Solutions of the callan-symanzik equation in a complex neighborhood of zero coupling

In this paper we investigate some of the consequences of having the vertex functions in a field theory or a model, that satisfy the Callan-Symanzik equation, also satisfy some analyticity and uniformity properties in the coupling constant $g$. The solutions of the Callan-Symanzik equation in a complex neighborhood of the origin are studied. The implications of analyticity in $g$ to scaling, anomalous dimensions, and Borel summability of large Euclidean momenta are pointed out. The input we start with, though not yet established in four-dimensional field theories, has been proved for two-dimensional ${\phi }^4$ theories. It also happens to be true in many "models" discussed in the literature in connection with Bjorken scaling.