An \'Etude in non-linear Dyson--Schwinger Equations

TL;DR

This paper employs the Hopf algebra structure of quantum field theory to derive nonperturbative results for Green functions, using recursion relations and Mellin transforms to connect weak and strong coupling regimes.

Contribution

It introduces a method combining Hopf algebra, renormalization group, and Mellin transforms to analyze nonperturbative aspects of quantum field theories.

Findings

Derived recursion relations for Green function expansions.

Established a functional equation linking weak and strong coupling.

Demonstrated the approach with a specific example.

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.