A unified theory for both massless, massive vector fields and non-triviality of physical Hilbert space of this theory

In this article, I talk about a model which is known and often used to unify both massless and massive vector free-field theory, discuss the importance of auxiliary scalar field introduced in the Lagrangian density that we consider in this theory which turns out to be unphysical in massive vector field case which I also show. But for massless theory in 3+1-dimensional case, I show that this auxiliary field makes the physical Hilbert space non-trivial in terms of introducing an extra physical degree of freedom or an extra non-trivial kind of polarized state which is absent in Maxwell's electrodynamics theory (free-massless vector field theory). And this state is also shown to be present in 1+1-dimensional theory whereas there is no on-shell photon in Maxwell's theory of electromagnetism, which is a massless vector field theory.