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Operatorial characterization of Majorana neutrinos

The Majorana neutrino $ψ_{M}(x)$ when constructed as a superposition of chiral fermions such as $ν_{L} + C\overline{ν_{L}}^{T}$ is characterized by $ ({\cal C}{\cal P}) ψ_{M}(x)({\cal C}{\cal P})^{\dagger} =iγ^{0}ψ_{M}(t,-\vec{x})$, and the CP symmetry describes the entire physics contents of Majorana neutrinos. Further specifications of C and P separately could lead to difficulties depending on the choice of C and P. The conventional $ {\cal C} ψ_{M}(x) {\cal C}^{\dagger} = ψ_{M}(x)$ with well-defined P is naturally defined when one constructs the Majorana neutrino from the Dirac-type fermion. In the seesaw model of Type I or Type I+II where the same number of left- and right-handed chiral fermions appear, it is possible to use the generalized Pauli-Gursey transformation to rewrite the seesaw Lagrangian in terms of Dirac-type fermions only; the conventional C symmetry then works to define Majorana neutrinos. In contrast, the "pseudo C-symmetry" $ν_{L,R}(x)\rightarrow C\overline{ν_{L,R}(x)}^{T}$ (and associated "pseudo P-symmetry"), that has been often used in both the seesaw model and Weinberg's model to describe Majorana neutrinos, attempts to assign a nontrivial charge conjugation transformation rule to each chiral fermion separately. But this common construction is known to be operatorially ill-defined and, for example, the amplitude of the neutrinoless double beta decay vanishes if the vacuum is assumed to be invariant under the pseudo C-symmetry.