We present a n umber of exact and asymptotic results concerning the perturbative tree-level amplitudes for the decay of an o -shell scalar into a large number of onshell scalars. These results are useful in the discussion of whether or not perturbative unitarity will be violated at su ciently high energies and/or multiplicities.

We discuss theoretical and phenomenological aspects of the use of boson propagators with energy-dependent widths in predictions for high-energy scattering processes. In general, gauge invariance is violated in such calculations. We discuss several approaches to restore gauge invariance, necessary for a reliable result. The most promising method is the addition of the relevant parts of the fermionic corrections, which fulfills all Ward identities. The numerical difference between this and other approaches is studied. A number of recommendations are given for LEP2 computations.

A study of zero-dimensional theories, based on exact results, is presented. First, relying on a simple diagrammatic representation of the theory, equations involving the generating function of all connected Green's functions are constructed. Second, exact solutions of these equations are obtained for several theories. Finally, renormalization is carried out. Based on the anticipated knowledge of the exact solutions the full dependence on the renormalized coupling constant is studied. ‡ andrevh@sci.kun.nl, ⋆ kleiss@sci.kun.nl, + Costas.Papadopoulos@cern.ch.