Excitation spectra of ${}_{\mathrm{\Xi }}{}^{15}\mathrm{C}$ and ${}_{\mathrm{\Xi }}{}^{12}\mathrm{Be}$ calculated with a $\mathrm{\Xi }N$ interaction from lattice QCD

Background:The accumulation of experimental data on $\mathrm{\Xi }$ hypernuclei has enabled testing the $\mathrm{\Xi }N$ interactions derived from first-principles lattice QCD simulations through many-body calculations.

Purpose:Based on the interactions derived from lattice QCD simulations, we calculate the spectrum of the ${}^{14}\mathrm{N}+{\mathrm{\Xi }}^{-}$ states in ${}_{\mathrm{\Xi }}{}^{15}\mathrm{C}$ as well as that of the ${}^{12}\mathrm{C}+{\mathrm{\Xi }}^{-}$ states in ${}_{\mathrm{\Xi }}{}^{13}\mathrm{B}$ and compare with observed data from emulsion experiments. In addition, we predict the spectrum of ${}_{\mathrm{\Xi }}{}^{12}\mathrm{Be}$ (${}^{11}\mathrm{B}+{\mathrm{\Xi }}^{-}$) using the same interaction.

Methods:Through the $G$-matrix calculations, we derive low-energy effective interactions from lattice QCD potentials, where we introduce odd-parity potentials based on the meson-exchange picture because they have not been obtained from lattice QCD simulations. Employing these interactions, we carry out molecular dynamics model calculations to obtain the spectra of ${}_{\mathrm{\Xi }}{}^{13}\mathrm{B}, {}_{\mathrm{\Xi }}{}^{15}\mathrm{C}$, and ${}_{\mathrm{\Xi }}{}^{12}\mathrm{Be}$.

Results:The $s$-wave states of ${\mathrm{\Xi }}^{-}$ in ${}_{\mathrm{\Xi }}{}^{15}\mathrm{C}$ are bound in the range of 5 to 10 MeV, while the $p$-wave states are in the range of 0 to 2 MeV. The same Hamiltonian predicts the bound $s$-wave states of ${}_{\mathrm{\Xi }}{}^{12}\mathrm{Be}$ in the range of 3 to 5 MeV. The obtained spectra show small spin-spin splitting and conversion widths.

Conclusions:The results for ${}_{\mathrm{\Xi }}{}^{13}\mathrm{B}$ and ${}_{\mathrm{\Xi }}{}^{15}\mathrm{C}$ are consistent with many of the experimental data, highlighting the capabilities of lattice QCD simulations. The binding of ${}_{\mathrm{\Xi }}{}^{12}\mathrm{Be}$ is deep enough to be observed by spectroscopy experiments. The difference between the lattice QCD and the meson-exchange models lies in the spin-spin splitting and conversion widths, underlining the importance of observations.

Figure 1

(a) The $s$- and $p$-wave states of ${}_{\mathrm{\Xi }}{}^{13}\mathrm{B}$ (${}^{13}\mathrm{C}+{\mathrm{\Xi }}^{-}$) calculated by using $\mathrm{\Xi }NG$-matrix interactions. For comparison, the experimental data [1] are also shown. (b) Same as (a) but for ${}_{\mathrm{\Xi }}{}^{15}\mathrm{C}$ (${}^{14}\mathrm{N}+{\mathrm{\Xi }}^{-}$). The experimental data are taken from Refs. [2, 3, 4, 5].

Figure 2

The intrinsic densities of the $s$- and $p$-wave states of ${}_{\mathrm{\Xi }}{}^{15}\mathrm{C}$ calculated by using the HALt13 ($s$-wave) and ESC08c $G$-matrix interactions. Upper panels show the densities on the $z=0$ plane, where solid lines show the $\mathrm{\Xi }$ single-particle wave function and dashed lines show the isosurface where the density of core nucleus becomes half of its maximum. Lower panels show the densities along the $y=z=0$ line.

Figure 3

The $s$-wave doublets of ${}^{11}\mathrm{B}(3/2_{\mathrm{gs}}^{-})\otimes {\mathrm{\Xi }}_s^{-}$ in ${}_{\mathrm{\Xi }}{}^{12}\mathrm{Be}$ calculated by using the $\mathrm{\Xi }NG$-matrix interactions.