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 $\nu$ are simultaneously transformed via[11]:

$$\nu^{\prime}= e^{i \alpha} \nu$$

$$\bar{\nu}^{\prime} =\bar{\nu} e^{- i \alpha}$$

$$e^{\prime} = e^{i \alpha} e$$ (11)

Considering $\alpha$ infinitesimal for the infinitesimal field transformation $\delta \Psi$, Noether's theorem yields lepton conservation. The standard model Lagrangian with the addition of a non standard-model Majorana mass term $m_{Maj}$:

$$L_{\rm Weyl}^{\rm SM} = \bar{\nu}_L \gamma_{\mu} {\partial \... ...right] + \left[ m_{Maj} {\bar{\nu}_L} (\nu_L)^c + H.C. \right]$$ (12)

The treatment of section 3 continues to hold. This means:
1. Lagrangian eq.(12) is phenomenologically equivalent to the Lagrangian

$$L_{\rm Maj}^{\rm Weyl-equivalent}= \bar{\nu}_M \gamma_{\mu} {\par... ... i g/\sqrt{2} \left[ W_{\mu}^- \bar{e}_L \gamma_{\mu} \nu_M(h=-1) + H.C.\right]$$

$$+ \left[ m_{Maj} \nu_M {\bar{\nu}_M} + H.C. \right]$$ (13)

2. assuming the validity of the standard-model gauge sector the neutrino is definitely not a Majorana field.
m$_{Maj}$ violates the invariance of eq.(12) under the transformation group eq.(11) because the Majorana mass term acquires a phase of e$^{2 i \alpha}$ 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:

$$\gamma_4 i {\partial \over {\partial t}} \nu_L(\rm {rest}) = m_{Maj} (\nu_L)^c(\rm {rest})$$ (14)

A solution to this equation in the Weyl representation is: $\nu_L(\rm {rest})$ = $\left( \begin{array}{c} 0 \\ \phi_0+i \sigma_2 \phi_0^*\end{array} \right)$ with $\phi_0= a e^{imt}$. This can be rewritten as:

$$\nu_L(rest) = {1\over \sqrt{2}} (\nu_D(\rm {rest})+\nu_D(\rm {rest})^c)_L = \nu_ L + (\nu^c)_L$$ (15)

where $\nu_D(\rm {rest})$ = $\left( \begin{array}{c} \phi_0 \\ \phi_0 \end{array} \right)$ describes a Dirac particle at rest. This result means: in the rest frame the Weyl spinor consists of the components ``left-handed neutrino $\nu_L$'' (helicity = $-$1) and ``left-handed antineutrino $(\nu^c)_L$'' (helicity=+1) which are not their respective charge conjugates. A Lorentz boost along the z-axis can be shown to transform $\nu_L(\rm {rest})$ (with helicity=0) into a state which is predominantly $\nu_L$ (helicity=$-$1) or $(\nu^c)_L$ (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.