instability while the red lines show the distribution of pure-$\mathrm{\Lambda }$ hypernuclei (disregarding the ${\mathrm{\Xi }}^{0,-}$ instability).