Abstract
We review the application of non abelian discrete groups to the theory of neutrino masses and mixing, which is strongly suggested by the agreement of the Tri-Bimaximal mixing pattern with experiment. After summarizing the motivation and the formalism, we discuss specific models, based on A4, S4 and other finite groups, and their phenomenological implications, including lepton flavor violating processes, leptogenesis and the extension to quarks. In alternative to Tri-Bimaximal mixing the application of discrete flavor symmetries to quark-lepton complementarity and Bimaximal Mixing is also considered.
Highlights
7.65+−00..2230 2.40+−00..1121 0.304+−00..002126 0.50+−00..0076 0.010+−00..001161 from the end point of the tritium beta decay spectrum, from cosmologysee, for example, Lesgourgues and Pastor2006͔͒, and from neutrinoless double beta decay0͒ ͓for a recent review, see, for example, Avignone et al ͑2008͔͒
If this experimental result is not a mere accident but a real indication that a dynamical mechanism is at work to guarantee the validity of TB mixing in the leading approximation, corrected by small nonleading terms, non-Abelian discrete flavor groups emerge as the main road to an understanding of this mixing pattern
We started by recalling some basic notions about finite groups and concentrated on those symmetries, such as A4 and S4, which are found to be the main candidates for obtaining TB mixing
Summary
Given the Pontecorvo-Maki–Nagakawa–SakataPMNSmixing matrix Usee Altarelli and Feruglio2004͒ for its general definition and parametrization, the general form of the neutrino mass matrix, in terms of thecomplex2͒ mass eigenvalues m1 , m2 , m3, in the basis where charged leptons are diagonal, is given by. The important case of TB mixing is obtained when sin2 212= 8 / 9 or x + y = w + z.3 In this case, the matrix m takes the form. Note that the mass matrix for TB mixing, in the basis where charged leptons are diagonal, as given in Eq ͑11͒, can be specified as the most general matrix which is invariant under - ͑or 2-3͒ symmetrysee Eqs. In the basis where charged-lepton masses are diagonal, from Eq ͑5͒ we derive the effective neutrino mass matrix in the BM case, m = m1⌽1⌽1T + m2⌽2⌽2T + m3⌽3⌽3T,. We are in a position to explain the role of finite groups and to formulate the general strategy to obtain one of the previous special mass matrices, for example, that of TB mixing. Along the same line, a model with - symmetry can be realized in terms of group S3 generated by products of A23 and Tsee, for example, Feruglio and Lin2008͔͒
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