Probing $\mathrm{\Xi }N$ interaction through inversion of spin-doublets in $\mathrm{\Xi }N\alpha \alpha$ nuclei

A new way to study the spin-isospin dependence of the $\mathrm{\Xi }N$ interaction is explored through the energy levels of $\mathrm{\Xi }N\alpha$ and $\mathrm{\Xi }N\alpha \alpha$ systems with $\alpha$ being a spectator to attract the $\mathrm{\Xi }N$ pair without changing its spin-isospin structure. By using the Gaussian expansion method (GEM) with the state-of-the-art $\mathrm{\Xi }N$ potential obtained from lattice QCD calculations, it is found that $\mathrm{\Xi }N\alpha \alpha$ has spin-doublet bound states with $J^{\pi }=1^{-}$ and $2^{-}$ in both isospin triplet and singlet channels. The inversion of the $1^{-}$-$2^{-}$ spin-doublet between the iso-triplet and the iso-singlet is found to be strongly correlated with the relative strengths of the $\mathrm{\Xi }N$ interaction in the ${}^{11}{\mathrm{S}}_0{,}^{13}{\mathrm{S}}_1{,}^{31}{\mathrm{S}}_0$ and ${}^{33}{\mathrm{S}}_1$ channels. The $(K^{-},K^{+})$ and $(K^{-},K^0)$ reactions on the ${}^{10}$B target are proposed to produce those bound states.