Low-energy hypernuclear spectra within a microscopic particle-rotor model with a relativistic point-coupling hyperon-nucleon interaction

We extend the microscopic particle-rotor model for hypernuclear low-lying states by including the derivative and tensor coupling terms in the point-coupling nucleon $\mathrm{\Lambda }$-particle $\left(N\mathrm{\Lambda }\right)$ interaction. Taking ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ as an example, we show that a good overall description for excitation spectra is achieved with four sets of effective $N\mathrm{\Lambda }$ interactions. We find that the $\mathrm{\Lambda }$ hyperon binding energy decreases monotonically with increasing strengths of the high-order interaction terms. In particular, the tensor coupling term decreases the energy splitting between the first $1/2^{-}$ and $3/2^{-}$ states and increases the energy splitting between the first $3/2^{+}$ and $5/2^{+}$ states in ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$.

Figure 1

The low-energy excitation spectra of ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ obtained with the microscopic particle-rotor model with (b) PCY-S1, (c) PCY-S2, (d) PCY-S3, and (e) PCY-S4. Part (f) shows the spectrum taken from Ref. [30], which was obtained by including only the leading-order (LO) $N\mathrm{\Lambda }$ interaction. The solid and the dashed lines represent the first and the second states for each spin-parity $\left(J^{\pi }\right)$, respectively. The experimental data shown in (a) are taken from Refs. [1, 38, 39]. The numbers with the arrows indicate the $B\left(E2\right)$ values for the $3/2_1^{+}\to 1/2_1^{+}$ and the $9/2_1^{+}\to 5/2_1^{+}$ transitions, given in units of $e^2{\mathrm{fm}}^4$. The dominant component of several hypernuclear states, together with its weight (in percent), is also given.

Figure 2

The same as in Fig. 1, but with the scaled $N\mathrm{\Lambda }$ interaction, in which the scaling factor is determined for each parameter set to reproduce the empirical $\mathrm{\Lambda }$ binding energy of ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$.

Figure 3

(a) and (c) Contour plots for the absolute value of the difference between the theoretical and the experimental hyperon binding energies of the ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ hypernucleus as a function of the coupling strength parameters $({\delta }_S^{N\mathrm{\Lambda }},{\delta }_V^{N\mathrm{\Lambda }})$ and $({\delta }_S^{N\mathrm{\Lambda }},{\alpha }_S^{N\mathrm{\Lambda }})$, respectively. In the former, ${\alpha }_V^{N\mathrm{\Lambda }}$ and ${\alpha }_S^{N\mathrm{\Lambda }}$ are fixed to the same values as in PCY-S2, while, in the latter, the value of ${\alpha }_V^{N\mathrm{\Lambda }}$ and ${\delta }_V^{N\mathrm{\Lambda }}$ is determined for each $({\alpha }_S^{N\mathrm{\Lambda }},{\delta }_S^{N\mathrm{\Lambda }})$ so as to keep the ratios ${\alpha }_V^{N\mathrm{\Lambda }}/{\alpha }_S^{N\mathrm{\Lambda }}$ and ${\delta }_V^{N\mathrm{\Lambda }}/{\delta }_S^{N\mathrm{\Lambda }}$ the same as those for PCY-S2. (b) and (d) Low-lying states in ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ calculated with the strength parameters denoted by the dots in panels (a) and (c), respectively.

Figure 4

(a) The $\mathrm{\Lambda }$ binding energy in ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ as a function of $|{\delta }_S^{N\mathrm{\Lambda }}+{\delta }_V^{N\mathrm{\Lambda }}|$, while keeping the same values of ${\alpha }_S^{N\mathrm{\Lambda }},{\alpha }_V^{N\mathrm{\Lambda }},{\alpha }_T^{N\mathrm{\Lambda }}$, and ${\delta }_V^{N\mathrm{\Lambda }}/{\delta }_S^{N\mathrm{\Lambda }}$ as the original values for the PCY-S1, PCY-S2, PCY-S3, and PCY-S4 parameter sets. $B_{\mathrm{\Lambda }}$ with the original value of ${\delta }_S^{N\mathrm{\Lambda }}$ and ${\delta }_V^{N\mathrm{\Lambda }}$ is denoted by the open circles for each parameter set. The experimental value is denoted by the thin solid line. (b) The energy levels of the low-lying states as a function of $|{\delta }_S^{N\mathrm{\Lambda }}+{\delta }_V^{N\mathrm{\Lambda }}|$ for the PCY-S1 parameter set. (c) and (d) The energy splitting between the $5/2_1^{+}$ and $3/2_1^{+}$ states and that between the $1/2_1^{-}$ and $3/2_1^{-}$ states, respectively, as a function of $|{\delta }_S^{N\mathrm{\Lambda }}+{\delta }_V^{N\mathrm{\Lambda }}|$.

Figure 5

Same as Fig. 4, but as a function of the tensor coupling strength $|{\alpha }_T^{N\mathrm{\Lambda }}|(=-{\alpha }_T^{N\mathrm{\Lambda }})$ for the PCY-S1 and PCY-S4 forces.