= Solution
{c}
Choose the <gauge covariant derivative> $D_\mu=\partial_\mu+ieA_\mu$. Write $\rho=f+h$, $r=1+h/f$ and $B_\mu=\partial_\mu\pi/f+eA_\mu$. The <polar parametrization of the Abelian Higgs field> gives
$$
D_\mu\phi=\frac{e^{i\pi/f}}{\sqrt2}\left[\partial_\mu h+i(f+h)B_\mu\right],
\qquad
\phi^\dagger\phi-\frac{f^2}{2}=fh+\frac{h^2}{2}.
$$
Hence the full <Lagrangian density> is
$$
\boxed{\mathcal L=-\frac14F_{\mu\nu}F^{\mu\nu}
+\frac12(\partial h)^2+\frac12\left(1+\frac hf\right)^2(\partial_\mu\pi+efA_\mu)^2
-\frac\lambda2f^2h^2-\frac\lambda2fh^3-\frac\lambda8h^4.}
$$
Assume $f>0$ and $\lambda>0$, so the minimum has $|\phi|=f/\sqrt2$. In this convention the local <gauge symmetry> acts as
$$
\boxed{\phi\mapsto e^{i\alpha}\phi,\qquad
A_\mu\mapsto A_\mu-\frac1e\partial_\mu\alpha,\qquad
h\mapsto h,\qquad\pi\mapsto\pi+f\alpha.}
$$
The combination $\partial_\mu\pi+efA_\mu$, the radial field $h$, and the <gauge field strength> are each invariant, proving invariance of the displayed <Lagrangian density>.
This parametrization follows the circle of classical minima: $h$ measures distance away from the <vacuum manifold>, whereas $\pi$ moves along a <gauge orbit>. It separates the massive radial <Higgs mode> from the would-be <Goldstone boson> and makes its removal by the <Higgs mechanism> transparent. At quadratic order,
$$
\mathcal L_2=-\frac14F^2+\frac12(\partial h)^2-\frac12\lambda f^2h^2
+\frac12(\partial\pi)^2+efA\cdot\partial\pi+\frac12e^2f^2A^2,
\qquad m_h^2=\lambda f^2,\quad m_A^2=e^2f^2.
$$
The <polar parametrization of the Abelian Higgs field> is a local description near the chosen nonzero <vacuum expectation value>; it is singular at $\phi=0$ and is not a global coordinate system for configurations with zeros or winding.
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