Makeenko-Migdal equation in a domained QCD vacuum

A solution for simple curves of the Makeenko-Migdal equation is presented which leads to string dynamics. By the nonrenormalized nature of the equation, any formal solution requires some extra physics input before it can be meaningfully interpreted. Then a 'minimal' renormalization is proposed both of the Wilson loop for a large number of colors and of the corresponding Makeenko-Migdal equation. This scheme is based on Witten's idea of a master field. The regularized solution contains a scale (related to the Regge-slope) which is connected to a gauge invariant domain size in the QCD vacuum, and the solution can only make sense for loops that are large relative to this scale. Using a vortex condensate model for the QCD vacuum, the length parameter is related to the QCD length scale.