Supergauge invariance and the Gell-Mann-Low eigenvalue

The connections among supergauge, scale and conformal invariance are elucidated on the example of a renormalizable field theory model. For the massless model at the Gell-Mann-Low eigenvalue, a non-perturbative argument leads to the contradictory result that it can only describe a free field theory. A study of the Callan-Symanzik equations for the massive model clarifies the situation from the point of view of perturbation theory. It is argued that the eigenvalue equation has no non-trivial solution and that the effective coupling constant increases without bound with energy. (7 refs).