Mean-field approaches for ${\mathrm{\Xi }}^{-}$ hypernuclei and current experimental data

Motivated by the recently observed hypernucleus (Kiso event) ${}_{\mathrm{\Xi }}{}^{15}\mathrm{C}$ (${}^{14}\mathrm{N}+{\mathrm{\Xi }}^{-}$), we identify the state of this system theoretically within the framework of the relativistic-mean-field and Skyrme–Hartree–Fock models. The $\mathrm{\Xi }N$ interactions are constructed to reproduce the two possibly observed ${\mathrm{\Xi }}^{-}$ removal energies, $4.38\pm 0.25$ MeV or $1.11\pm 0.25$ MeV. The present result is preferable to be ${}^{14}\mathrm{N}(\mathrm{g}.\mathrm{s}.)+{\mathrm{\Xi }}^{-}\left(1p\right)$, corresponding to the latter value.

Figure 1

(top) BHF s.p. potentials $U_{\mathrm{\Xi }}\left(k\right)$ (real and imaginary parts) in symmetric nuclear matter at ${\rho }_N=0.17{\text{fm}}^{-3}$, obtained with the Nijmegen ESC08b $YN$ model [32]. (bottom) Dependence on nuclear density of the BHF s.p. potential $U_0\equiv U_{\mathrm{\Xi }}(k=0)$ (black solid and dashed curves), and the Skyrme SL0, SL2, and SL3 mean fields $V_{\mathrm{\Xi }}$ (dotted red, green, and blue curves) in homogeneous nuclear matter, Eq. (22). The subscript $s\left(p\right)$ for each interaction denotes the potential obtained by the assumption of $\mathrm{\Xi }$ occupation in the $s\left(p\right)$ orbit in Table 1. See the captions to Table 1 for details.

Figure 2

The potential between ${}^{14}\mathrm{N}$ and ${\mathrm{\Xi }}^{-}$ obtained by the ${\mathrm{RMF}}^{\sigma \omega \rho }$ model. The black solid line is shown with all components, $V_{\sigma +\omega }+V_{\rho }+V_C$. To see the contributions to $V_{\mathrm{\Xi }}$, the self-coupling $V_{\rho }$ defined by $V_{\rho }^{\left(S\right)}, V_{\rho }, V_{\sigma +\omega }$, and the Coulomb potential $V_C$ are shown separately.

Figure 3

Local ${\mathrm{\Xi }}^{-}$ mean field $V_{\mathrm{\Xi }}$ and Coulomb field $V_C$ (short-dashed red curves) in ${}_{\mathrm{\Xi }}{}^{12}\mathrm{Be}$, obtained with (a), (c) the SHF and (b), (d) RMF models with various forces in Table 1. The Woods–Saxon mean field deduced in Ref. [7] is shown for comparison by the dash-dotted blue curve. Panels (a) and (b) show $V_{\mathrm{\Xi }}$ for the ${\mathrm{\Xi }}^{-}$ occupation of a $s$ orbit, while ${\mathrm{\Xi }}^{-}$ occupies the $p$ orbit in panels (c) and (d). In panels (a) and (c), results of different sets of Skyrme parameters SL0, SL2, and SL3 are shown by solid, dashed, and dashed-dotted curves, respectively. In the RMF cases (b) and (d), the components $V_{\sigma +\omega }$ (long-dash-dotted magenta curve), $V_{\rho }$ (dash-dot-dotted green curve), and the spurious potential $V_{\rho }^{\left(s\right)}$ (dashed orange curve) are shown separately. The sum of $V_{\sigma +\omega }, V_{\rho }, V_C$, and $V_{\rho }^{\left(s\right)}$ is denoted by $V_{\mathrm{\Xi }}{\mathrm{RMF}}^{\sigma \omega \rho }$ (solid curve), while the sum without the isovector potential $V_{\rho }$ is given by $V_{\mathrm{\Xi }}{\mathrm{RMF}}^{\sigma \omega }$ (long dashed curve).