We establish that the Abelian gauge theory first considered by Higgs in which the gauge vector boson acquires a finite mass due to the spontaneous breakdown of symmetry is renormalizable, in the sense that the Bogoliubov-Parasiuk-Hepp program can be executed in such a way that the Ward-Takahashi identities are satisfied. This paper contains the global study of the Ward-Takahashi identities and low-energy theorems of the model. We show that the Goldstone boson and the scalar excitation of zero mass associated with the Landaugauge vector propagator are unphysical, and disappear from the S matrix.
Benjamin W. Lee (1972) studied this question.