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We derive a system of coupled flow equations for the proper-vertices of thebackground effective average action and we give an explicit representation ofthese by means of diagrammatic and momentum space techniques. This explicitrepresentation can be used as a new computational technique that enables theprojection of the flow of a large new class of truncations of the backgroundeffective average action. In particular, these can be single- or bi-fieldtruncations of local or non-local character. As an application we studynon-abelian gauge theories. We show how to use this new technique to calculatethe beta function of the gauge coupling (without employing the heat kernelexpansion) under various approximations. In particular, one of theseapproximations leads to a derivation of beta functions similar to thoseproposed as candidates for an "all-orders" beta function. Finally, we discusssome possible phenomenology related to these flows.

RG flow equations for the proper-vertices of the background effective average action