The Symmetries And The Neutrinos Mixing Models

In order to interpret the continuous energy spectra of the β decay, Pauli proposed the existence of the neutrino, which was considered as the neutral massless fermion. However, Pontecorvo and Gribov proposed that if the neutrino is massive and there is the mixing of the different massive neutrinos the propagation of the neutrinos may bring the flavor oscillation. A variety of the neutrino experiments, such as the Homestake, Kamiokande and etc, have confirmed the flavor oscillation of the neutrinos and determined the mixing parameters θ12,θ23and the mass parameters ΔmAm2, ΔmSlo2. The nonzero mixing angle θ13is being confirmed by the experiments.The flavor oscillation of the neutrinos reveals that the mass scale of the neutrino is tiny, which can not be interpreted in the standard model of the particle physics. As a phenomenological assumption, the seesaw mechanism introduces the right-handed neutrino with the large mass scale to depress the mass scale of the neutrino. There is no obvious evidences for the existence of the right-handed neutrino with the large mass scale. Even though we accept the seesaw mechanism as the interpretation of the tiny mass of the neutrino, the introductions of the neutrino mass and the mixing of the massive neutrinos ask for the new examinations of the familiar concepts and laws in the standard model. The concept of the flavor state, the conservation law of the energy-momentum and the time-dependent perturbation theory become non-simple and nontrivial in the background of the neutrinos mixing and the neutrinos oscillation. First, because of the time-dependence of the flavor charge of the neutrino, it is difficult to define the universal flavor state. We can only employ the flavor state with the limited and the approximate conditions. One can even avoid the flavor state for less theoretical controversies and treat the neutrino as the intermedial particle in the discussion of the neutrinos oscillation. Second, in order to coordinate the flavor conservation in the weak interaction and the flavor violation in the neutrino propagation, the conventional time-dependent perturbation theory is limited in the calculation of the reaction probability where the neutrino is involved. In the calculation of the probability involving the neutrino, the time scale in the time-dependent perturbation theory should be much less than Is. Otherwise, it will bring the obvious flavor violation in the weak interaction which is in contradiction with the micro-causality. Third, because of the uncertainty of the energy-momentum of the neutrino, the conventional constraint of the conservation of the energy-momentum does not mean the kinetic entanglement of the neutrino and the recoiling particle. The kinetic entanglement is the result of the special experiment design and the special detection. On the other hand, the analysis of the symmetry demonstrates that the flavor oscillation involves the limit of the uncertainties of the energy and the momentum of the neutrino. Both the momentum and the energy are uncertain for the flavor neutrino. The improvement of the resolution of the measurement of the momentum or the energy can destroy the flavor oscillation of the neutrino.The pattern of the neutrino oscillation and the neutrino mixing can not be interpreted with the conventional symmetry group in the standard model. Thus, the discrete non-Abel flavor symmetry group is introduced in the neutrino model. These discrete group such as the A4, S4, T’and so on can interpret the classical mixing pattern, namely the tribimaximal mixing pattern. With the confirmation of the nonzero mixing angle θ13by the experiments, especially the convincing results of the neutrinos experiments at the Daya Bay, the modification or the perturbation is needed in these models with the discrete flavor group. However, even though these models can interpret the conventional mixing pattern and give the constraints on the undetermined mixing parameters and the mass parameters, there is the special fine-tuning of the vacuum expected value of the scalar field in these models. The naturalness of these model is not convincing. As a more economical procedure of the model construction, one can introduce the mass matrix of the neutrinos directly without the employment of the Higss mechanism and the extra scalar fields. The mass matrix satisfies the special discrete flavor symmetry. By diagonalization of the mass matrix, we can obtain the expected mixing matrix of the neutrinos. Although this method avoids the fine-tuning of the scalar field, the discrete symmetry group that induces the same mixing matrix is not unique. We can not discriminate these different symmetry groups. So lots of work is needed to deepen the interpretation of the neutrino mixing pattern.Besides the conventional mixing pattern, some anomalous oscillation signals were reported in the experiments. Some experiments signals, such as that of the MINOS, reveal that the oscillation of the neutrino is different to that of the anti-neutrino. Other signals, such as that of the LSND and MiniBooNE, reveal that there is the extra large mass scale of the order leV2besides the tiny mass scales in the solar neutrino and the atmosphere neutrino. As a plausible interpretation, the introduction of the Lorentz invariance violation and the CPT violation can illustrate the difference of the oscillation of the neutrinos and the anti-neutrinos. Further more, because of the energy-dependence of the mixing matrix element in the model of the Lorentz invariance violation, some of the other anomalous oscillation can be interpreted. However, because of the introduction of the Lorentz invariance violation, we should modify the dispersion relation of the neutrino besides the energy-dependence mixing element, which means there may be superluminal neutrinos. Except the MINOS experiment and the OPERA experiment, kinds of the neutrinos experiments by far reveal that there is no obvious event of the superluminal neutrinos. In particular, very recently, the OPERA collaboration has claimed the flaw in their experiments. Therefore, the plausible neutrino model with the Lorentz violation should satisfies the condition that the modification of the dispersion relation is not obvious compared with the conventional relation. In the regime without the superluminal neutrinos, the method testing the Lorentz-violation models focuses on the energy-dependence of the mixing element of the neutrinos. The constraints on the Lorentz-violation models by far are very strict. However, because of the model-dependence of the constraints, it is still possible to construct the viable Lorentz-violation models of the neutrinos.Generally speaking, the symmetry analysis is one of the most important methods in the studying of the phenomenology of the neutrino oscillation, which is enlightening in the construction of the mixing models of the neutrinos. It is believed that the symmetry analysis will be still an effective method in the future phenomenological researches of the neutrinos.