We report on the nonlocal gauge invariant operator of dimension two, FµN (D²)-1 FµN. We are able to localize this operator by introducing a suitable set of (anti)commuting antisymmetric tensor fields. Starting from this, we succeed in constructing a local gauge invariant action containing a mass parameter, and we prove the renormalizability to all orders of perturbation theory of this action in the linear covariant gauges using the algebraic renormalization technique. We point out the existence of a nilpotent BRST symmetry. Despite the additional (anti)commuting tensor fields and coupling constants, we prove that our model in the limit of vanishing mass is equivalent with ordinary massless Yang-Mills theories by making use of an extra symmetry in the massless case. We also present explicit renormalization group functions at two loop order in the ${\overline{\mbox{MS}}}$ scheme.