The Gribov-Zwanziger action in the presence of the gauge invariant, nonlocal mass operator $Tr\int d^{4}x{F}_{\mu \nu }(D^2{)}^{-1}{F}_{\mu \nu }$ in the Landau gauge

We prove that the nonlocal gauge invariant mass dimension two operator $F_{\mu \nu }(D^2{)}^{-1}{F}_{\mu \nu }$ can be consistently added to the Gribov-Zwanziger action, which implements the restriction of the path integral’s domain of integration to the first Gribov region when the Landau gauge is considered. We identify a local polynomial action and prove the renormalizability to all orders of perturbation theory by employing the algebraic renormalization formalism. Furthermore, we also pay attention to the breaking of the BRST invariance, and to the consequences that this has for the Slavnov-Taylor identity.