= Solution
For $\mu^2<0$, minimize the <Higgs potential> to obtain $v^2=-\mu^2/\lambda$. In <unitary gauge>, the physical scalar fluctuation is real and
$$
\mathcal L_\Phi=\frac12(\partial\sigma)^2
+\frac{g^2}{8}(v+\sigma)^2W_j^\mu W_{j\mu}
-\frac12(2\lambda v^2)\sigma^2-\lambda v\sigma^3-\frac\lambda4\sigma^4+\text{constant}.
$$
Comparing with $m_W^2W_j^\mu W_{j\mu}/2$ and $-m_\sigma^2\sigma^2/2$ gives
$$
\boxed{m_{W_1}=m_{W_2}=m_{W_3}=gv/2,\qquad m_\sigma=\sqrt{2\lambda}\,v=\sqrt{-2\mu^2}.}
$$
All three gauge generators are broken by this complex doublet; this pure SU(2) model has no surviving massless <photon>. Three scalar angular degrees of freedom supply the longitudinal vector <polarizations>, leaving one real <Higgs boson>. The count is $3\times2+4=3\times3+1=10$. The corresponding fundamental-doublet example also appears in <Gauge Theory by David Tong>.<Gauge Theory by David Tong>[Section 2.7, example “SU(2) with Fundamental Matter”, Higgs-phase spectrum]{ref}
The remaining interactions are $g^2v\sigma W_jW_j/4$, $g^2\sigma^2W_jW_j/8$, $-\lambda v\sigma^3$ and $-\lambda\sigma^4/4$. They give vertices with two equal-species vector legs and one or two scalar legs, or three or four scalar legs, as drawn above. The coefficients in the Lagrangian must be differentiated with respect to identical fields when obtaining fully normalized <Feynman rules>; the written coefficients alone are not all the vertex factors.
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