Microscopic study of low-lying spectra of $\mathrm{\Lambda }$ hypernuclei based on a beyond-mean-field approach with a covariant energy density functional

We present a detailed formalism of the microscopic particle-rotor model for hypernuclear low-lying states based on a covariant density functional theory. In this method, the hypernuclear states are constructed by coupling a hyperon to low-lying states of the core nucleus, which are described by the generator coordinate method (GCM) with the particle number and angular momentum projections. We apply this method to study in detail the low-lying spectrum of ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ and ${}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$ hypernuclei. We also briefly discuss the structure of ${}_{\mathrm{\Lambda }}{}^{155}\mathrm{Sm}$ as an example of heavy deformed hypernuclei. It is shown that the low-lying excitation spectra with positive-parity states of the hypernuclei, which are dominated by $\mathrm{\Lambda }$ hyperon in the $s$ orbital coupled to the core states, are similar to that for the corresponding core states, while the electric quadrupole transition strength, $B\left(E2\right)$, from the $2_1^{+}$ state to the ground state is reduced according to the mass number of the hypernuclei. Our study indicates that the energy splitting between the first $1/2^{-}$ and $3/2^{-}$ hypernuclear states is generally small for all the hypernuclei which we study. However, their configurations depend much on the properties of a core nucleus, in particular on the sign of deformation parameter. That is, the first $1/2^{-}$ and $3/2^{-}$ states in ${}_{\mathrm{\Lambda }}^{13}\mathrm{C}$ are dominated by a single configuration with $\mathrm{\Lambda }$ particle in the $p$-wave orbits and thus provide good candidates for a study of the $\mathrm{\Lambda }$ spin-orbit splitting. On the other hand, those states in the other hypernuclei exhibit a large configuration mixing and thus their energy difference cannot be interpreted as the spin-orbit splitting for the $p$ orbits.

Figure 1

A schematic picture of the microscopic particle-rotor model for $\mathrm{\Lambda }$ hypernucleus, in which $\mathit{r}$ denotes the coordinate of the $\mathrm{\Lambda }$ hyperon. In this approach, the nuclear core states are described microscopically with the multireference density functional theory.

Figure 2

The energy curve of the mean-field state and the particle-number and angular momentum projected ($I^{\pi }=0^{+},2^{+}$, and $4^{+}$) states for ${}^{12}\mathrm{C}$ as a function of the intrinsic quadrupole deformation $\beta$. The filled squares indicate the lowest three GCM solutions for each $I^{\pi }$, which are plotted at their average deformation $\overline{\beta }\equiv {\sum }_{\beta }{\left|g_n^I\left(\beta \right)\right|}^2\beta$, where $g_n^I\left(\beta \right)$ is the collective wave functions defined by Eq. (20). The results are obtained with the PC-F1 force.

Figure 3

The collective wave functions $g_n^I$ given by Eq. (20), for the first three states in ${}^{12}\mathrm{C}$ with spin parities of $0^{+},2^{+},4^{+}$, and $6^{+}$.

Figure 4

The vector transition density, ${\rho }_{\lambda ,V}^{I_n,I_n^{\prime }}$, given by Eq. (13a), and the scalar transition density, ${\rho }_{\lambda ,S}^{I_n,I_n^{\prime }}$, given by Eq. (13b), in the low-lying states ($n=1$) of ${}^{12}\mathrm{C}$. These are plotted by the solid and the dashed lines, respectively. The insets show the difference of the vector and scalar transition densities.

Figure 5

The charge form factor for the transition from the ground state to (a) the ground state, (b) the excited $2_1^{+}$ state, and (c) the excited $4_1^{+}$ state in ${}^{12}\mathrm{C}$ calculated with the GCM method with the PC-F1 force, in comparison with the available data [43, 44]. $R$ on the top panel is the root-mean-square charge radius of ${}^{12}\mathrm{C}$.

Figure 6

The charge form factors between the two bands with $n=1$ and $n=2$ in ${}^{12}\mathrm{C}$ calculated with the GCM method with the PC-F1 force, in comparison with the available data from Ref. [47].

Figure 7

The spectrum of ${}^{12}\mathrm{C}$ obtained with several methods, that is, the GCM with the PC-F1 interaction (the present calculation), GCM with the Skyrme SLy4 force [45], antisymmetrized molecular dynamics (AMD) [48], and the $\alpha$ cluster model [47]. The excitation energies are given in units of MeV. The solid and the dashed arrows are the quadrupole transition strength $B\left(E2\right)$ ($e^2 {\mathrm{fm}}^4$) and the electric monopole transition matrix element $\left|M\right(E0\left)\right|$ ($e {\mathrm{fm}}^2$), respectively. The experimental data are taken from Refs. [50, 51].

Figure 8

(a) A contour plot of the absolute value of the difference between the theoretical and the experimental hyperon binding energies of ${}_{\mathrm{\Lambda }}^{13}\mathrm{C}$ hypernucleus as a function of the coupling strength parameters ${\alpha }_S^{N\mathrm{\Lambda }}$ and ${\alpha }_V^{N\mathrm{\Lambda }}$ in the $N\mathrm{\Lambda }$ interaction. The calculated energies are obtained with the microscopic particle-rotor model. (b) Energy levels of the $3/2^{+}, 3/2^{-}$, and $1/2^{-}$ states in ${}_{\mathrm{\Lambda }}^{13}\mathrm{C}$ calculated with the strength parameters denoted by the dots in the panel (a).

Figure 9

The potential energy surfaces of hypernucleus ${}_{\mathrm{\Lambda }}^{13}\mathrm{C}$ obtained with the single-channel calculation (the dot-dashed lines) and the coupled-channel calculation (the solid lines) with spin parity of (a) $J^{\pi }=1/2^{+}$, (b) $J^{\pi }=1/2^{-}$, and (c) $J^{\pi }=3/2^{-}$ as a function of deformation parameter $\beta$. For comparison, the figure also shows the energy surface for the nuclear core with spin-parity of $I^{\pi }=0^{+}$ (the dashed lines). The energy surfaces for hypernuclei are shifted by a constant amount as indicated in each panel.

Figure 10

The low-energy excitation spectra of ${}^{12}\mathrm{C}$ [(a)–(c)] and ${}_{\mathrm{\Lambda }}^{13}\mathrm{C}$ [(d)–(l)]. For ${}^{12}\mathrm{C}$, the full GCM calculations are compared with the experimental data taken from Ref. [52]. The columns (h) and (i) show the positive-parity states in ${}_{\mathrm{\Lambda }}^{13}\mathrm{C}$, while the columns (j) and (k) show the negative-parity states. The experimental data of ${}_{\mathrm{\Lambda }}^{13}\mathrm{C}$ are taken from Refs. [53, 54]. For comparison, the results of single-channel calculation for ${}_{\mathrm{\Lambda }}^{13}\mathrm{C}$ with the $\mathrm{\Lambda }$ particle in the $s_{1/2}, p_{1/2}$ and $p_{3/2}$ orbitals are also plotted in the columns (d)–(g).

Figure 11

(a) The transition densities ${\alpha }_V{\rho }_{\lambda ,V}^{22}+{\alpha }_S{\rho }_{\lambda ,S}^{22}$ ($\lambda =0$ and $\lambda =2$) and (b) the potential $U_V^{kk}\left(r\right)+U_S^{kk}\left(r\right)$ in Eq. (28) for the hypernuclear states $J=1/2^{-},3/2^{-},5/2^{-},7/2^{-}$ with the ${\mathrm{\Lambda }}_{p3/2}\otimes 2_1^{+}$ configuration as a function of the radial coordinate $r$.

Figure 12

Excitation energy of low-lying states in ${}_{\mathrm{\Lambda }}^{13}\mathrm{C}$ as a function of the cutoff of core states $n$ [(a) and (b)] and the cutoff of core angular momentum $I$ [(c) and (d)] for the coupled-channels calculations.

Figure 13

A comparison of low-energy excitation spectra of ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ obtained with the present microscopic particle-rotor model (MPRM), the multi-channel algebraic scattering (MCAS) approach [55], the $3\alpha +\mathrm{\Lambda }$ cluster model [56], and the experimental data [1].

Figure 14

A comparison of the yrast rotational states of ${}^{20}\mathrm{Ne}$ obtained with several methods. The results of the cluster model and the AMD are taken from Refs. [57] and [14], respectively.

Figure 15

Energy curve $E_J\left(\beta \right)$ for the $J^{\pi }=1/2^{+}$ (solid line) and $1/2^{-}$ (dot-dashed line) states in ${}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$ as a function of the deformation $\beta$ of the core nucleus. These are obtained with PC-F1 (left panel) and PC-PK1 (right panel) forces. In order to make a comparison easy, each hypernuclear curve is shifted by a constant value so that the energy at the absolute minimum coincides with that for ${}^{20}\mathrm{Ne}$ with $I^{\pi }=0^{+}$. The difference between the energy curve of ${}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$ and that of ${}^{20}\mathrm{Ne}$ is shown in the panels (c) and (d) for the PC-F1 and PC-PK1 forces, respectively.

Figure 16

The low-energy excitation spectra of ${}^{20}\mathrm{Ne}$ and ${}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$ obtained with the microscopic particle-rotor model calculations.

Figure 17

Same as Fig. 11, but for ${}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$.

Figure 18

A comparison of low-energy excitation spectra of ${}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$ obtained with the cluster model [57], the AMD [14], and the present microscopic particle-rotor model (MPRM) calculations with the PC-F1 and PC-PK1 forces.

Figure 19

Same as Fig. 9, but for ${}^{154}\mathrm{Sm}$ and ${}_{\mathrm{\Lambda }}{}^{155}\mathrm{Sm}$.

Figure 20

Low-energy excitation spectra of ${}^{154}\mathrm{Sm}$ [(a)–(f)] and ${}_{\mathrm{\Lambda }}{}^{155}\mathrm{Sm}$ [(g)–(i)].