Proof that L $_{\rm Maj}^{\rm Weyl-equivalent}$ $\neq$ L $_{\rm Maj}^{\rm SM}$

Let us calculate the Lagrangian L $_{\rm Maj}^{SM}$ for massless Majorana neutrinos that is predicted by the Standard model. Applying P$_L$ onto eq.(6) one gets:

$$\nu_L = \sqrt{2} P_L \nu_M(h=-1)$$ (8)

One can also show that [21,22]:

$$\bar{\nu}_L \gamma_{\mu} {\partial \over {\partial x_\mu}} \... ...1) \gamma_{\mu} {\partial \over {\partial x_\mu}} \nu_M(h=-1)$$ (9)

Replacing the kinetic term in Lagrangian (4) using eq.(9) and $\nu_L$ in the interaction term using eq.(8) one gets:

$$L_{\rm Maj}^{\rm SM}= \bar{\nu}_M(h=+1) \gamma_{\mu} {\part... ...W_{\mu}^- \bar{e}_L \gamma_{\mu} P_L \nu_M(h=-1) + H.C.\right]$$ (10)

The charged-current coupling constant in eq.(10) is seen to be a factor $\sqrt{2}$ larger than in eq.(7) the two Lagrangians are thus different.
The numerical value $g={e \over{sin(\theta_w)}}$ is determined in the standard-model gauge theory by considering only neutral-current (for $sin(\theta_w)$) and electromagnetic (for $e$) reactions of the electron, i.e. without reference to neutrino properties. One numerically different coupling constant in the two otherwise identical Lagrangians eq.(7) and eq.(10) is a difference which persists to the phenomenological level (i.e. the application of Feynman rules). In other words: if the neutrino is a Majorana particle and its gauge interactions are the one of the standard model, charged-current reactions of the neutrino would have a factor 2 larger cross section than observed. If we assume the strict validity of the standard model gauge sector a priori (see assumption A in the introduction) the observed neutrino, if massless must be a Weyl neutrino, i.e. definitely not its own antiparticle. This conclusion rests only on the quantitative consideration of the charged current ``source'' term $i g W_{\mu}^- \bar{e}_L \gamma_{\mu} \nu_L$; as long as only kinetic, mass and the form of the interaction term are considered (as is done in all equivalence proofs in the literature!) Majorana and Weyl fields are seen to be completely equivalent.