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The content of two additional Ward identities exhibited by the $U(1)$ Higgsmodel is exploited. These novel Ward identities can be derived only when a pairof local composite operators providing a gauge invariant setup for the Higgsparticle and the massive vector boson is introduced in the theory from thebeginning. Among the results obtained from the above mentioned Ward identities,we underline a new exact relationship between the stationary condition for thevacuum energy, the vanishing of the tadpoles and the vacuum expectation valueof the gauge invariant scalar operator. We also present a characterization ofthe two-point correlation function of the composite operator corresponding tothe vector boson in terms of the two-point function of the elementary gaugefields. Finally, a discussion on the connection between the cartesian and thepolar parametrization of the complex scalar field is presented in the light ofthe Equivalence Theorem. The latter can in the current case be understood inthe language of a constrained cohomology, which also allows to rewrite theaction in terms of the aforementioned gauge invariant operators. We alsocomment on the diminished role of the global $U(1)$ symmetry and its breaking.

The Abelian Higgs model under a gauge invariant looking glass:  exploiting new Ward identities for gauge invariant operators and the  Equivalence Theorem