  We prove the renormalizability to all orders of a refined Gribov-Zwanziger
type action in linear covariant gauges in four-dimensional Euclidean space. In
this model, the Gribov copies are taken into account by requiring that the
Faddeev-Popov operator is positive definite with respect to the transverse
component of the gauge field, a procedure which turns out to be analogous to
the restriction to the Gribov region in the Landau gauge. The model studied
here can be regarded as the first approximation of a more general
nonperturbative BRST invariant formulation of the refined Gribov-Zwanziger
action in linear covariant gauges obtained recently in [Phys. Rev. D 92, no. 4,
045039 (2015) and arXiv:1605.02610 [hep-th]]. A key ingredient of the set up
worked out in [Phys. Rev. D 92, no. 4, 045039 (2015) and arXiv:1605.02610
[hep-th]] is the introduction of a gauge invariant field configuration
$\mathbf{A}_{\mu}$ which can be expressed as an infinite non-local series in
the starting gauge field $A_\mu$. In the present case, we consider the
approximation in which only the first term of the series representing
$\mathbf{A}_{\mu}$ is considered, corresponding to a pure transverse gauge
field. The all order renormalizability of the resulting action gives thus a
strong evidence of the renormalizability of the aforementioned more general
nonperturbative BRST invariant formulation of the Gribov horizon in linear
covariant gauges.
