Renormalization group flow equations for the proper vertices of the background effective average action

We derive a system of coupled flow equations for the proper vertices of the background effective average action and we give an explicit representation of these by means of diagrammatic and momentum space techniques. This explicit representation can be used as a new computational technique that enables the projection of the flow of a large new class of truncations of the background effective average action. In particular, these can be single- or bifield truncations of local or nonlocal character. As an application we study non-Abelian gauge theories. We show how to use this new technique to calculate the beta function of the gauge coupling ( without employing the heat kernel expansion) under various approximations. In particular, one of these approximations leads to a derivation of beta functions similar to those proposed as candidates for an "all- orders" beta function. Finally, we discuss some possible phenomenology related to these flows.