The local composite operator $A_{\mu }^2$ is added to the Zwanziger action, which implements the restriction to the Gribov region $\Omega$ in Euclidean Yang-Mills theories in the Landau gauge. We prove that Zwanziger’s action with the inclusion of the operator $A_{\mu }^2$ is renormalizable to all orders of perturbation theory, obeying the renormalization group equations. This allows us to study the dimension two gluon condensate $⟨A_{\mu }^2⟩$ by the local composite operator formalism when the restriction to the Gribov region $\Omega$ is taken into account. The resulting effective action is evaluated at one-loop order in the $\overline{\mathrm{MS}}$ scheme. We obtain explicit values for the Gribov parameter and for the mass parameter due to $⟨A_{\mu }^2⟩$, but the expansion parameter turns out to be rather large. Furthermore, an optimization of the perturbative expansion in order to reduce the dependence on the renormalization scheme is performed. The properties of the vacuum energy, with or without the inclusion of the condensate $⟨A_{\mu }^2⟩$, are investigated. In particular, it is shown that in the original Gribov-Zwanziger formulation, i.e. without the inclusion of the operator $A_{\mu }^2$, the resulting vacuum energy is always positive at one-loop order, independently from the choice of the renormalization scheme and scale. In the presence of $⟨A_{\mu }^2⟩$, we are unable to come to a definite conclusion at the order considered. In the $\overline{\mathrm{MS}}$ scheme, we still find a positive vacuum energy, again with a relatively large expansion parameter, but there are renormalization schemes in which the vacuum energy is negative, albeit the dependence on the scheme itself appears to be strong. Concerning the behavior of the gluon and ghost propagators, we recover the well-known consequences of the restriction to the Gribov region, and this in the presence of $⟨A_{\mu }^2⟩$, i.e. an infrared suppression of the gluon propagator and an enhancement of the ghost propagator. Such a behavior is in qualitative agreement with the results obtained from the studies of the Schwinger-Dyson equations and from lattice simulations.

• Received 22 February 2005

DOI:https://doi.org/10.1103/PhysRevD.72.014016