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We revisit the time evolution of the lepton family number for a SU(2) doublet consisting of a neutrino and a charged lepton. The lepton family number is defined through the weak basis of the SU(2) doublet, where the charged lepton mass matrix is real and diagonal. The lepton family number carried by the neutrino is defined by the left-handed current of the neutrino family. For this work we assume the neutrinos have Majorana mass. This Majorana mass term is switched on at time $t=0$ and the lepton family number is evolved. Since the operator in the flavor eigenstate is continuously connected to that of the mass eigenstate, the creation and annihilation operators for the two eigenstates are related to each other. We compute the time evolution of all lepton family numbers by choosing a specific initial flavor eigenstate for a neutrino. The evolution is studied for relativistic and nonrelativistic neutrinos. The nonrelativistic region is of particular interest for the Cosmic Neutrino Background predicted from big bang models. In that region we find the lepton family numbers are sensitive to Majorana and Dirac phases, the absolute mass, and mass hierarchy of neutrinos.
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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.
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