Operatorial characterization of Majorana neutrinos

The Majorana neutrino $psi_{M}(x)$ when constructed as a superposition of chiral fermions such as $ u_{L} + Coverline{ u_{L}}^{T}$ is characterized by $ ({cal C}{cal P}) psi_{M}(x)({cal C}{cal P})^{dagger} =igamma^{0}psi_{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} psi_{M}(x) {cal C}^{dagger} = psi_{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 $ u_{L,R}(x)rightarrow Coverline{ u_{L,R}(x)}^{T}$ (and associated pseudo P-symmetry), that has been often used in both the seesaw model and Weinbergs 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.