Higgs boson production via positron-electron collision 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... ?
 A: The $^*$ notation does not mean excited in this case, it means "off shell" (i.e. virtual or having the "wrong" mass).
At the second vertex the $Z^0$ is put "on-shell" by the emission of a Higgs (note, however, that it will decay very quickly in any case).
The lepton pair can annihilate directly to the Higgs, but the event is experimentally identical to annihilation to photons or $Z^0$s (because the thing that makes a coupling possible is that both side have compatible quantum numbers, so that (at tree level) all three possibilities decay to very similar end states).
The reaction pictured is experimentally identifiable because the on-shell $Z$ decays to an lepton pair with a mass of 90 GeV and the Higgs decays to a limited choice of end states that are mostly reconstructable and add up to the Higgs mass.
A surprising amount of collider physics is not so much about what can happen as about what can be uniquely shown to have happened.
Finally, I would certainly not describe this reaction as a "decay".
A: 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.
