Borel summability and the renormalization group

The assumptions and results of recent work by Lipatov and others on the behavior of perturbation theory at high orders are delineated and used in conjunction with the Callan-Symanzik equation. Some consequences related to Borel summability and scaling are obtained for $g{\phi }^4$ field theory in four dimensions. Our main result is to show that the above work implies that for this theory a contradiction exists between Borel summability in which the Borel sum fully determines the Euclidean Green's functions in a fixed interval, $0<g<\epsilon$ and the existence of a nontrivial ultraviolet-stable fixed point for $0<g<\infty$.