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

TL;DR

This paper discusses an extension of the Pinch Technique to produce renormalization-group invariant Green's functions, demonstrating their properties, relation to the running charge, and illustrating the concepts with a scalar theory example.

Contribution

It introduces PT-RGI Green's functions, showing their gauge independence, IR finiteness, and connection to the conventional running charge, with a practical approximation method.

Findings

PT-RGI Green's functions are gauge- and process-independent.

The conventional running charge is derived from the PT-RGI three-gluon vertex.

A good approximation to the full PT-RGI vertex uses modified propagators with a constant mass.

Abstract

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.