The asymptotic behaviour of cubic field theories is investigated in th
e Regge Limit using the techniques of environmentally friendly renorma
lization, environmentally friendly in the present context meaning asym
metric in its momentum dependence. In particular we consider the cross
over between large and small energies at fixed momentum transfer for a
model scalar theory of the type phi(2) psi. The asymptotic forms of t
he crossover scaling functions are exhibited for all two particle scat
tering processes in this channel to one loop in a renormalization grou
p improved perturbation theory. (C) 1997 Published by Elsevier Science
B.V.