Study of the Renormalization Group for Small Coupling Constants

A renormalizable quantum field theory is said to be stagnant if it is asymptotically free. We study the renormalization group for small coupling constants. In particular the pseudoscalar-fermion theory with an internal-symmetry group $G$ and in which the coupling matrix furnishes a representation of $G$ is considered. A representation is said to be stagnant if the associated theory is stagnant. We show that Cartan's four families $A$, $B$, $C$, $D$ and the exceptional algebra ${\mathrm{G}}_2$ possess no stagnant representation. On the basis of this result we conjecture that there are no asymptotically free quantum field theories in four dimensions. Some possible asymptotic behaviors of field theories are also described.