Operatorial characterization of Majorana neutrinos

Abstract

The Majorana neutrino psi _{M}(x) when constructed as a superposition of chiral fermions such as nu _{L} + Coverline{nu _{L}}^{T} is characterized by (mathcal{C}mathcal{P}) psi _{M}(x)(mathcal{C}mathcal{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 mathcal{C} psi _{M}(x) mathcal{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” nu _{L,R}(x)rightarrow Coverline{nu _{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.