One of the suggested diagrams for the Higgs production is the following:
so basically an electron-positron pair annihilates and forms an (excited?) Z boson, which then decays into another (less excited?) Z boson and a Higgs boson.
Why can't the electron-positron pair decay directly into a Higgs boson?
Charge and lepton number would be conserved anyway, and if the pair has enough energy to produce the $Z^*$ boson in the first place it should have enough energy to produce the Higgs boson… ?
Best Answer
The electron-positron pair can produce directly a Higgs boson, but this process is very suppressed, because the coupling between the leptons and the Higgs is proportional to the tiny mass $m_e$:
$$g_{\rm Hee}=-i\frac{ m_e}{v},$$ where $v\approx 246 \,\rm{GeV}.$
On the other hand, the process $e^+ e^-\to H Z$ is more likely to happen, because the coupling between $H$ and $Z$ is proportional to the $W$ mass:
$$ g_{\rm hZZ}^{\mu\nu}=i g \frac{M_W}{\cos^2\theta_W} g^{\mu\nu}. $$ In the latter case we also have to take in account the propagator of the $Z$ boson, which introduces a suppression factor of order $1/m_Z^2$, but at the end we still have a larger cross section.