On-shell counterterms and nonlinear invariances

Abstract
For theories with nonlinear invariances it is not sufficient to restrict one's attention to on-shell invariants in order to discuss the infinities of the S matrix. We give examples of models in which S-matrix divergences cannot be absorbed by invariant on-shell counterterms, and describe the general situation. We point out the possible relevance of the nonclosure of the algebra of symmetry transformations.