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

We investigate the dimension two condensate $< {\bar ϕ}^{ab}_i ϕ^{ab}_i -{\bar ω}^{ab}_i ω^{ab}_i >$ 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 $< {\bar ϕ}^{ab}_i ϕ^{ab}_i -{\bar ω}^{ab}_i ω^{ab}_i >$ 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 $γ^2$. 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.