Weyl neutrinos with Majorana masses

Lepton conservation is induced, according Noether's theorem, by the invariance of (4) under the following continuous transformation group. The charged lepton field e and neutrino field are simultaneously transformed via[11]:

Considering infinitesimal for the infinitesimal field transformation , Noether's theorem yields lepton conservation. The standard model Lagrangian with the addition of a non standard-model Majorana mass term :

The treatment of section 3 continues to hold. This means:

1. Lagrangian eq.(12) is phenomenologically equivalent to the Lagrangian

2. assuming the validity of the standard-model gauge sector the neutrino is definitely not a Majorana field.

m violates the invariance of eq.(12) under the transformation group eq.(11) because the Majorana mass term acquires a phase of e under transformation (11).

What is the mechnanism with which a Weyl field, which is never its own charge conjugate, violates lepton number? Consider the state of a Weyl field with a Majorana mass in an inertial frame at which it is a rest:

(14) |

(15) |

With a Majorana mass term a Weyl field can thus violate lepton conservation,