Spin and pseudospin symmetries in the single-$\mathrm{\Lambda }$ spectrum

We explore spin and pseudospin symmetries in the single-$\mathrm{\Lambda }$ spectrum within the framework of relativistic mean-field theory. We find that the spin symmetry of the single-$\mathrm{\Lambda }$ spectrum maintains, which is consistent with experimental results of small spin-orbit splitting, whereas the approximate pseudospin symmetry is quite similar to that of nucleons. More interestingly, the $\omega \mathrm{\Lambda }\mathrm{\Lambda }$ tensor coupling has opposite effects on these symmetries which makes the spin symmetry better but the pseudospin symmetry worse. This can be understood by the negative value of the $\omega \mathrm{\Lambda }\mathrm{\Lambda }$ tensor potential, which reduces the spin-orbit potential while increasing the pseudospin-orbit potential.

Figure 1

Single-particle spectra for the $\mathrm{\Lambda }$ hyperon in ${}_{\mathrm{\Lambda }}{}^{209}\mathrm{Pb}$ for spin and pseudospin doublets. Potential $V+S$ is shown as the blue solid line. The $\omega \mathrm{\Lambda }\mathrm{\Lambda }$ tensor coupling is omitted.

Figure 2

(a) Reduced SO splitting $\mathrm{\Delta }{E}_{\mathrm{SO}}=({\varepsilon }_{j_{<}}-{\varepsilon }_{j_{>}})/(2l+1)$ and (b) reduced PSO splitting $\mathrm{\Delta }{E}_{\mathrm{PSO}}=({\varepsilon }_{j_{<}}-{\varepsilon }_{j_{>}})/(2\tilde{l}+1)$ versus their average s.p. energies $E_{\mathrm{av}}=({\varepsilon }_{j_{<}}+{\varepsilon }_{j_{>}})/2$ in s.p. spectrum for $\mathrm{\Lambda }$ hyperon of ${}_{\mathrm{\Lambda }}{}^{209}\mathrm{Pb}$. For the spin doublets, $j_{<}=l-1/2$ and $j_{>}=l+1/2$; and for the pseudospin doublets, $j_{<}=\tilde{l}-1/2$ and $j_{>}=\tilde{l}+1/2$. Spin (pseudospin) doublets with the same $l$ ($\tilde{l}$) are linked by lines. The $\omega \mathrm{\Lambda }\mathrm{\Lambda }$ tensor coupling is omitted.

Figure 3

Comparison of the $\mathrm{\Lambda }$ hyperon effective CB $V_{\mathrm{CB}}^{\mathrm{eff}}$ (dashed and dot-dashed lines) with the effective SO potential $V_{\mathrm{SO}}^{\mathrm{eff}}$ (solid line) for (a) $p_{1/2}$ and (b) $p_{3/2}$ states in ${}_{\mathrm{\Lambda }}{}^{209}\mathrm{Pb}$. The dashed lines are for $1p_{1/2}$ and $1p_{3/2}$ states, and the dot-dashed lines are for $2p_{1/2}$ and $2p_{3/2}$ states. The $\omega \mathrm{\Lambda }\mathrm{\Lambda }$ tensor coupling is omitted.

Figure 4

Comparison of the $\mathrm{\Lambda }$ hyperon effective PCB $V_{\mathrm{PCB}}^{\mathrm{eff}}$ (short dot-dashed lines) and the corrected effective PSO potential $V_{\mathrm{PSO}}^{\mathrm{c}\text{−}\mathrm{eff}}$ (solid lines) multiplied by the square of the wave function $F$ of the lower component for the (a) $s_{1/2}$ and (b) $d_{3/2}$ states in ${}_{\mathrm{\Lambda }}{}^{209}\mathrm{Pb}$. All the potentials are multiplied by a factor of ${10}^3$. The $\omega \mathrm{\Lambda }\mathrm{\Lambda }$ tensor coupling is omitted.

Figure 5

Single-particle wave functions for the $\mathrm{\Lambda }$ hyperon of the $1p$, $2p$, and $1f$ spin doublets (upper panels) and $1\tilde{p}$, $2\tilde{p}$, and $1\tilde{f}$ pseudospin doublets (lower panels) in ${}_{\mathrm{\Lambda }}{}^{209}\mathrm{Pb}$ calculated by the RMF theory. The $\omega \mathrm{\Lambda }\mathrm{\Lambda }$ tensor coupling is omitted.

Figure 6

Single-particle spectrum for $\mathrm{\Lambda }$ hyperon in ${}_{\mathrm{\Lambda }}{}^{209}\mathrm{Pb}$ as a function of the $\omega \mathrm{\Lambda }\mathrm{\Lambda }$ tensor coupling strength $f_{\omega \mathrm{\Lambda }\mathrm{\Lambda }}/g_{\omega \mathrm{\Lambda }}$.

Figure 7

Same as Fig. 2, but with different $\omega \mathrm{\Lambda }\mathrm{\Lambda }$ tensor coupling strengths $f_{\omega \mathrm{\Lambda }\mathrm{\Lambda }}/g_{\omega \mathrm{\Lambda }}$.