= Solution
Up to a vacuum-energy constant and the topological Abelian $F\widetilde F$ term, the most general renormalizable <Abelian Higgs model> is
$$
\mathcal L=-\frac14F_{\mu\nu}F^{\mu\nu}
+|D_\mu\varphi|^2-m^2|\varphi|^2-\lambda|\varphi|^4,
\qquad D_\mu=\partial_\mu-igA_\mu.
$$
It has three local dynamical parameters, $g,m^2,\lambda$. Stability requires $\lambda>0$, and spontaneous symmetry breaking occurs for $m^2<0$. Writing
$$
\varphi(x)=\frac{v+h(x)}{\sqrt2}e^{i\pi(x)/v},
\qquad v^2=-\frac{m^2}{\lambda},
$$
and choosing <unitary gauge> removes $\pi$. The physical masses are
$$
\boxed{m_A^2=g^2v^2,\qquad m_h^2=2\lambda v^2=-2m^2}.
$$
The interactions include $g^2vhA_\mu A^\mu$, $g^2h^2A_\mu A^\mu/2$, $-\lambda vh^3$, 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>.
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