Multi-instantons at high energies: Premature unitarization and supercritical behavior

We investigate the role of multi-instantons at high energies in weakly coupled theories. We perform an explicit computation of their effect and we find that they become as important as the one-instanton contribution at energies where the latter is still exponentially small ("premature unitarization"). The computation relies on various simplifying assumptions, and we discuss them. We argue, however, that premature unitarization is a phenomenon of a very general nature, which should survive our simplifying assumptions. It this is the case, there exists a critical energy where a change of regime is forced, and the dilute-instanton-gas approximation totally collapses. Possible scenarios in the critical and supercritical domain are discussed at the qualitative level.