An Étude in non-linear Dyson–Schwinger Equations

✍ Scribed by Dirk Kreimer; Karen Yeats

• Book ID: 103869875
• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 226 KB
• Volume: 160
• Category: Article
• ISSN: 0920-5632
• DOI: 10.1016/j.nuclphysbps.2006.09.036

✦ Synopsis

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 = ln q 2 /μ 2 and start from an expansion in the scale GR(α, L) = 1 + P k γ k (α)L k . We derive recursion relations between the γ 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.