Study of the all orders multiplicative renormalizability of a local matter confining Gribov-Zwanziger action in the MAG.

We address the issue of the all order multiplicative renormalizability of $SU(2)$ Gribov-Zwanziger theories quantized in the maximal Abelian gauge in presence of confined matter fields. The non-linear character of the maximal Abelian gauge requires the introduction of quartic interaction terms in the Faddeev-Popov ghosts as well as in the localizing Zwanziger fields, extended a well known feature of this gauge. We show that, when scalar matter fields are introduced, a second quartic interaction term between scalar fields, Faddeev-Popov ghosts and Zwanziger-like fields naturally arises. A BRST invariant action accounting for those quartic interaction terms is identified and proven to be multiplicative renormalizable to all orders by means of the algebraic renormalization procedure.