= Solution
The four tree-level diagrams for <electron-positron annihilation into W-boson pairs> are
$$
\boxed{s\text{-channel }\gamma^*,\quad s\text{-channel }Z^*,\quad
t\text{-channel }\nu_e,\quad s\text{-channel }h^*.}
$$
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2018-ww-production-diagrams.png]
{title=The four tree-level Standard Model diagrams for electron-positron annihilation into W pairs}
The first two diagrams contain the <electromagnetic interaction> and <weak interaction> neutral electron currents and the <gauge boson self-interaction> vertices $\gamma W^+W^-$ and $ZW^+W^-$. The third has two <weak charged current> vertices joined by an <electron neutrino> propagator. In the fourth, the incoming <Electron>-<Positron> pair couples to the <Higgs boson>, which then couples to $W^+W^-$.
The small factor is at the initial Higgs-electron vertex. With the usual electroweak vacuum value $v\simeq246\,\mathrm{GeV}$, <electron Yukawa suppression of Higgs production> gives
$$
y_e=\frac{\sqrt2m_e}{v}\simeq2.9\times10^{-6},\qquad
g_{hee}=\frac{m_e}{v}\simeq2.1\times10^{-6}.
$$
These are tiny compared with ordinary <electroweak interaction> gauge couplings. The <decay amplitude> for the <Higgs boson> diagram carries this small coefficient, and its separate squared contribution carries its square. Thus \b[the Higgs diagram is negligible because the electron Yukawa coupling is proportional to the electron mass]. It vanishes in the massless-electron approximation, even though the <Higgs boson>-<W boson> coupling itself is not small. The remaining three gauge diagrams must be added coherently, including their interference.
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