$\Xi^-$ emission probabilities at $(K^-,K^+)$ reaction points and the $\Xi N$ interaction

$\Xi^-$ escaping and scattering probabilities in the $(K^-,K^+)$ reaction with nuclear targets is investigated on the basis of the eikonal approximation. Calculations are performed using the $S=-2$ sector of the Nijmegen model-D interaction, whose hard core radii are treated phenomenologically. %The $\Xi^- N$, $\Xi^- p$ and $\Xi^- n$ elastic cross sections are The $\Xi^- N$ elastic cross sections are derived from the $^9$Be$(K^-,K^+)$ data of the BNL-E906 experiment, which is useful to determine the hard-core radii in $P$-states. The $\Xi$ single-particle potential energy is investigated with the G-matrix calculation in nuclear matter, which is sensitive to the hard-core radii in $S$-states. The important constraint for the $\Xi N$ interaction is obtained by combining the two analyses.