^{3}.

A distinctive feature observed in lattice simulations of confining non-Abelian gauge theories, such as quantum chromodynamics, is the presence of a dynamical mass for the gauge field in the low-energy regime of the theory. In the Gribov-Zwanziger framework in the Landau gauge, such mass is a consequence of the generation of the dimension-two condensates

For more than fifty years, the standard way to perform the quantization of a gauge field theory in the continuum has been the Faddeev-Popov procedure

Each of these gauge choices can, at least in principle, facilitate the understanding of some given physical aspect of the theory. In any case, one expects that the actual physical observables do not depend on the choice of the gauge condition, so that every gauge choice should lead to the same physical results. Although this has been shown to be true in perturbative calculations, the proof of gauge equivalence is not as straightforward at the nonperturbative level. Finally, although it is possible to successfully formulate a gauge-invariant version of Yang-Mills theory on a discretized spacetime in the nonperturbative regime

As discussed by Gribov in his seminal work

A solution to the Gribov problem would be to evaluate the path integral in such a way that only one representative for each gauge orbit is accounted for. This subset of the gauge field configuration space is the

A practical way to implement the restriction of the path integral to the Gribov region has been proposed by Zwanziger in

An interesting feature of the gluon propagator in the GZ framework is that it violates reflection positivity, which is one of the Osterwalder-Schrader axioms of Euclidean field theory

In spite of the success of the RGZ approach in the Landau gauge, its original formulation does not allow for an extension to other gauges such as the linear covariant gauges or the

In this work, we explore the instability of the Gribov-Zwanziger theory that gives rise to the refined GZ theory in a gauge-parameter independent manner. We do this by considering a

According to the gauge principle, a crucial step towards a well-defined theory at the quantum level is the establishment of the BRST invariance of the action. Let us briefly review the BRST-invariant formulation of the Gribov-Zwanziger action, which will allow us to extend previous results in the Landau gauge to a family of other gauges continually connected to it. Following

An important property of the field

This now allows us to generalize the GZ theory in other gauges different from the Landau gauge, i.e., the linear covariant gauge

Since the horizon function displays BRST invariance, it follows immediately that the new Gribov-Zwanziger action

Note that the

Putting all ingredients in a single action, we obtain the BRST-invariant Gribov-Zwanziger action in the

It is important to point out here that the expression above, Eq.

Finally, let us point out that the auxiliary field

In order to study the dynamical generation of the condensates

In order to show that nonzero condensates appear already at leading order, let us consider the quadratic terms in the action

Within the quadratic approximation

It is convenient to start with the auxiliary fields

After the final integration in the gluon field, the gauge-dependent terms present in the longitudinal part of the gluon action

Now that we have calculated the vacuum energy density in the quadratic approximation, we may proceed to calculate the condensates

Note that these condensates are calculated within the BRST invariant formulation of the Gribov-Zwanziger theory, hence the action

Note that the sources

Even though our calculation already shows that the condensates

From the expression

Let us finally note that the Gribov parameter is independent of the gauge parameters and is thus allowed to enter explicitly findings for physical observables. This can be immediately seen from the defining equation of

Due to the BRST invariance, it turns out that the condensates

In this work, we have extended previous discussions on the generation of dimension-two condensates in the Landau gauge Gribov-Zwanziger framework to a two-parameter family of gauges, namely, the

Such gauge independence reinforces the fact that both Gribov parameter and condensates enter the correlation functions of physical operators, i.e., the correlation functions of local gauge invariant quantity as, for example, the glueball spectrum (cf. e.g.,

An interesting point raised by these developments is whether one can probe directly these gauge-invariant dimension-two condensates on the lattice. For that, one would need to write these nonlocal operators in terms of lattice variables, a task which is not straightforward. Nevertheless, as we have emphasized, the current description of the RGZ formalism in

The authors would like to thank the Brazilian agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação Carlos Chagas de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) for financial support. This paper is also part of the project Instituto Nacional de Ciência e Tecnologia–Física Nuclear e Aplicações (INCT-FNA) Process No. 464898/2014-5. B. W. M. is supported by CNPq project Universal (Grant No. 431796/2016-5) and FAPERJ (Grant No. E-26/202.649/2018). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Financial Code 001 (M. N. F.).

For completeness, let us introduce the BRST transformations of the fields present in the GZ action

As is well-known, the Faddeev-Popov action

Finally, note that the addition of the BRST-invariant sources

An analogous argument can also be cast in the other mentioned gauges.

With an important difference that the operators which condense in our case are not elementary fields that enter the Lagrangian, as would be the case for spontaneous magnetization in a spin model.