Absence of Majorana-Weyl fermions in d=4 and the theory of Majorana fermions

It is customary to identify ${\psi }_{+}={\nu }_R+C{\overline{{\nu }_R}}^T$ with a Majorana fermion on the basis of chirality changing charge conjugation $\tilde{C}:{\nu }_R\to C{\overline{{\nu }_R}}^T$ and parity $\tilde{P}:{\nu }_R\to i{\gamma }^0{\nu }_R$. The theorem on the absence of a Majorana-Weyl fermion in $d=4$ states $\tilde{C}{\gamma }_5{\tilde{C}}^{-1}=-{\gamma }_5$ with $\tilde{C}=C{\gamma }_4^T$, and thus the charge conjugation of the equivalent Majorana ${\psi }_{+}=(\frac{1+{\gamma }_5}{2}){\nu }_R+(\frac{1-{\gamma }_5}{2})C{\overline{{\nu }_R}}^T$ vanishes without subsidiary ${\gamma }_5\to -{\gamma }_5$, namely, not defined in field theory. To be consistent with the theorem, it is common to use a doublet representation of chirality preserving charge conjugation $\widehat{C}:{\nu }_{R,L}\to C{\overline{{\nu }_{L,R}}}^T$ and parity $\widehat{P}:{\nu }_{R,L}\to i{\gamma }^0{\nu }_{L,R}$ in theory containing both ${\nu }_{R,L}$. In the type I seesaw model, the latter formulation is applicable but ${\psi }_{+}={\nu }_R+C{\overline{{\nu }_R}}^T$ is not a Majorana fermion. An analogue of the Bogoliubov transformation converts ${\psi }_{\pm }={\nu }_{R,L}\pm C{\overline{{\nu }_{R,L}}}^T$, which are obtained by a precise diagonalization of the seesaw model, to Majorana fermions ${\psi }_{M_{1,2}}=(\psi \pm C{\bar{\psi }}^T)\mathrm{/}\sqrt{2}$ with a Dirac-type fermion $\psi$, as originally defined by Majorana. A chiral projection $[(1+{\gamma }_5)\mathrm{/}2]{\psi }_{M_1}$ of a Majorana fermion is not a chiral fermion, which ensures the presence of the neutrino-less double beta decay.