Nullification in scalar theories with derivative couplings

We discuss the structure of scalar field theories having the property that all on-shell S-matrix elements vanish in tree approximation. It is shown that there exists a large class of such theories, with derivative couplings, which are all locally related to a free theory by a nonlinear transformation. It is also shown that a field-dependent wave-function renormalization provides all necessary counterterms so that all on-shell S-matrix elements vanish also at the one-loop level.