= Solution
{c}
<Unitary gauge> fixes the scalar phase to zero. Locally choose $\alpha=-\pi/f$ in part <3/b>; then $\pi'=0$ and $A'_\mu=A_\mu+\partial_\mu\pi/(ef)$. The remaining fields are the radial <Higgs mode> and a massive vector. Its quadratic <Lagrangian density> is $-F^2/4+m_A^2A^2/2$ and its <gauge-boson propagator> is
$$
\boxed{D^U_{\mu\nu}(k)=\frac{-i}{k^2-m_A^2+i0}
\left(g_{\mu\nu}-\frac{k_\mu k_\nu}{m_A^2}\right).}
$$
For $k^2\ne0$, introduce the transverse and longitudinal <orthogonal projectors> $P^T_{\mu\nu}=g_{\mu\nu}-k_\mu k_\nu/k^2$ and $P^L_{\mu\nu}=k_\mu k_\nu/k^2$. Omitting the infinitesimal in this algebraic decomposition,
$$
D^U=-\frac{iP^T}{k^2-m_A^2}+\frac{iP^L}{m_A^2}.
$$
At large <momentum> in <Euclidean space>, the transverse coefficient falls as $k^{-2}$ but the longitudinal coefficient approaches a constant of order $m_A^{-2}$. Thus <power counting> applied directly to <unitary gauge> <Feynman diagrams> suggests increasingly severe <ultraviolet divergences>. This hides the <renormalizability> of physical <scattering amplitudes>; it does not disprove it. A finite-$\xi$ <R-xi gauge> exposes the better <propagator> scaling while preserving the fields needed to track the cancellations.
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