Low-lying level structure of $\mathrm{\Lambda }$ hypernuclei and spin dependence of the $\mathrm{\Lambda }N$ interaction with antisymmetrized molecular dynamics

$\mathrm{\Lambda }N$ spin-spin and spin-orbit splittings in low-lying excitation spectra are investigated for $p$-shell $\mathrm{\Lambda }$ hypernuclei on the basis of the microscopic structure calculation within the antisymmetrized molecular dynamics, where the $\mathrm{\Lambda }NG$-matrix interaction derived from the baryon-baryon interaction model ESC (extended soft core) is used. It is found that the ground-state spin-parity is systematically reproduced in the $p$-shell $\mathrm{\Lambda }$ hypernuclei by tuning the $\mathrm{\Lambda }N$ spin-spin and spin-orbit interactions so as to reproduce the experimental data of ${}_{\mathrm{\Lambda }}{}^4\mathrm{H},{}_{\mathrm{\Lambda }}{}^7\mathrm{Li}$, and ${}_{\mathrm{\Lambda }}{}^9\mathrm{Be}$. Furthermore, we also focus on the excitation energies of the excited doublets as well as the energy shifts of them by the addition of a $\mathrm{\Lambda }$ particle.

Figure 1

Calculated and observed excitation spectra of the core nuclei ${}^6\mathrm{Li},{}^8\mathrm{Li},{}^{10}\mathrm{B},{}^{12}\mathrm{C},{}^{13}\mathrm{C},{}^{14}\mathrm{N}$, and ${}^{15}\mathrm{O}$. Numbers shown in the spectra are excitation energies (MeV). In panels (a), (b), (c), (d), and (f), the calculated results with the Gogny D1S modified so as to reproduce $Ex({}^6\mathrm{Li};3^{+})$ (mod. D1S) are shown together with those with the original parameter set of the Gogny D1S (D1S). The observed data are taken from Ref. [44] for ${}^6\mathrm{Li}$, Ref. [45] for ${}^{11}\mathrm{C}$, Ref. [46] for ${}^{12}\mathrm{C}$, Ref. [47] for ${}^{14}\mathrm{N}$ and ${}^{15}\mathrm{O}$, and Ref. [48] for the others.

Figure 2

Same as Fig. 1 but for ${}^8\mathrm{Be},{}^9\mathrm{Be}$, and ${}^9\mathrm{B}$. The calculated results with Volkov No. 2 (Volkov) are compared with those with Gogny D1S (D1S). The observed data are taken Ref. [48].

Figure 3

(a) Comparison of calculated (solid) and observed (dotted) excitation spectra of ${}_{\mathrm{\Lambda }}{}^4\mathrm{H}$. Calculated spectrum with the original parameter set of $\mathrm{ESC}14+\mathrm{MBE}$ is shown in the left (original), while that using $\mathrm{ESC}14+\mathrm{MBE}$ with tuning the $\mathrm{\Lambda }N$ even-state spin-spin force is shown in the middle $\left(+\mathrm{\Delta }{V}_{\sigma }^E\right)$. Numbers shown in the spectra are excitation energies (MeV). (b) Same as (a) but for ${}_{\mathrm{\Lambda }}{}^7\mathrm{Li}$. Calculated spectrum using $\mathrm{ESC}14+\mathrm{MBE}$ with the tuning in the $\mathrm{\Lambda }N$ even-state (even and odd state) spin-spin force(s) are shown in the second (third) from the left, while that without tuning is displayed in the left. Experimental data are taken from Refs. [50, 51, 52, 53].

Figure 4

Calculated (solid) and observed (dotted) excitation spectra. Numbers shown in the spectra are excitation energies (MeV). Observed data of the excitation spectra are also shown for ${}_{\mathrm{\Lambda }}{}^7\mathrm{Li}$ [52, 53], ${}_{\mathrm{\Lambda }}{}^9\mathrm{Be}$ [54], ${}_{\mathrm{\Lambda }}{}^{10}\mathrm{Be}$ [55], ${}_{\mathrm{\Lambda }}{}^{10}\mathrm{B}$ [56, 57], ${}_{\mathrm{\Lambda }}{}^{11}\mathrm{B}$ [34], ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{C}$ [35], ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ [1, 58], ${}_{\mathrm{\Lambda }}{}^{15}\mathrm{N}$ [56], and ${}_{\mathrm{\Lambda }}{}^{16}\mathrm{O}$ [59].

Figure 5

(a) Calculated energy spectra of ${}_{\mathrm{\Lambda }}{}^4\mathrm{H}$ and ${}_{\mathrm{\Lambda }}{}^4\mathrm{He}$ using $\text{ESC}+\text{MBE}$ with $k_F=0.92{\mathrm{fm}}^{-1}$. (b) Same as in (a) but with the phenomenological charge symmetry breaking (CSB) force explained in text. (c) Experimental data taken from Refs. [63, 73].