Σ−Capture in Deuterium

Transition rates for the final states ${\Sigma }^0+n+n$ and $\Lambda +n+n$ are calculated in impulse approximation for ${\Sigma }^{-}d$ capture from rest, taking into account the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ $n-n$ attraction and the tensor term in the ${\Sigma }^{-}p\to \Lambda n$ transition amplitude. Using the ${\Sigma }^{-}p\to {\Sigma }^{0}n$ and ${\Sigma }^{-}p\to \Lambda n$ amplitudes calculated for global symmetry by de Swart and Dullemond, the branching ratio $\frac{{\Sigma }^0}{({\Sigma }^0+\Lambda )}$ obtained is 0.24, compared with the observed ratio 0.037±0.022. The uncertainties in the comparison between the observed and calculated ratios are discussed in detail, especially those concerned with the validity of the impulse approximation, in view of the near-resonant $\Sigma N$ interaction in the global symmetry model, and the question of the appropriate form of the spin average, in view of the strong electromagnetic spin-orbit forces in the ${\Sigma }^{-}d$ atom. It is concluded that the appropriate spin average is $\frac{2}{3}{\left[\frac{{\Sigma }^0}{({\Sigma }^0+\Lambda )}\right]}_{S=\frac{3}{2}}+\frac{1}{3}{\left[\frac{{\Sigma }^0}{({\Sigma }^0+\Lambda )}\right]}_{S=\frac{1}{2}}$ within corrections of about 10%. Agreement with the experimental ratio appears unlikely with global symmetry, even when these uncertainties are taken into account.