How to Describe Neutrino Mixing and CP Violation

We present a classification of possible parametrizations of the 3 × 3 lepton flavor mixing matrix in terms of the rotation and phase angles. A particular parametrization, which is most convenient to describe the observables of neutrino oscillations and that of the neutrinoless double beta decay, is studied in detail. The possibility to determine the Diracand Majorana-type CP-violating phases are also discussed. Electronic address: [email protected] 1 The accumulating evidence for atmospheric and solar neutrino oscillations strongly suggests that neutrinos are massive and lepton flavors are mixed [1]. In general the flavor mixing among N different lepton families is described by a N×N unitary matrix V , whose number of independent parameters depends on the nature of neutrinos. If neutrinos are Dirac particles, V can be parametrized in terms of N(N − 1)/2 rotation angles and (N − 1)(N − 2)/2 phase angles. If neutrinos are Majorana particles, however, a full parametrization of V requires N(N − 1)/2 rotation angles and the same number of phase angles. The flavor mixing of charged leptons and Dirac neutrinos is completely analogous to that of quarks, for which a number of different parametrizations have been proposed and classified in the literature [2]. One of the purposes of this paper is to classify the parametrizations of flavor mixing between charged leptons and Majorana neutrinos with N = 3. Regardless of the phase-assignment freedom, we find that there are nine distinct ways to describe the 3× 3 lepton flavor mixing matrix. Although different representations of lepton flavor mixing are mathematically equivalent, one of them is very likely to describe the underlying physics of lepton mass generation and CP violation in a more transparent way, or is particularly convenient in the analyses of experimental data on neutrino oscillations and lepton-number-violating processes. We point out that there does exist such a parametrization, which allows for simple connections between the measurable quantities and the flavor mixing angles. In addition, the phase assignment of this “standard” parametrization assures that the single Dirac-type CP-violating phase is associated only with neutrino oscillations, and the two Majorana-type CP-violating phases are associated only with the neutrinoless double beta decay. Some remarks are made on the difficulty to separately determine the Majorana-type CP-violating phases. Let us start to parametrize the 3× 3 lepton flavor mixing V , which is defined to link the neutrino flavor eigenstates (νe, νμ, ντ ) to the neutrino mass eigenstates (ν1, ν2, ν3):