= Solution
{c}
After the quadratic mixing cancellation, the vector inverse quadratic operator is
$$
K=-(k^2-m_A^2)P^T-\left(\frac{k^2}{\xi}-m_A^2\right)P^L.
$$
Inverting its two <orthogonal projectors> and multiplying by $i$ gives
$$
\boxed{D^{(\xi)}_{\mu\nu}(k)=\frac{-i}{k^2-m_A^2+i0}
\left[g_{\mu\nu}-(1-\xi)\frac{k_\mu k_\nu}{k^2-\xi m_A^2+i0}\right].}
$$
Equivalently $D^{(\xi)}=-iP^T/(k^2-m_A^2+i0)-i\xi P^L/(k^2-\xi m_A^2+i0)$. At fixed finite $\xi$, both sectors fall as $k^{-2}$ at large <momentum> in <Euclidean space>; $\xi=1$ gives $-ig_{\mu\nu}/(k^2-m_A^2+i0)$, and taking $\xi\to\infty$ at fixed $k$ recovers <unitary gauge>. The latter limit is not uniform in <ultraviolet> <momentum>.
The expected <power counting> conclusion is that this $k^{-2}$ behaviour, together with polynomial interactions of <mass dimension> at most four in Cartesian scalar variables and the <gauge symmetry> identities, permits <renormalization of spontaneously broken gauge theories> using a finite set of physical parameters. \b[The propagator alone is not sufficient to establish that claim in the polar variables of this question.] In fact, part <3/b> contains $h(\partial\pi)^2/f$ and $h^2(\partial\pi)^2/(2f^2)$, of dimensions five and six, and the nonlinear <gauge fixing> term contains $-\xi e^2h^4\pi^2/(2f^2)$, also of dimension six. Ordinary <power counting> is consequently not manifest in these variables.
For a representation where it is manifest, write $\phi=(f+H+i\chi)/\sqrt2$ and choose the usual polynomial <R-xi gauge> function $\partial\cdot A-\xi ef\chi$. It yields the same free vector <propagator>, cancels the quadratic mixing, and leaves only dimension-at-most-four interactions, including its <Faddeev-Popov ghosts>. This supplies the missing vertex condition in the source's proposed inference; the <equivalence theorem for field redefinitions> relates physical predictions near the <vacuum>. Good <propagator> falloff is the necessary ingredient illustrated by the requested formula, rather than a complete proof for arbitrary parametrizations or nonlinear choices of <gauge fixing>.
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