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

TL;DR

This paper explores new Ward identities in the Abelian Higgs model using gauge invariant operators, establishing relationships between vacuum conditions, correlators, and the Equivalence Theorem, with implications for gauge symmetry and parametrization.

Contribution

It introduces novel Ward identities involving gauge invariant operators, linking vacuum conditions, correlation functions, and the Equivalence Theorem in the Abelian Higgs model.

Findings

Derived new Ward identities for gauge invariant operators.

Established a relationship between vacuum energy conditions and operator expectation values.

Characterized vector boson correlators in terms of elementary gauge fields.

Abstract

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