Asymptotically Free Solutions of the Scalar Mean Field Flow Equations

The flow equations of the renormalisation group permit to analyse rigorously the perturbative n-point functions of renormalisable quantum field theories including gauge theories. In this paper, we want to do a step towards a rigorous nonperturbative analysis of the flow equations (FEs). We restrict to massive scalar (one-component) fields and analyse a mean field limit where the Schwinger functions are considered to be momentum independent and thus are replaced by their zero momentum values. We analyse smooth solutions of the system of FEs for the n-point functions for different sets of boundary conditions. We will realise that allowing for nonvanishing irrelevant terms permits to construct asymptotically free nontrivial smooth solutions of the scalar field mean field FEs.