Outline of the paper

The review of the precise definition of the Majorana-neutrino field in section 2 makes clear that mathematically Weyl and Majorana fields are different. Langrangians predicted by the standard model for two mathematically different fermion fields - even if they have the same degrees of freedom - must not necessarily lead to the same phenomenology². Below I derive the Lagrangians giving rise to the same phenomenology (briefly outlined in the next paragraph ``1.'') and the Lagrangians derived from the standard model (briefly outlined in following paragraph ``2.'') and find they are not identical.
1. In section 3 - as a review of work from several authors in the late 1950s - it is shown that a unitary similarity transformation (the ``Pauli I'' transformation) between Weyl and Majorana neutrino fields exists. One concludes that Weyl neutrinos with a standard-model (SM) Lagrangian L $_{\rm Weyl}^{\rm SM}$ can be unitarily transformed to a Majorana neutrino with a Lagrangian L $_{\rm Maj}^{\rm Weyl-equivalent}$ which is uniquely specified by the condition that any similarity transformation leaves the form of all field equations unmodified. A Majorana neutrino obeying this Lagrangian is then phenomenologically completely equivalent to a Weyl neutrino.
2. Since the late 1970s the interaction of the left-handed Weyl neutrino $\nu_L$ is uniquely specified by the standard model without any reference to the observed neutrino's properties, (analogous to the top quark, whose weak-interaction properties were all specified before its actual discovery). Because a Majorana-neutrino field can be decomposed to Weyl-neutrino components via the ``Pauli I'' transformation, one can derive the standard-model Lagrangian of a Majorana neutrino L $_{\rm Maj}^{SM}$. One finds that L $_{\rm Maj}^{\rm SM}$ $\neq$ L $_{\rm Maj}^{\rm Weyl-equivalent}$ In other words: the ``Pauli I'' transformation - that brings a Weyl to a phenomenologically equivalent Majorana neutrino - is not SU(2) invariant.
If the observed neutrino were of Majorana type it had to be phenomenologically equivalent to a Weyl neutrino with standard-model interactions to not contradict experimental facts (e.g. about neutrino cross sections). Under assumption A. in the introduction we can then conclude that the observed neutrino must be of Weyl type. The end of section 3 discusses why this conclusion is not in contradiction with the pubished literature on the ``Majorana Dirac confusion theorem''.
Section 5 of the paper is devoted to showing in detail how a Weyl neutrino with a Majorana mass term can violate lepton number without ever being its own anti particle and section 6 concludes.