Within the framework of the world-line formalism we write down in detail a two-loop Euler-Heisenberg type action for gluon loops in Yang-Mills theory and discuss its divergence structure.We exactly perform all the world-line moduli integrals at two loops by inserting a mass parameter, and then extract divergent coefficients to be renormalized.

Using the world-line method we resum the scalar one-loop effective action. This is based on an exact expression for the one-loop action obtained for a background potential and a Taylor expansion of the potential up to quadratic order in x-space. We thus reproduce results of Masso and Rota very economically. An alternative resummation scheme is suggested using "center of mass" based loops which is equivalent under the assumption of vanishing third and higher derivatives in the Taylor expansion but leads to simplified expressions. In an appendix some general issues concerning the relation between world-line integrals with fixed end points versus integrals with fixed center are clarified. We finally note that this method is also very valuable for gauge field effective actions where it is based on the Euler-Heisenberg type resummation. *

The dynamics of weakly coupled, non-abelian gauge fields at high temperature is non-perturbative if the characteristic momentum scale is of order |k| ∼ g 2 T . Such a situation is typical for the processes of electroweak baryon number violation in the early Universe. Bödeker has derived an effective theory that describes the dynamics of the soft field modes by means of a Langevin equation. This effective theory has been used for lattice calculations so far [18,19]. In this work we provide a complementary, more analytic approach based on Dyson-Schwinger equations.Using methods known from stochastic quantisation, we recast Bödeker's Langevin equation in the form of a field theoretic path integral. We introduce gauge ghosts in order to help control possible gauge artefacts that might appear after truncation, and which leads to a BRST symmetric formulation and to corresponding Ward identities.A second set of Ward identities, reflecting the origin of the theory in a stochastic differential equation, is also obtained. Finally Dyson-Schwinger equations are derived.