Chirality for massive neutrinos - right neutrinos - Majorana

[chym]

My question concerns the chirality vs helicity for massive neutrinos.
I know that as the mass is really light we can usually approximate helicity = chirality.
But I would like to consider the exact case with the mass :
i.e left handed (chirality )neutrino propagates with both left and right helicity (if boost).
So it means that right (helicity) neutrinos exist in a "beam" of left chiral neutrinos...
If neutrinos are Majorana, these neutrinos with right helicity could interact exactly as right anti-neutrino are doing... So why this has not been already observed ? I know double beta decay is searched for that, but in more simple processes... like inverse beta decay ?
thank you

 

[Orodruin]

Neutrino masses are really really tiny. The processes you mention, where you can observe the Majorana nature, are suppressed by the chirality flip so roughly $(m_\nu/E)^2$ where $E$ is a relevant energy scale, usually of order weak scale. This makes such processes exceedingly rare.