$\Lambda$ spin-orbit splittings deduced from DWIA analysis of the $^{89}$Y$(\pi^+,K^+)^{89}_{\Lambda}$Y reaction

High resolution measurements of hypernuclear $\gamma$ rays from the $^9$Be$(\pi^+,K^+\gamma )^9_{\Lambda}$Be and $^{13}$C$(K^-,\pi^-\gamma )^{13}_{\Lambda}$C reactions have shown clearly that the $\Lambda N$ spin-orbit force is very small. In heavier cases such as $^{89}_{\Lambda}$Y, however, the $(\pi^+,K^+)$ experiments show a series of strong peaks having doublet-like substructure which apparently suggests sizable $\Lambda N$ spin-orbit splittings. In order to resolve this discrepancy, detailed structure calculations of $^{89}_{\Lambda}$Y have been performed by taking nuclear core excitation into account. The obtained wave functions have been used to estimate the $(\pi^+,K^+)$ reaction cross sections within DWIA. Based on a careful analysis of the peak structure, a theoretical explanation is given for the first time how to understand the doublet substructure in a series of observed major peaks in $^{89}_{\Lambda}$Y. In the discussion we conclude small $\Lambda$ spin-orbit force which is consistent with the two light hypernuclear cases.