Operatorial characterization of Majorana neutrinos

Abstract

Abstract The Majorana neutrino $$\psi _{M}(x)$$ ψM(x) when constructed as a superposition of chiral fermions such as $$\nu _{L} + C\overline{\nu _{L}}^{T}$$ νL+CνL¯T is characterized by $$ (\mathcal{C}\mathcal{P}) \psi _{M}(x)(\mathcal{C}\mathcal{P})^{\dagger } =i\gamma ^{0}\psi _{M}(t,-\vec {x})$$ (CP)ψM(x)(CP)†=iγ0ψM(t,-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 $$ \mathcal{C} \psi _{M}(x) \mathcal{C}^{\dagger } = \psi _{M}(x)$$ CψM(x)C†=ψM(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” $$\nu _{L,R}(x)\rightarrow C\overline{\nu _{L,R}(x)}^{T}$$ νL,R(x)→CνL,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.