Triaxially deformed relativistic point-coupling model for $\Lambda$ hypernuclei: A quantitative analysis of the hyperon impurity effect on nuclear collective properties

Background: The impurity effect of hyperons on atomic nuclei has received a renewed interest in nuclear physics since the first experimental observation of appreciable reduction of $E2$ transition strength in low-lying states of the hypernucleus ${}_{\Lambda }^7\mathrm{Li}$. Many more data on low-lying states of $\Lambda$ hypernuclei will be measured soon for $sd$-shell nuclei, providing good opportunities to study the $\Lambda$ impurity effect on nuclear low-energy excitations.

Purpose: We carry out a quantitative analysis of the $\Lambda$ hyperon impurity effect on the low-lying states of $sd$-shell nuclei at the beyond-mean-field level based on a relativistic point-coupling energy density functional (EDF), considering that the $\Lambda$ hyperon is injected into the lowest positive-parity (${\Lambda }_s$) and negative-parity (${\Lambda }_p$) states.

Method: We adopt a triaxially deformed relativistic mean-field (RMF) approach for hypernuclei and calculate the $\Lambda$ binding energies of hypernuclei as well as the potential-energy surfaces (PESs) in the $(\beta ,\gamma )$ deformation plane. We also calculate the PESs for the $\Lambda$ hypernuclei with good quantum numbers by using a microscopic particle rotor model (PRM) with the same relativistic EDF. The triaxially deformed RMF approach is further applied in order to determine the parameters of a five-dimensional collective Hamiltonian (5DCH) for the collective excitations of triaxially deformed core nuclei. Taking ${}_{\Lambda }{}^{25,27}\mathrm{Mg}$ and ${}_{\Lambda }{}^{31}\mathrm{Si}$ as examples, we analyze the impurity effects of ${\Lambda }_s$ and ${\Lambda }_p$ on the low-lying states of the core nuclei.

Results: We show that ${\Lambda }_s$ increases the excitation energy of the $2_1^{+}$ state and decreases the $E2$ transition strength from this state to the ground state by $12\%\text{to}17\%$. On the other hand, ${\Lambda }_p$ tends to develop pronounced energy minima with larger deformation, although it modifies the collective parameters in such a way that the collectivity of the core nucleus can be either increased or decreased.

Conclusions: The quadrupole deformation significantly affects the $\Lambda$ binding energies of deformed hypernuclei. A beyond-mean-field approach with the dynamical correlations due to restoration of broken symmetries and shape fluctuation is essential in order to study the $\Lambda$ impurity effect in a quantitative way.

Figure 1

(a) The convergence feature of potentials $V_0^B\left(\mathit{r}\right)+S^B\left(\mathit{r}\right)$, (b) $V_0^B\left(\mathit{r}\right)-S^B\left(\mathit{r}\right)$ (with $B=n,p,\Lambda$), and (c) the tensor density ${\alpha }_T^{\left(N\Lambda \right)}{\rho }_T^{\Lambda }\left(\mathit{r}\right)$ with respect to the maximum number of major shell ($N_f$) in the 3DHO basis for the triaxial RMF calculation for ${}^{17}{}_{\Lambda }\mathrm{O}$. These quantities are evaluated at $x=y=0.36$ fm and plotted as a function of $z$. For a comparison, the results of spherical RMF calculations are also shown by the lines.

Figure 2

(a) The $\Lambda$ binding energies in single-$\Lambda$ hypernuclei with $\Lambda$ in different single-particle states obtained with the spherical and the triaxial RMF calculations, and their comparison with the available data from Refs. [20, 71, 72, 73]. The parameter set PC-F1 and PCY-S1 are adopted for the $NN$ and the $N\Lambda$ interactions, respectively. (b) The quadrupole deformation $\beta$ for each $\Lambda$ hypernuclei obtained with the triaxial RMF calculation. See text for more details.

Figure 3

The total energy of ${}^{50}\mathrm{V}$, ${}_{\Lambda s}^{51}\mathrm{V}$, ${}_{\Lambda p}^{51}\mathrm{V}$, and ${}_{\Lambda d}^{51}\mathrm{V}$ as a function of deformation parameter $\beta$ with the triaxially deformed RMF calculations. The energy of hypernuclei is shifted by normalizing the minimum energy to that of ${}^{50}\mathrm{V}$. The energy difference between the hypernuclei and ${}^{50}\mathrm{V}$ at $\beta =0$ is indicated with the numbers.

Figure 4

The potential-energy surfaces obtained with the triaxially deformed RMF calculations for (a) ${}^{24}\mathrm{Mg}$, ${}_{\Lambda s}^{25}\mathrm{Mg}$, ${}_{\Lambda p}^{25}\mathrm{Mg}$, and the core nucleus inside ${}_{\Lambda s}^{25}\mathrm{Mg}$; (c) ${}^{26}\mathrm{Mg}$, ${}_{\Lambda s}^{27}\mathrm{Mg}$, ${}_{\Lambda p}^{27}\mathrm{Mg}$, and the core nucleus inside ${}_{\Lambda s}^{27}\mathrm{Mg}$; (e) ${}^{30}\mathrm{Si}$, ${}_{\Lambda s}^{31}\mathrm{Si}$, ${}_{\Lambda p}^{31}\mathrm{Si}$, and the core nucleus inside ${}_{\Lambda s}^{30}\mathrm{Si}$ as a function of deformation parameter $\beta$. All the energies are normalized to the global minimum. The deformation-dependent $\Lambda$ separation energy defined by Eq. (17) as a function of deformation parameter $\beta$ for (b) ${}_{\Lambda s}^{25}\mathrm{Mg}$ and ${}_{\Lambda p}^{25}\mathrm{Mg}$; (d) ${}_{\Lambda s}^{27}\mathrm{Mg}$ and ${}_{\Lambda p}^{27}\mathrm{Mg}$; and (f) ${}_{\Lambda s}^{31}\mathrm{Si}$ and ${}_{\Lambda p}^{31}\mathrm{Si}$. The binding energies of ${\Lambda }_s$ and ${\Lambda }_p$ for each case are given above the figures.

Figure 5

(a) The potential-energy surfaces for ${}^{30}\mathrm{Si}$, ${}_{\Lambda s}^{31}\mathrm{Si}$, ${}_{\Lambda p}^{31}\mathrm{Si}$, as a function of deformation parameter $\beta$ obtained with the deformed RMF calculations using the same PC-F1 force for the $NN$ interaction but different $N\Lambda$ interaction (PCY-S1 and PCY-S4 [62], respectively). The energies are normalized to the global minima. (b) The $\Lambda$ separation energy $S_{\Lambda }\left(\beta \right)$ as a function of deformation parameter $\beta$.

Figure 6

(a) The total energy of ${}^{24}\mathrm{Mg}$, ${}_{\Lambda s}^{25}\mathrm{Mg}$, and ${}_{\Lambda p}^{25}\mathrm{Mg}$ as a function of deformation parameter $\gamma$, where the deformation $\beta$ is fixed at the value of the global minimum of the energy surface. The energies are normalized to the energy of the global minimum. (b) Same as in panel (a), but for ${}^{26}\mathrm{Mg}$, ${}_{\Lambda s}^{27}\mathrm{Mg}$, and ${}_{\Lambda p}^{27}\mathrm{Mg}$.

Figure 7

The potential-energy surfaces of (a) ${}^{24}\mathrm{Mg}$, (b) ${}_{\Lambda s}^{25}\mathrm{Mg}$, (c) ${}_{\Lambda s}^{25}\mathrm{Mg}$ (cn), and (d) ${}_{\Lambda p}^{25}\mathrm{Mg}$ in the $(\beta ,\gamma )$ plane. The energies are normalized to the global minimum. The energy difference between (e) ${}_{\Lambda s}^{25}\mathrm{Mg}$ and its core nucleus and (f) between ${}_{\Lambda p}^{25}\mathrm{Mg}$ and its core nucleus are also plotted. Two neighboring contour lines are separated by 0.5 MeV.

Figure 8

Same as Fig. 7, but for ${}^{26}\mathrm{Mg}$, ${}_{\Lambda s}^{27}\mathrm{Mg}$, and ${}_{\Lambda p}^{27}\mathrm{Mg}$.

Figure 9

Same as Fig. 7, but for ${}^{30}\mathrm{Si}$, ${}_{\Lambda s}^{31}\mathrm{Si}$, and ${}_{\Lambda p}^{31}\mathrm{Si}$.

Figure 10

The single-particle energies of $\Lambda$ hyperon in ${}_{\Lambda s}^{25}\mathrm{Mg}$ as a function of deformation parameters $\beta$ and $\gamma$ from the calculation with the PC-F1 and PCY-S1 for the $NN$ and $N\Lambda$ interactions, respectively. In panel (a), the energy levels from the calculation without the tensor potential $U_T$ (11b) are also plotted with the dashed lines.

Figure 11

Comparison of the particle-number and angular-momentum projected PES (solid curve) of ${}^{24}\mathrm{Mg}$ with the PESs with spin-parity of $I^{\pi }=1/2^{+}$ (dashed curve) and $1/2^{-}$ (dash-dotted curve) for ${}^{25}{}_{\Lambda }\mathrm{Mg}$ obtained with the microscopic particle-rotor model.

Figure 12

(a) The collective potential $V_{\mathrm{coll}}$, (b) the moment of inertia along $x$ axis, ${\mathcal{I}}_x$, (c) the mean squared radius of protons, and (d) the mass parameters $B_{\beta \beta }$ as a function of quadrupole deformation $\beta$ for ${}^{24}\mathrm{Mg}$ and ${}^{25}{}_{\Lambda }\mathrm{Mg}$ (cn) obtained with the fine-dimensional collective Hamiltonian (5DCH) approach.

Figure 13

(a), (b) The low-spin spectra of the ground-state band for ${}^{24}\mathrm{Mg}$ and (c), (d) the nuclear core of ${}^{25}{}_{\Lambda }\mathrm{Mg}$ obtained with the 5DCH method. The $B\left(E2\right)$ values are in units of ${\mathrm{e}}^2$ ${\mathrm{fm}}^4$. The spectrum of ${}^{24}\mathrm{Mg}$ is compared with the corresponding experimental data, taken from Ref. [80].

Figure 14

Same as Fig. 12, but for ${}^{26}\mathrm{Mg}$ and nuclear core inside ${}_{\Lambda }^{27}\mathrm{Mg}$.

Figure 15

Same as Fig. 13, but for ${}^{26}\mathrm{Mg}$ and ${}^{27}{}_{\Lambda }\mathrm{Mg}$ (cn). The experimental data for ${}^{26}\mathrm{Mg}$ are taken from Refs. [82, 83].

Figure 16

Same as Fig. 12, but for ${}^{30}\mathrm{Si}$ and nuclear core inside ${}^{31}{}_{\Lambda }\mathrm{Si}$.

Figure 17

Same as Fig. 13, but for ${}^{30}\mathrm{Si}$ and ${}^{31}{}_{\Lambda }\mathrm{Si}$ (cn). The experimental data for ${}^{30}\mathrm{Si}$ are taken from Refs. [82, 83].

Figure 18

The impurity effect of ${\Lambda }_s$ and ${\Lambda }_p$ hyperon on the excitation energies $E_x\left(2_1^{+}\right)$ and $E_x\left(2_2^{+}\right)$ for the lowest two $2^{+}$ states, the proton root-mean-squared radius $r_p$ of the ground state ($0_1^{+}$), and the $E2$ transition strength $B(E2:2_1^{+}\to 0_1^{+})$ in ${}^{24,26}\mathrm{Mg}$ and ${}^{30}\mathrm{Si}$ from the 5DCH calculations based on the PC-F1 and PCY-S1 forces for the $NN$ and $N\Lambda$ interactions, respectively. For comparison, the impurity effect of ${\Lambda }_s$ and ${\Lambda }_p$ on ${}^{30}\mathrm{Si}$ calculated with the $N\Lambda$ interaction of PCY-S4 [62] is also plotted in the last column.