Dimension two condensates in the Gribov-Zwanziger theory in the Coulomb gauge

TL;DR

This paper studies a specific condensate in the Gribov-Zwanziger framework for Yang-Mills theories in Coulomb gauge, revealing different divergence behaviors in 3D and 4D that influence the gluon correlation functions.

Contribution

It provides the first explicit calculation of the dimension two condensate in Coulomb gauge within the Gribov-Zwanziger approach, highlighting divergence issues and their implications for gluon propagator behavior.

Findings

3D condensate has a non-integrable IR divergence.

4D condensate shows a logarithmic UV divergence.

Results suggest different infrared behaviors in 3D and 4D Yang-Mills theories.

Abstract

We investigate the dimension two condensate $< {\bar \phi}^{ab}_i \phi^{ab}_i -{\bar \omega}^{ab}_i \omega^{ab}_i >$ within the Gribov-Zwanziger approach to Euclidean Yang-Mills theories in the Coulomb gauge, in both 3 and 4 dimensions. An explicit calculation shows that, at the first order, the condensate $< {\bar \phi}^{ab}_i \phi^{ab}_i -{\bar \omega}^{ab}_i \omega^{ab}_i >$ is plagued by a non-integrable IR divergence in 3D, while in 4D it exhibits a logarithmic UV divergence, being proportional to the Gribov parameter $\gamma^2$. These results indicate that in 3D the transverse spatial Coulomb gluon two-point correlation function exhibits a scaling behaviour, in agreement with Gribov's expression. In 4D, however, they suggest that, next to the scaling behaviour, a decoupling solution might emerge too.