Sum rule approach to a soft dipole mode in $\Lambda$ hypernuclei

Applying the sum rule approach, we investigate the energy of a soft dipole motion in $\Lambda$ hypernuclei, which results from a dipole oscillation of a $\Lambda$ hyperon against the core nucleus. To this end, we systematically study single-$\Lambda$ hypernuclei, from ${}_{\Lambda }^{16}$O to ${}_{\Lambda }^{208}$Pb, for which the ground-state wave function is obtained in the frame-work of Hartree-Fock method with several Skyrme-type $\Lambda N$ interactions. Our results indicate that the excitation energy of the soft dipole $\Lambda$ mode, $E_{sd\Lambda }$, decreases as the mass number increases. We find that the excitation energy is well parametrized as $E_{sd\Lambda }=26.6A^{-1/3}+11.2A^{-2/3}$ MeV for the SkM* $NN$ interaction and $E_{sd\Lambda }=21.3A^{-1/3}+22.6A^{-2/3}$ MeV for the SIII $NN$ interaction as a function of mass number $A$.

Figure 1

The root-mean-square radius $\sqrt{{\langle r^2\rangle }_{\Lambda }}$ for the $\Lambda$ hyperon in single-$\Lambda$ hypernuclei as a function of the mass number $A$, obtained with the Skyrme-Hartree-Fock calculations with six different $\Lambda N$ interactions.

Figure 2

The non-energy-weighted sum rule for the soft dipole $\Lambda$ mode as a function of mass number $A$ calculated with six different $\Lambda N$ interactions.

Figure 3

The energy-weighted sum rule for the soft dipole $\Lambda$ mode in single-$\Lambda$ hypernuclei as a function of mass number $A$.

Figure 4

The excitation energy of the soft $\Lambda$ dipole mode, $E_{sd\Lambda }$, obtained with the sum rule approach as $E_{sd\Lambda }=m_1^{\Lambda }/m_0^{\Lambda }$. The thick solid line indicates the results of the empirical formula given by Eq. (13). The SkM${}^{*}$ $NN$ interaction [40] is used.

Figure 5

Same as Fig. 4, but calculated with the SIII $NN$ interaction [43].

Figure 6

The strength distributions for the $\Lambda$ dipole mode induced by the operator given by Eq. (7) for double $\Lambda$ hypernuclei (a) ${}_{\Lambda \Lambda }^{18}$O and (b) ${}_{\Lambda \Lambda }^{210}$Pb. The dashed and solid lines are the results of the Skyrme-HF and the RPA, respectively. The dipole strengths are smeared by a Lorentzian function with a width of 1 MeV.