The previous Faddeev calculation of the two-alpha plus $\Lambda$ system for ${}_{\Lambda }{}^9\mathrm{B}\mathrm{e}$ is extended to incorporate the spin-orbit components of the ${\mathrm{SU}}_6$ quark-model (QM) baryon-baryon interactions. We employ the Born kernel of the QM $\Lambda N$ $LS$ interaction and generate the spin-orbit component of the $\Lambda \alpha$ potential by $\alpha$-cluster folding. The Faddeev calculation in the $jj$-coupling scheme implies that the direct use of the QM Born kernel for the $\Lambda N$ $LS$ component is not good enough to reproduce the small experimental value $\Delta E_{\ell s}^{\mathrm{expt}}=43\pm 5\mathrm{keV}$ for the $5∕2^{+}$-$3∕2^{+}$ splitting. This procedure predicts 3–5 times larger values in the models FSS and fss2. The spin-orbit contribution from the effective meson-exchange potentials in fss2 is argued to be unfavorable to the small $\ell s$ splitting, through the analysis of the Scheerbaum factors for the single-particle spin-orbit potentials calculated in the $G$-matrix formalism.

• Received 13 July 2004

DOI:https://doi.org/10.1103/PhysRevC.70.047002