Asymptotics of the S-matrix and unitarisation

The manner in which the elastic scattering amplitude obeys unitarity, how it enters
the circle of unitarity, and what its asymptotic limit is, remains a problem for models
which include terms that rise fast with s. We have checked that the features of cross
sections which come from unitarisation are present for most unitarisation schemes, e.g.
those that saturate the profile function or those that describe multiple exchanges via an
analytic formula. We have also obtained a scheme which interpolates between different
classes of the unitarisation and found corresponding non-linear equations. Considering
different forms of energy dependence of the scattering amplitude, and a variety of
unitarisation schemes, we show that, in order to reproduce the data, the fits choose an
amplitude that corresponds to an asymptotic value S = 0.