Generator coordinate method for hypernuclear spectroscopy with a covariant density functional

We apply the generator coordinate method (GCM) to single-$\mathrm{\Lambda }$ hypernuclei in order to discuss the spectra of hypernuclear low-lying states. To this end, we use the same relativistic point-coupling energy functional both for the mean-field and the beyond-mean-field calculations. This relativistic GCM approach provides a unified description of low-lying states in ordinary nuclei and in hypernuclei, and is thus suitable for studying the $\mathrm{\Lambda }$ impurity effect. We carry out an illustrative calculation for the low-lying spectrum of ${}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$, in which the interplay between the hypernuclear collective excitations and the single-particle excitations of the unpaired $\mathrm{\Lambda }$ hyperon is taken into account in a full microscopic manner.

Figure 1

(a) The total energy curves for ${}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$ obtained in the mean-field approximation as a function of quadrupole deformation $\beta$. These are calculated by putting the $\mathrm{\Lambda }$ hyperon in different single-particle orbitals shown in the lower panel. For comparison, the energy curve for the core nucleus ${}^{20}\mathrm{Ne}$ is also plotted. (b) The single-particle energies of the $\mathrm{\Lambda }$ hyperon in ${}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$ as a function of quadrupole deformation. These are labeled with the ${\mathrm{\Omega }}^{\pi }$ number, that is, the projection of the angular momentum onto the $z$ axis in the body fixed frame.

Figure 2

The projected energy curves for ${}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$ obtained by putting the $\mathrm{\Lambda }$ hyperon on the three lowest single-particle orbitals labeled by ${\mathrm{\Omega }}^{\pi }(=K^{\pi })$. The corresponding mean-field energy curves are also shown for a comparison. The solutions of the GCM calculations are indicated by the squares and the horizontal bars placed at the average deformation.

Figure 3

The low-lying excitation spectra of ${}^{20}\mathrm{Ne}$ (a) and ${}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$ [(b)–(d)] constructed with the GCM approach. The numbers with the arrows indicate the $E2$ transition strengths, given in units of $e^2{\mathrm{fm}}^4$. The experimental data for ${}^{20}\mathrm{Ne}$ are taken from Ref. [31].