An Étude in non-linear Dyson--Schwinger Equations

Abstract

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 $G_R(\alpha ,L)$ in such circumstances which depend on a single scale $L=\ln q^2/{\mu }^2$ and start from an expansion in the scale $G_R(\alpha ,L)=1+{\sum }_k{\gamma }_k(\alpha )L^k$. We derive recursion relations between the ${\gamma }_k$ 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.

6 pages. Two typos corrected

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