Optimized $\delta$-Expansion and Triviality or Non-Triviality of Field Theories

Abstract

We use a very simple version of the optimized (linear) $\delta $ - expansion by scaling the free part of the Lagrangian with a variational parameter. This method is well suited to calculate the renormalized coupling constant in terms of the free one and the cutoff. One never has to calculate any new Feynman graphs but simply can modify existing results from the literature. We find that $\Phi _4^4$ -theory as well as QED are free in the limit where the cutoff goes to infinity. In contrast to this, the structure of Yang-Mills theories enforces a special choice of the Lagrangian of the $\delta $ - expansion. Together with the change in the sign of the $\beta $ - function, this leads to a different behavior and allows Yang-Mills theory to become non trivial.