Time evolution of lepton number carried by Majorana neutrinos

Abstract

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 avor 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 avor 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.

Lepton Family Numbers and Non-Relativistic Majorana Neutrinos

“…2 corresponds to the region where the azimuthal angle φ lies within the range [0, π). This region is named A region in [1]. The other green-colored hemisphere corresponds to φ within [π, 2π) and is named the region A.…”

“…The explicit expression for a α (q, t) and b α (q, t) can be found in Eq. (30-33) of [1]. The lepton family number at t ≥ 0 is obtained by replacing a α (q)and b α (q) in Eq.…”

“…With the lepton family number, the neutrino flavor transition and the neutrino and anti-neutrino transition effects is included in a single framework and the formula for time evolution of lepton numbers that is valid for the broad range of the momentum was obtained [1]. We reviewed this approach by focusing on the definition of the lepton family numbers, the several subtle points of the derivation such as the continuity condition and the momentum regions.…”

“…By considering the fact that the temperature of CNB is comparable to the neutrinos mass with 10 −4 (eV), it is desirable to formulate the lepton number violation and the lepton flavor violation in a single framework which is valid even in the non-relativistic region. In our previous work [1], such framework has been developed for Majorana neutrino and the formula for time variation of lepton number which is valid for the wide range of the momentum, i.e., from the relativistic case to non relativistic one, was derived. In the framework, the lepton family number operators in the Heisenberg representation were derived by turning on the Majorana mass term in the Hamiltonian at t = 0.…”

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In this talk, we have reviewed the recent development on the time evolution of lepton family number carried by Majorana neutrinos [1]. This article focuses on the subtle points of the derivation of the lepton family numbers and their time evolution.

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