= Solution
Take $\tau^a=\sigma^a/2$, $Y_\phi=1/2$, metric $g_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$, and a <Higgs potential> with $\mu_h^2>0$, $\lambda>0$. The gauge-scalar <electroweak interaction> is
$$
\mathcal L_{\mathrm{bos}}=-\frac14W^a_{\mu\nu}W^{a\mu\nu}-\frac14B_{\mu\nu}B^{\mu\nu}+(D_\mu\phi)^\dagger D^\mu\phi-V(\phi),\qquad V(\phi)=-\mu_h^2\phi^\dagger\phi+\lambda(\phi^\dagger\phi)^2.
$$
Here $B_{\mu\nu}=\partial_\mu B_\nu-\partial_\nu B_\mu$. With the printed plus-sign convention for $D_\mu$, define $[D_\mu,D_\nu]=igW^a_{\mu\nu}\tau^a+ig'Y_\phi B_{\mu\nu}$; thus $W^a_{\mu\nu}=\partial_\mu W^a_\nu-\partial_\nu W^a_\mu-g\epsilon^{abc}W^b_\mu W^c_\nu$. The minus sign in the nonlinear <gauge field strength> follows from this convention.
The minima have $\phi^\dagger\phi=v^2/2$, with $v^2=\mu_h^2/\lambda$. A <gauge transformation> rotates the vacuum to $(0,v)^T/\sqrt2$. In <unitary gauge>, the three angular <Goldstone bosons> are removed, leaving $\phi=(0,v+H)^T/\sqrt2$ and one real <Higgs boson>. The generator $Q=T^3+Y$ annihilates the vacuum, because its lower component has $T^3=-1/2$ and <hypercharge> $1/2$. This identifies the unbroken electromagnetic <gauge group>.
The quadratic terms from the <Higgs field> <kinetic term> are
$$
\frac12(\partial H)^2+\frac{g^2v^2}{8}\left[(W^1)^2+(W^2)^2\right]+\frac{v^2}{8}(gW^3-g'B)^2.
$$
Define the <Weinberg angle> and physical fields by
$$
\boxed{\tan\theta_W=\frac{g'}g},\qquad \boxed{W^\pm_\mu=\frac{W^1_\mu\mp iW^2_\mu}{\sqrt2}}.
$$
The neutral combinations are
$$
\boxed{\begin{aligned}Z_\mu&=\cos\theta_W W^3_\mu-\sin\theta_W B_\mu,\\ A_\mu&=\sin\theta_W W^3_\mu+\cos\theta_W B_\mu.\end{aligned}}
$$
The neutral <gauge-boson mass matrix> is $(v^2/4)\begin{pmatrix}g^2&-gg'\\-gg'&g'^2\end{pmatrix}$. It has <eigenvalues> zero and $(g^2+g'^2)v^2/4$, with the zero <eigenvector> giving $A_\mu$. Expanding the <Higgs potential> about its minimum gives $V=\text{constant}+\lambda v^2H^2+\lambda vH^3+\lambda H^4/4$. Therefore the <tree-level electroweak gauge-boson masses> and scalar mass are
$$
\boxed{m_A=0,\qquad m_{W^+}=m_{W^-}=\frac{gv}{2},\qquad m_Z=\frac v2\sqrt{g^2+g'^2},\qquad m_H=\sqrt{2\lambda}\,v.}
$$
Also $m_W=m_Z\cos\theta_W$ and $e=g\sin\theta_W=g'\cos\theta_W$. The three removed <Goldstone bosons> supply the longitudinal <polarization vector> degrees of freedom of the massive <electroweak gauge bosons>.
Replacing $v^2$ by $(v+H)^2$ in the neutral mass term gives all the <Higgs boson couplings to Z bosons> in <unitary gauge>:
$$
\mathcal L_{HZ}=\frac{m_Z^2}{v}HZ_\mu Z^\mu+\frac{m_Z^2}{2v^2}H^2Z_\mu Z^\mu.
$$
There are exactly the cubic $HZZ$ and quartic $HHZZ$ elementary <tree-level Feynman diagrams>. Differentiating with respect to the identical fields gives <Feynman rules> $2im_Z^2g_{\mu\nu}/v$ and $2im_Z^2g_{\mu\nu}/v^2$, respectively. There is no elementary $ZHH$ vertex for the real neutral radial field.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-305-higgs-vertices.png]
{title=Higgs interaction vertices}
{description=The two elementary <Higgs boson couplings to Z bosons>, with dashed scalar legs and wavy vector legs.}
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