Bödeker's Effective Theory: From Langevin Dynamics to Dyson-Schwinger Equations

The dynamics of weakly coupled, non-abelian gauge fields at high temperature is non-perturbative if the characteristic momentum scale is of order |k| \sim 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 to leading logarithmic order by means of a Langevin equation. This effective theory has been used for lattice calculations so far. 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 argue that a physically reasonable truncation of the Dyson-Schwinger equations requires the introduction of gauge ghosts, which in general is not mandatory in stochastic quantisation. This leads to a BRST symmetric formulation and to corresponding Ward-Takahashi identities. A second BRST symmetry reflecting the origin in a stochastic differential equation has to be sacrificed to establish the gauge BRST symmetry. The (stochastic) Ward identities can still be obtained by referring to the underlying structure and are shown to produce a cancellation among several terms of the gauge Ward identity. To clarify some issues, we derive the Feynman rules and perform some perturbative calculations. Finally, we deduce the Dyson-Schwinger equations and suggest a truncation scheme that approximately respects the gauge and stochastic Ward identities

Resummed effective action in the world-line formalism

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.Comment: 15 page