The $Lambda p$ interaction close to the $Sigma N$ threshold is considered. Specifically, the pronounced structure seen in production reactions like $K^-d to pi^- Lambda p$ and $ppto K^+ Lambda p$ around the $Sigma N$ threshold is analyzed. Modern interaction models of the coupled $Lambda N - Sigma N$ systems generate such a structure either due to the presence of a (deuteron-like) unstable bound state or of an inelastic virtual state. % A determination of the position of the prominent peak as observed in various experiments for the two aforementioned reactions leads to values that agree quite well with each other. Furthermore, the deduced mean value of $2128.7pm 0.3$ MeV for the peak position coincides practically with the threshold energy of the $Sigma^+ n$ channel. This supports the interpretation of the structure as a genuine cusp, signaling an inelastic virtual state in the $^3S_1-^3D_1$ partial wave of the $Sigma N$ isospin 1/2 channel. % There is also evidence for a second peak (or shoulder) in the data sets considered which appears at roughly 10-15 MeV above the $Sigma N$ threshold. However, its concrete position varies significantly from data set to data set and, thus, a theoretical interpretation is difficult.
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