Abelian embedding formulation of the Stueckelberg model and its power-counting renormalizable extension

We elucidate the geometry of the polynomial formulation of the non-Abelian Stueckelberg mechanism. We show that a natural off-shell nilpotent Becchi-Rouet-Stora-Tyutin (BRST) differential exists allowing to implement the constraint on the $\sigma$ field by means of BRST techniques. This is achieved by extending the ghost sector by an additional U(1) factor (Abelian embedding). An important consequence is that a further BRST-invariant but not gauge-invariant mass term can be written for the non-Abelian gauge fields. As all versions of the Stueckelberg theory, also the Abelian embedding formulation yields a nonpower-counting renormalizable theory in $D=4$. We then derive its natural power-counting renormalizable extension and show that the physical spectrum contains a physical massive scalar particle. Physical unitarity is also established. This model implements the spontaneous symmetry breaking in the Abelian embedding formalism.