The unitary equivalence of Weyl and Majorana neutrino fields, also called ``Dirac-Majorana confusion theorem''

For the present argument it is sufficient to consider only charged currents in the low energy limit. The standard-model (SM) Lagrangian for massless Weyl neutrino fields is then:

$$L_{\rm Weyl}^{\rm SM}= \bar{\nu}_L \gamma_{\mu} {\partial \o... ...left[W_{\mu}^- \bar{e}_L \gamma_{\mu} P_L \nu_L + H.C. \right]$$ (4)

here $g={e \over{sin(\theta_w)}}$. Let us now answer the following question: What Lagrangian `` $L_{\rm Maj}^{\rm Weyl-equivalent}$'' must hold Majorana neutrino, so that it shows a phenomenology identical to the one of the Weyl neutrino with Lagrangian (4)?
Pauli[4] specified the following ``Pauli I'' transformation ``U$_1$'' which transforms a neutrino field $\nu$ into $\nu^{\prime}$:

$$\nu^{\prime}=U_1 \nu U_1^{-1}={1\over \sqrt{2}}(\nu - \gamma_5 \nu^c)$$ (5)

This transformation ⁷ can be easily shown to be unitary but does not conserve a SU(2) invariance of a Lagrangian. Similarity transformations leave the form of operator equations (i.e. in particular the field equations and anticommutation relations) unmodified and the expectation values of field operators do not change under a unitary transformation of field operator together with the field states[11,4]. Therefore the phenomenology remains unchanged if one replaces $\nu$ by $\nu^{\prime}$ everywhere.
For the special case $\nu=\nu_L$ eq.(5) reads (h=helicity)⁸:

$$\nu^{\prime}={1\over \sqrt{2}} (\nu_L - \gamma_5 (\nu_L)^c)= {1\over \sqrt{2}} (\nu_L + (\nu_L)^c) = \nu_M(h=-1)$$ (6)

From the invariance of the field equations, the Majorana Lagrangian is obtained by replacing $\nu_L$ with $\nu_M$ in equation (4)

$$L_{\rm Maj}^{\rm Weyl-equivalent}= \bar{\nu}_M(h=+1) \gamma_{\mu} {\partial \over {\partial x_\mu}} \nu_M(h=-1)$$

$$+ i g/\sqrt{2} \left[ W_{\mu}^- \bar{e}_L \gamma_{\mu} P_L \nu_M(h=-1) + H.C.\right]$$ (7)

The ``Pauli I'' transformation eq.(5) does not include the electron field and is therefore not equivalent to a mere representation change of the theory. It is thus not at all clear if this Lagrangian still obeys the standard model (see next section). In the late 1950s (i.e. long before the formulation of the standard model) - with no reason whatsoever to exclude the validity of L $_{\rm Maj}^{\rm Weyl-equivalent}$ for neutrinos - various authors [13,14,15,16,17,18,19] could only conclude that massless $\nu_L$ and $\nu_M$ (helicity=$-$1 states) (and analogously $\bar{\nu}_L$ and $\bar{\nu}_M$ (helicity=+1 states)) are phenomenologically completely equivalent (this conclusion was later also called ``Dirac - Majorana confusion theorem''[3]). The ``Dirac - Majorana confusion theorem'' was never discussed in the literature under the assumption of quantitative validity of the standard model. The difference between Majorana and Weyl neutrino is of a purely quantitative character (a factor $\sqrt{2}$) all qualitative properties are the same (e.g. in the massless case both Majorana and Weyl neutrinos conserve lepton number). Kayser[3] and Zraek[20] state the confusion theorem's validity under the assumption that the weak interaction is left handed (``qualitative validity'' of the standard model), a correct statement which is not in contradiction with the present paper.