Gauge-invariant variables and Slavnov-Taylor decomposition in spontaneously broken theories

A renormalizable extension of the Abelian Higgs-Kibble (HK) model supplemented by a dimension 6 derivative-dependent operator is presented. The dim.6 operator is controlled by the parameter z and violates powercounting renormalizability. At z = 0 one recovers the usual power-counting renormalizable Abelian HK model. A field-theoretic representation of the physical Higgs scalar by a gauge-invariant variable is used in order to formulate the theory by a novel differential equation controlling the dependence of the quantized theory on z. We show that the Slavnov-Taylor identities hold true separately in the grading induced by the number of internal physical Higgs propagators. This is at variance with the ordinary formalism and is crucial to the consistent definition of the quantum field theory at z \neq 0 via the solution to the differential equation.