= Solution
A real antisymmetric generator of the connected rotation group is
$$
T=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad R(\alpha)=e^{\alpha T}.
$$
Choose $D_\mu\boldsymbol\phi=(\partial_\mu-gA_\mu T)\boldsymbol\phi$. Under $\boldsymbol\phi\mapsto R(\alpha(x))\boldsymbol\phi$ and $A_\mu\mapsto A_\mu+\partial_\mu\alpha/g$, the <gauge covariant derivative> transforms in the same way as the field. Include the canonical <gauge kinetic term> $-F_{\mu\nu}F^{\mu\nu}/4$ so the vector's mass <normalization> is specified.
In <unitary gauge>, $\boldsymbol\phi=(v+h,0)^T$. The relevant <Lagrangian density> is
$$
\mathcal L=-\tfrac14F^2+\tfrac12(\partial h)^2
+\tfrac12g^2(v+h)^2A_\mu A^\mu-\lambda(2vh+h^2)^2.
$$
Reading the quadratic vector term as $m_A^2A^2/2$ gives
$$
\boxed{m_A=|gv|.}
$$
The angular scalar supplies the longitudinal polarization of the massive vector; it is not a separate physical massless particle after gauging. The physical scalar still has $m_h^2=8\lambda v^2$.
The interactions among the physical fields are
$$
\boxed{\mathcal L_{\rm int}=-4\lambda vh^3-\lambda h^4
+g^2vhA_\mu A^\mu+\tfrac12g^2h^2A_\mu A^\mu.}
$$
Thus the complete physical vertex list is $hhh$, $hhhh$, $AAh$, and $AAhh$. An Abelian <gauge kinetic term> generates no $AAA$ or $AAAA$ self-interaction. Gauge-dependent Goldstone vertices in another gauge do not add physical particle species to this list. These statements concern the broken phase $v\ne0$; when $v=0$ the unitary-gauge broken-phase parametrization is unavailable.
\Image[/past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2015-abelian-higgs-physical-vertices.png]
{title=All physical interaction vertices in the broken two-real-scalar gauge model}
{description=Original vertex inventory in unitary gauge: three and four radial scalars, two vectors with one radial scalar, and two vectors with two radial scalars. Dashed lines denote h and wavy lines A.}
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