On The Origin of Neutrino Mass and Mixing in the Standard Model
  One can describe cosmological relic neutrinos by adding Lagrange multipliers
to the Standard Model Lagrangian for them. The two possible Lagrange
multipliers are a chemical potential, which fixes the mean
neutrino/anti-neutrino asymmetry, and a Majorana mass, which fixes the mean
spin-entropy. Because these neutrinos originated from a thermal bath, their
entropy should be maximal, implying that each state in the background is a
symmetric superposition of a neutrino and anti-neutrino. Therefore the Standard
Model must be augmented by a flavor-diagonal Majorana neutrino mass matrix.
This impacts the propagator via tadpole diagrams due to self-interactions. In
the low-energy limit, neutrino self-interactions are entirely off-diagonal
because same-flavor four-fermion operators vanish by Pauli exclusion. These
interactions must be diagonalized when propagating through a bath of neutrinos,
using the U(3) global flavor symmetry. U(3) gets broken broken down to SO(3) by
Majorana masses, and down to $A_4$ if the three masses are different. Thus our
universe today contains tri-bimaximal mixing and Majorana neutrinos. Neutrino
mixing is due to the mismatch between the flavor-diagonal Majorana mass matrix
arising at finite density and the self-interaction diagonal finite density
propagator. The mass hierarchy is inverted and Majorana phases are absent.
Lepton number is conserved and the neutrino-less double beta decay experiment
absorbs a pair of neutrinos from the relic background and will prove their
Majorana nature.
