$\mathrm{\Sigma }$-nuclear spin-orbit coupling from two-pion exchange

Using SU(3) chiral perturbation theory we calculate the density-dependent complex-valued spin-orbit coupling strength $U_{\Sigma ls}(k_f)+ i W_{\Sigma ls}(k_f)$ofa$\Sigma$hyperoninthenuclearmedium\mathrm{.}Theleadinglong-range$\Sigma N$interactionarisesfromiteratedone-pionexchangewitha$\Lambda$ora$\Sigma$ hyperon in the intermediate state. We find from this unique long-range dynamics a sizeable ``wrong-sign’’ spin-orbit coupling strength of $U_{\mathrm{\Sigma }ls}(k_{f0})\simeq -20$ MeVfm${}^2$ at normal nuclear matter density ${\rho }_0=0.16$fm${}^{-3}$. The strong $\mathrm{\Sigma }N\to \mathrm{\Lambda }N$ conversion process contributes at the same time an imaginary part of $W_{\Sigma ls}(k_{f0}) \simeq -12$MeVfm$^2$. When combined with estimates of the short-range contribution the total $\mathrm{\Sigma }$-nuclear spin-orbit coupling becomes rather weak.