  This paper presents a complete algebraic proof of the renormalizability of
the gauge invariant $d=4$ operator $F_{\mu\nu}^2(x)$ to all orders of
perturbation theory in pure Yang-Mills gauge theory, whereby working in the
Landau gauge. This renormalization is far from being trivial as mixing occurs
with other $d=4$ gauge variant operators, which we identify explicitly. We
determine the mixing matrix $Z$ to all orders in perturbation theory by using
only algebraic arguments and consequently we can uncover a renormalization
group invariant by using the anomalous dimension matrix $\Gamma$ derived from
$Z$. We also present a future plan for calculating the mass of the lightest
scalar glueball with the help of the framework we have set up.
