On the threshold branch-point of many-body channels

Abstract

The analytic behaviour of the elastic and break-up scattering amplitudes in a soluble one-dimensional model has been examined by Nussenzveig. It was found that the break-up threshold gave rise to rather curious cube-root branch-points in the scattering amplitude. In this paper, we shall examine the scattering amplitude in a soluble model which reduces Nussenzveig’s model to the special case where the incident and ionized particles have equal masses. It will be shown that the threshold branch-point is a function of the ratio of the masses of the two particles and that the cube-root occurs only when the masses are equal. In general, there are an infinite number of Riemann sheets associated with the threshold branch-point. An examination into the physical origin of such a threshold behaviour will also be made to determine if a more complicated branch-point than a simple square-root may exist for a more realistic potential model.