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The QCD running charge and its RGI three-gluon vertex parent in the Pinch Technique

We give a brief review of an elementary extension of the Pinch Technique (PT) that yields renormalization-group invariant (RGI) Green's functions, called PT-RGI. These are also gauge- and process-independent, show dimensional transmutation, and are the natural ingredients of skeleton expansions of physical processes. Because of a dynamically-generated gluon mass all PT-RGI Green's functions are IR-finite. Next we show from the ghost-free Ward identities of the PT how the conventional running charge is recovered from the full PT-RGI three-gluon vertex, which depends on three momenta. The usual running charge, depending on only one momentum, is not necessarily a good substitute for this PT-RGI three-gluon vertex. We show that at one dressed loop a good approximation to the full dressed loop PT-RGI three-gluon vertex, both in the UV and in the IR, comes from input propagators and vertices that are free, except that the propagator is modified by introducing a (constant) mass term. Finally,we illustrate these ideas in the much simpler context of a scalar theory with cubic interactions in d=6, which is asymptotically free.