As discussed in the last lecture, the precise properties of neutrinos are hard to measure. This has led to some hypotheses about neutrinos that have yet to be proven true or false. Probably the most famous is by Majorana.
What exactly do we know about neutrinos?
Majorana proposed that the neutrino was its own anti-particle, ν=νbar, and the only distinguishable difference is that there are two substates, νL and νR. If the neutrino is nearly massless, then these two states are nearly independent, yielding results that agree with experiment.
The new effect that is implied and searched for is neutrinoless double-β decay. A small number of nuclei can undergo double-β decay, for instance 76Ge --> 76Se + 2e- + 2νbar. This decay is allowed and observed.
If the neutrino is a Majorana particle, then it is predicted that the neutrino from the first decay can be absorbed (in its anti-particle form) to initiate the second decay. The result is a decay with no neutrinos emitted, neutrinoless double-β decay. The signature is the emission of two electrons with a fixed total energy. This hasn't yet been observed.
In the remaining discussion, we will assume that neutrinos are "standard" Dirac particles.
For many years, the Standard Model assumed that neutrinos are massless, only because all evidence was consistent with this assumption, and the resulting equations were somewhat simplified. But what happens if we drop this assumption? Then neutrinos can have small, but finite masses. Additionally, they may have non-diagonal couplings to W bosons, in the same way that the down-type quarks have non-diagonal couplings. The down-type quark couplings involve the CKM matrix, and in a similar spirit, the neutrino couplings involve a complex, 3×3 matrix known as the MNS matrix.
As with the quarks, the result is that the mass eigenstates, and the weak eigenstates are not the same. Consider an example with two neutrinos, weak eigenstates νe and νμ, and mass eigenstates ν1 and ν2, related by a 2×2 matrix: