  This paper presents a complete algebraic analysis of the renormalizability of
the $d=4$ operator $F^2_{\mu\nu}$ in the Gribov-Zwanziger (GZ) formalism as
well as in the Refined Gribov-Zwanziger (RGZ) version. The GZ formalism offers
a way to deal with gauge copies in the Landau gauge. We explicitly show that
$F^2_{\mu\nu}$ mixes with other $d=4$ gauge variant operators, and we determine
the mixing matrix $Z$ to all orders, thereby only using algebraic arguments.
The mixing matrix allows us to uncover a renormalization group invariant
including the operator $F^2_{\mu\nu}$. With this renormalization group
invariant, we have paved the way for the study of the lightest scalar glueball
in the GZ formalism. We discuss how the soft breaking of the BRST symmetry of
the GZ action can influence the glueball correlation function. We expect
non-trivial mass scales, inherent to the GZ approach, to enter the pole
structure of this correlation function.
