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

It is customary to identify $ψ_{+}=ν_{R} + C\overline{ν_{R}}^{T}$ with a Majorana fermion on the basis of chirality changing charge conjugation $\tilde{C}: ν_{R}\rightarrow C\overline{ν_{R}}^{T}$ and parity $\tilde{P}: ν_{R}\rightarrow iγ^{0}ν_{R}$. The theorem on the absence of a Majorana-Weyl fermion in $d=4$ states $\tilde{C}γ_{5}\tilde{C}^{-1}= -γ_{5}$ with $\tilde{C}=Cγ_{4}^{T}$, and thus the charge conjugation of the equivalent Majorana $ψ_{+}=(\frac{1+γ_{5}}{2})ν_{R} + (\frac{1-γ_{5}}{2})C\overline{ν_{R}}^{T}$ vanishes without subsidiary $γ_{5}\rightarrow - γ_{5}$, 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 $\hat{C}:ν_{R,L}\rightarrow C\overline{ν_{L,R}}^{T}$ and parity $\hat{P}: ν_{R,L}\rightarrow iγ^{0}ν_{L,R}$ in theory containing both $ν_{R,L}$. In the type I seesaw model, the latter formulation is applicable but $ψ_{+}=ν_{R} + C\overline{ν_{R}}^{T}$ is not a Majorana fermion. An analogue of the Bogoliubov transformation converts $ψ_{\pm}=ν_{R, L} \pm C\overline{ν_{R, L}}^{T}$, which are obtained by a precise diagonalization of the seesaw model, to Majorana fermions $ψ_{M_{1,2}}=(ψ\pm C\overlineψ^{T})/\sqrt{2}$ with a Dirac-type fermion $ψ$, as originally defined by Majorana. A chiral projection $[(1+γ_{5})/2] ψ_{M_{1}}$ of a Majorana fermion is not a chiral fermion, which ensures the presence of the neutrino-less double beta decay.