What experimental measurement could be used to show that a neutrino is a Majorana and not a Dirac particle? I've just been reading something on the concept that neutrinos could be Majorana particles and not Dirac fermions.
I was wondering what experimental measurement could show/prove that neutrinos are in fact Majorana particles?
 A: First, let's clear up some terminology: the usual statement "Majorana fermions are their own antiparticles" is correct, but confusing because the words we usually use to describe neutrinos are made for Dirac fermions.
If neutrinos had no mass at all, there would be two independent types of neutrino: a left-handed and a right-handed neutrino. These particles can be distinguished by the reactions they participate in. For example, in $\beta$ decay, only right-handed neutrinos come out, while in $\beta^+$ decay, only left-handed neutrinos come out.
If neutrinos are Dirac, the 'right-handed neutrino' above is interpreted as an anti-neutrino, and that's what we call it in the Standard Model, e.g. we say antineutrinos are produced in $\beta$ decay. 
If neutrinos are Majorana, it means that the 'right-handed neutrino' above is not an antiparticle. It does not mean the left-handed and right-handed neutrinos are the same thing, since they participate in different reactions. However, a Majorana mass term would allow neutrinos to oscillate between the left-handed and right-handed states.

Therefore, we can prove the neutrino is Majorana by observing a left-handed neutrino turning into a right-handed neutrino; or, in the Dirac language, a neutrino "turning into an antineutrino". So in principle, either of the below experiments could work:


*

*Aim a beam of neutrinos at a target, and look for a reaction that could only be caused by an antineutrino.

*Aim two beams of neutrinos at each other, and look for pair annihilation (only possible if some neutrinos turn into antineutrinos).


These experiments are infeasible, because neutrinos don't interact much and it's hard to remove background. A bigger problem is that neutrinos move very close to $c$, and hence experience a lot of time dilation; that means the transition between neutrinos and antineutrinos is very slow.

The actual experimental setup is to try to detect neutrinoless double beta decay, or '$0\nu \beta \beta$'. This is the process where two beta decays happen at once, making two antineutrinos, and then one antineutrino turns into a neutrino and the pair annihilates.
It's easy to tell if $0\nu\beta\beta$ occurred, because at the end of the reaction, there would be no neutrinos to carry away energy. Then you expect to see two electrons come out back-to-back with energy almost exactly equal to $E/2$, where $E$ is the energy released, giving a very sharp bump in the cross section. For technical reasons, $0\nu \beta\beta$ is also kinematically favored over $2\nu\beta\beta$ by a factor of $10^6$, making it easier to detect.
A: I will just try to add what knzhou said about first possible experiment to identify whether neutrinos are majorana particles or Dirac .
So consider a situation where muon neutrino at rest in the middle of a room with spin down.Consider there are two targets one is up target(located above the neutrino) and one is down target(located below the neutrino). Suppose the neutrino accelerated in upward direction and hit the target, can produce a muon particle(because accelerated upward direction so left handed particle,imply neutrino particle).Now consider the other situation where the particle accelerated downward and hit the target ,produces a muon antiparticle this imply neutrino is a majorana particle(because RH neutrino haven't observed in nature,so it should be a RH anti-neutrino,so that will result in a muon anti-particle) and if neutrino is a Dirac particle muon anti-particle will not get produce by this way.
