 Solution

Up to a vacuum-energy constant and the topological Abelian $F\tilde{F}$ term, the most general renormalizable Abelian Higgs model isIt has three local dynamical parameters, $g,m^2,\lambda$. Stability requires $\lambda >0$, and spontaneous symmetry breaking occurs for $m^2<0$. Writingand choosing unitary gauge removes $\pi$. The physical masses areThe interactions include $g^{2}vh{A}_{\mu }{A}^{\mu }$, $g^2{h}^2{A}_{\mu }{A}^{\mu }/2$, $-\lambda v{h}^3$, and $-\lambda h^4/4$. The original massless vector's two polarizations and the complex scalar's two real components become a massive vector with three polarizations and one massive scalar. This rearrangement, in which the gauge field absorbs the would-be Goldstone mode and acquires mass without explicitly breaking gauge invariance, is the Higgs mechanism.