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# Weyl neutrinos with Majorana masses

It could be that Lagrangian (4) for a Weyl neutrino contains a small Majorana mass term. In this section I first review the formal reason why this leads to a lepton-number violating theory and then analyze how this is possible for a particle that is never its own antiparticle.
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 :
 (12) The treatment of section 3 continues to hold. This means:
1. Lagrangian eq.(12) is phenomenologically equivalent to the Lagrangian (13) 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) A solution to this equation in the Weyl representation is: = with . This can be rewritten as:
 (15) where = describes a Dirac particle at rest. This result means: in the rest frame the Weyl spinor consists of the components left-handed neutrino '' (helicity = 1) and left-handed antineutrino '' (helicity=+1) which are not their respective charge conjugates. A Lorentz boost along the z-axis can be shown to transform (with helicity=0) into a state which is predominantly (helicity= 1) or (helicity=+1). I.e. depending on the inertial frame, a massive Weyl particle is predominantly particle or antiparticle. In no frame it is its own antiparticle, however.
With a Majorana mass term a Weyl field can thus violate lepton conservation, without being its own antiparticle.   Next: Summary Up: The non-equivalence of Weyl Previous: Proof that L L
Rainer Plaga 2001-08-03