Approximate Differential Equations for Renormalization Group Functions in Models Free of Vertex Divergencies

I introduce an approximation scheme that allows to deduce differential equations for the renormalization group $\beta$-function from a Schwinger–Dyson equation for the propagator. This approximation is proven to give the dominant asymptotic behavior of the perturbative solution. In the supersymmetric Wess–Zumino model and a ${\varphi }_6^3$ scalar model which do not have divergent vertex functions, this simple Schwinger–Dyson equation for the propagator captures the main quantum corrections.