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 $^{16}_{\;\,\Lambda}$O to $^{208}_{\;\;\;\Lambda}$Pb, for which the ground state wave function is obtained in the framework 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 as a function of mass number $A$.