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Yang-Mills theory in the maximal Abelian gauge in presence of scalar matter fields

We address the issue of the all order multiplicative renormalizability of SU(2) Yang-Mills theories quantized in the maximal Abelian gauge in presence of scalar matter fields. The non-linear character of the maximal Abelian gauge requires the introduction of quartic interaction terms in the Faddeev-Popov ghosts, a well known feature of this gauge. We show that, when scalar matter fields are introduced, a second quartic interaction term between scalar fields and Faddeev-Popov ghosts 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.