The running coupling from the four-gluon vertex in Landau gauge Yang-Mills theory

TL;DR

This paper calculates the running coupling from the four-gluon vertex in Landau gauge Yang-Mills theory using Dyson-Schwinger equations, revealing a small infrared fixed point and supporting the effective theory driven by the Faddeev-Popov determinant.

Contribution

It provides a numerical determination of the four-gluon vertex functions and the resulting running coupling, highlighting differences from ghost-gluon coupling and supporting infrared effective theory concepts.

Findings

Reproduces asymptotic freedom in the ultraviolet

Finds a small nonzero infrared fixed point for the coupling

Explains agreement of Dyson-Schwinger and functional renormalization group results in the infrared

Abstract

We consider the running coupling from the four-gluon vertex in Landau gauge, SU($N_c$) Yang-Mills theory as given by a combination of dressing functions of the vertex and the gluon propagator. We determine these functions numerically from a coupled set of Dyson-Schwinger equations. We reproduce asymptotic freedom in the ultraviolet momentum region and find a coupling of order one at mid-momenta. In the infrared we find a nontrivial (i.e. nonzero) fixed point which is three orders of magnitude smaller than the corresponding fixed point in the coupling of the ghost-gluon vertex. This result explains why the Dyson-Schwinger and the functional renormalization group equations for the two point functions can agree in the infrared, although their structure is quite different. Our findings also support Zwanziger's notion of an infrared effective theory driven by the Faddeev-Popov determinant.