How super-renormalizable interactions cure their infrared divergences

Perturbative expansions in models which possess super-renormalizable interactions of massless fields are beset by severe infrared divergences. We show that the complete theory is well defined and has no such divergences; rather the exact amplitudes are nonanalytic functions of the coupling constant and cannot be expanded in its powers. Typically, logarithms of the coupling constant occur, as well as analytic pieces. The analytic portions cannot be found in perturbation theory; they are determined by matrix elements of composite operators. But the nonanalytic behavior is completely fixed in terms of the theory's other parameters. The present investigation should be relevant to a study of physical quantum chromodynamics at its finite-temperature phase transitions.