An Etude in non-linear Dyson-Schwinger Equations

We show how to use the Hopf algebra structure of quantum field theory to derive nonperturbative results for the short-distance singular sector of a renormalizable quantum field theory in a simple but generic example. We discuss renormalized Green functions GR(α,L) in such circumstances which depend on a single scale L=lnq2/μ2 and start from an expansion in the scale GR(α,L)=1+-bar kγk(α)Lk. We derive recursion relations between the γk which make full use of the renormalization group. We then show how to determine the Green function by the use of a Mellin transform on suitable integral kernels. We exhibit our approach in an example for which we find a functional equation relating weak and strong coupling expansions