Note on spin-orbit interactions in nuclei and hypernuclei

A detailed comparison is made between the spin-orbit interactions in $\Lambda$ hypernuclei and ordinary nuclei. We argue that there are three major contributions to the spin-orbit interaction: 1) a short-range component involving scalar and vector mean fields; 2) a ''wrong-sign'' spin-orbit term generated by the pion exchange tensor force in second order; and 3) a three-body term induced by two-pion exchange with excitation of virtual $\Delta(1232)$-isobars (a la Fujita-Miyazawa). For nucleons in nuclei the long-range pieces related to the pion-exchange dynamics tend to cancel, leaving room dominantly for spin-orbit mechanisms of short-range origin (parametrized e.g. in terms of relativistic scalar and vector mean fields terms). In contrast, the absence of an analogous $2\pi$-exchange three-body contribution for $\Lambda$ hyperons in hypernuclei leads to an almost complete cancellation between the short-range (relativistic mean-field) component and the ''wrong-sign'' spin-orbit interaction generated by second order $\pi$-exchange with an intermediate $\Sigma$ hyperon. These different balancing mechanisms between short- and long-range components are able to explain simultaneously the very strong spin-orbit interaction in ordinary nuclei and the remarkably weak spin-orbit splitting in $\Lambda$ hypernuclei.