= Solution
Write $\phi=(v+h/\sqrt2)e^{i\vartheta}$. The covariant kinetic term contains
$$
(v+h/\sqrt2)^2(\partial_\mu\vartheta+eA_\mu)
(\partial^\mu\vartheta+eA^\mu).
$$
A local <U(1) gauge symmetry> can remove $\vartheta$ in <unitary gauge>. The angular fluctuation no longer appears as an independent physical massless <Goldstone boson>; its degree of freedom supplies the longitudinal polarization of the massive vector field. At quadratic order,
$$
\mathcal L^{(2)}\supset-\frac14F_{\mu\nu}F^{\mu\nu}+e^2v^2A_\mu A^\mu
=-\frac14F^2+\frac12m_A^2A^2,\qquad
\boxed{m_A^2=2e^2v^2}.
$$
The factor of two follows from the convention that $v$ is the complex field's vacuum value. This <Higgs mechanism> changes a two-polarization massless electromagnetic field into a three-polarization massive <Proca field>; the remaining radial scalar keeps $m_h^2=2\lambda v^2$. Static fields in the homogeneous broken <phase> are screened on a scale of order $m_A^{-1}$ in these units. The gauge <symmetry> is a redundancy of the description; the physical statement is the changed spectrum and disappearance of an independent massless angular mode.
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