= Solution
\b[Spectrum and symmetry.] Choose a <vacuum expectation value> $\phi_0=(0,v/\sqrt2)^T$. In <unitary gauge> the scalar field is $\phi=(0,(v+h)/\sqrt2)^T$, with real $h$. The <Pauli matrix multiplication law> gives $\{\tau^a,\tau^b\}=\delta^{ab}I/2$. Since $A^aA^b$ is symmetric in gauge indices, only this symmetric product contributes. The kinetic and potential terms therefore become
$$
(D_\mu\phi)^\dagger D^\mu\phi
=\frac12(\partial_\mu h)^2+\frac{g^2}{8}(v+h)^2A_\mu^aA^{a\mu},
$$
$$
V(h)=\frac{\lambda v^2}{2}h^2+\frac{\lambda v}{2}h^3+\frac{\lambda}{8}h^4.
$$
Comparison with $\frac12m_A^2A_\mu^aA^{a\mu}$ and $\frac12m_h^2h^2$ gives
$$
\boxed{m_{A^1}=m_{A^2}=m_{A^3}=\frac{gv}{2},\qquad m_h=\sqrt{\lambda}\,v.}
$$
These are the <scalar interactions after complete SU2 breaking>. The scalar mass uses the specified potential normalization: the prefactor $\lambda/2$ is important.
A nonzero complex doublet has trivial stabilizer in $SU(2)$: a matrix fixing $(0,v)^T$ must have second column $(0,1)^T$, and unitarity and determinant one then fix the first column too. Thus all three gauge generators are broken. Three scalar angular directions are absorbed by the <Higgs mechanism>, giving each <gauge boson> a longitudinal polarization. The physical degrees of freedom count is
$$
\boxed{3\times2+4=3\times3+1=10,\qquad\text{no massless physical fields}.}
$$
There is no surviving electromagnetic $U(1)$ in this theory: it contains only the gauged $SU(2)$, not the electroweak product group. The would-be <Goldstone bosons> are not additional physical massless particles. This is <complete breaking by an SU2 scalar doublet>.
\b[Scalar interactions.] In <unitary gauge>, all interaction terms containing the physical scalar are
$$
\boxed{\mathcal L_{\mathrm{scalar,int}}
=\frac{g^2v}{4}hA_\mu^aA^{a\mu}
+\frac{g^2}{8}h^2A_\mu^aA^{a\mu}
-\frac{\lambda v}{2}h^3-\frac{\lambda}{8}h^4.}
$$
These give $hAA$, $hhAA$, $hhh$, and $hhhh$ <Feynman vertices>. Their factors are respectively $ig^2v\,\delta^{ab}g_{\mu\nu}/2$, $ig^2\delta^{ab}g_{\mu\nu}/2$, $-3i\lambda v$, and $-3i\lambda$, including the identical-leg multiplicities. Gauge-dependent descriptions also contain the absorbed scalar modes; the above are all physical-scalar interactions in the chosen gauge.
\b[Gauge self-interactions.] Write $f_{\mu\nu}^a=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a$. Expanding the field-strength square gives the cubic and quartic pieces
$$
\boxed{\mathcal L_3
=\frac g2\epsilon^{abc}f_{\mu\nu}^aA^{b\mu}A^{c\nu}
=g\epsilon^{abc}(\partial_\mu A_\nu^a)A^{b\mu}A^{c\nu},}
$$
$$
\boxed{\mathcal L_4
=-\frac{g^2}{4}\epsilon^{abc}\epsilon^{ade}
A_\mu^bA_\nu^cA^{d\mu}A^{e\nu}.}
$$
The sign of the cubic term follows from the minus sign in the given nonlinear field strength. The quartic term can also be written using $\epsilon^{abc}\epsilon^{ade}=\delta^{bd}\delta^{ce}-\delta^{be}\delta^{cd}$. The cubic <Feynman vertex> joins three wavy gauge lines, has one momentum factor, and is proportional to $g\epsilon^{abc}$. The quartic <Feynman vertex> joins four wavy lines, is proportional to $g^2$, and has products of structure constants with metric contractions. There are no higher pure-gauge vertices in this Lagrangian. These are the <SU2 gauge self-interaction vertices>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-47-gauge-vertices.png]
{title=Cubic and quartic gauge-boson vertices from the SU2 field-strength square}
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