An Étude in non-linear Dyson--Schwinger Equations

Details An Étude in non-linear Dyson--Schwinger Equations An Étude in non-linear Dyson--Schwinger Equations

2006-05-09

arxiv.org/abs/hep-th/0605096v2

Physics

High Energy Physics

High Energy Physics - Theory

6 pages. Two typos corrected

Scientific paper

10.1016/j.nuclphysbps.2006.09.03

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.

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