= Solution
Using <5/c> and omitting $\partial_\mu\Phi$, the <covariant derivative> is
$$
D_\mu\Phi=\begin{pmatrix}-igfW^+_\mu/\sqrt2\\ iGfZ_\mu/2\end{pmatrix},
\qquad G=\sqrt{g^2+g'^2}.
$$
The two components are orthogonal, so their squared magnitudes add; $(W^+_\mu)^\dagger=W^-_\mu$. Thus the <Higgs field> kinetic contribution to the <Lagrangian density> is
$$
\boxed{\mathcal L=(v^2+2vh+h^2)\left(\frac{g^2}{4}W^+_\mu W^{-\mu}+\frac{G^2}{8}Z_\mu Z^\mu\right).}
$$
The two $v^2$ terms are <gauge boson> <mass> terms. Comparing with $m_W^2W^+W^-+\tfrac12m_Z^2Z^2$ gives $\boxed{m_W=gv/2,\ m_Z=Gv/2}$ and $m_W/m_Z=\cos\theta_W$. The factor $1/2$ for the real <Z boson> field is essential.
The two $2vh$ terms are the cubic <Higgs boson coupling to W bosons> and <Higgs boson coupling to Z bosons>. They produce the three-leg $hW^+W^-$ and $hZZ$ <Feynman vertices>, allowing <Higgs decay to two W bosons>, <Higgs decay to two Z bosons> when kinematically available, and the corresponding production mechanisms with virtual <gauge bosons>. The two $h^2$ terms are quartic $hhW^+W^-$ and $hhZZ$ interactions, including vector-boson scattering into a <Higgs boson> pair. Written in terms of the <masses>,
$$
\mathcal L\supset m_W^2W^+W^-+\tfrac12m_Z^2Z^2
+\frac{2m_W^2}v hW^+W^-+\frac{m_Z^2}v hZ^2
+\frac{m_W^2}{v^2}h^2W^+W^-+\frac{m_Z^2}{2v^2}h^2Z^2.
$$
The corresponding <Feynman vertex> factors are $2im_V^2g_{\mu\nu}/v$ for one $h$ and $2im_V^2g_{\mu\nu}/v^2$ for two $h$, for $V=W,Z$, after differentiating with respect to identical fields. These six contributions contain no <photon> <mass> or direct $hAA$ interaction because $Q\Phi=0$.
\Image[/past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2022-higgs-vertices.png]
{title=Mass insertions and cubic and quartic Higgs-gauge vertices}
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