Majorara mass and fermion number violation

Question

How can it be shown that the Majorana mass violates the fermion number by two units? Can even a Noether charge be defined in presence of Majorana mass term?

Answer 1

In the Standard Model, fermion number is not conserved. Lepton number is conserved, because of an accidental symmetry. One cannot write down a renormalizable, gauge and Lorentz invariant operator that violates lepton number conservation in the Standard Model.

A Majorana neutrino would violate lepton number conservation by two units. To see this, consider, e.g. neutrinoless double beta decay. You can draw Feynman diagrams with two $W^-e^-v_e$ vertices in which two incoming $W$-bosons each decay into an electron and an electron-neutrino. The two electron-neutrinos annihilate (possible because they are Majorana particles), leaving a final state of two elecrons, violating lepton number conservation by $2$ units.

You can see that Majorana neutrinos violate lepton number conservation by $2$ units from the mass term. The mass term, $$ \mathcal{L} = \frac12 m \psi^T C^{-1}\psi, $$ is not invariant under the $U(1)$ lepton number symmetry, $\psi\to\exp(iL\theta)\psi$. It picks up a phase of twice the lepton number of the neutrino, i.e. $\Delta L=2$ rather than $\Delta L=0$. A Majorana neutrino cannot be charged under a $U(1)$ symmetry.

Because there is not a lepton number $U(1)$ symmetry, there is no conserved Noether charge corresponding to lepton number.

Answer 2

You can think of a Majorana mass as arising from coupling of the neutrino to a condensate of a (lepton number) charge 2 higgs field. Then you can write down a conserved Noether current provided you include the charge current of the Higgs.

Answer 3

Fermion number is conserved in the standard model, Fermion propagation paths never end in a Feynman diagram. The examples given are beyond the standard model. However, in various Grand Unification models Fermion number violation is predicted. Neutrinoless Double beta decay in based on SO(10) which is a GUT model

Answer 4

It should be pointed out that there is zero experimental/ observation evidence for Fermion number conservation violation. Fermion number violation is predicted in Grand Unification theories and SUSY but , at least based on current evidence, there is no reason to think nature is fundamentally Super symmetric or that grand unification occurs in nature. The SUSY models that have useful features like explaining the Higgs mass have been ruled out by the LHC. Also despite the best efforts proton decay and neutrino-less double beta decay, both predictions of Grand Unification , have never been observed. And then there is super string/ M theory which predicts nothing at all. Maybe it's time to rethink where particle physics has been going.