Woolly Cusps

Abstract

Recent experiments on ($K^{-}$, $p$) scattering led us to the study of the elastic scattering 1+2 → 1+2 in the energy region where the inelastic process 1+2 → 3+4 sets in, for the case that particle 3 is unstable. We call "woolly cusp" the phenomenon which corresponds to the sharp cusp in the stable case. The procedure followed is to consider the inelastic channel to be of the three-body type where the three-body states are parametrized by a Breit-Wigner formula around a mean mass $m^{*}$ of particle 3. The connection between a woolly and a sharp cusp is made evident. The problem is studied in terms of a two-channel $S$-wave $K$ matrix. In the two-channel approximation the woolly cusp necessarily shows a decrease in the elastic cross section $\sigma$ above a characteristic energy. As a function of energy, $\sigma$ will either show a maximum, or an inflection point. In either case, the energy at which this happens may lie above or below the inelastic threshold for the fictitious case that particle 3 has a sharp mass $m^{*}$. The sign and magnitude of the elastic scattering phase shift at this "$m^{*}$ point" approximately determines which case is actually realized.