The charged fermion masses of the three generations exhibit the two strong hierarchies m 3 m 2 m 1 . We assume that also neutrino masses satisfy m ν3 > m ν2 > m ν1 and derive the consequences of the hierarchical spectra on the fermionic mixing patterns. The quark and lepton mixing matrices are built in a general framework with their matrix elements expressed in terms of the four fermion mass ratios, m u /m c , m c /m t , m d /m s and m s /m b , and m e /m μ , m μ /m τ , m ν1 /m ν2 and m ν2 /m ν3 , for the quark and lepton sector, respectively. In this framework, we show that the resulting mixing matrices are consistent with data for both quarks and leptons, despite the large leptonic mixing angles. The minimal assumption we take is the one of hierarchical masses and minimal flavor symmetry breaking that strongly follows from phenomenology. No special structure of the mass matrices has to be assumed that cannot be motivated by this minimal assumption. This analysis allows us to predict the neutrino mass spectrum and set the mass of the lightest neutrino well below 0.01 eV. The method also gives the 1σ allowed ranges for the leptonic mixing matrix elements. Contrary to the common expectation, leptonic mixing angles are found to be determined solely by the four leptonic mass ratios without any relation to symmetry considerations as commonly used in flavor model building. Still, our formulae can be used to build up a flavor model that predicts the observed hierarchies in the masses -the mixing follows then from the procedure which is developed in this work.
S 3 models offer a low energy approach to describe the observed pattern of masses and mixing, of both quarks and leptons. In this work, we first revisit an S 3 model with only one Higgs electroweak doublet, where the flavour symmetry must be broken in order to produce an acceptable pattern of masses and mixing for fermions. Then, we analyse different S 3 models, where the flavour symmetry is preserved as an exact, but hidden symmetry of the low energy spectra, after the electroweak symmetry breaking. The latter models require the addition of two more Higgs electroweak doublets which are accommodated in an S 3 doublet. We also explore the consequences of adding a fourth Higgs electroweak doublet, thus occupying all three irreducible representations of S 3 . We show how the various S 3 -invariant mass matrices of the different models can reproduce the two texture zeroes and Nearest Neighbour Interaction matrix forms, which have been found to provide a viable and universal treatment of mixing for both quarks and leptons. We also find analytical and exact expressions for the CKM matrix of the models in terms of quark mass ratios. Finally, we compare the expressions of the CKM matrix of the different S 3 models with the most up to date values of masses and mixing in the quark sector, via a χ 2 analysis. We find that the analytical expressions we derived reproduce remarkably well the most recent experimental data of the CKM matrix, suggesting that S 3 is a symmetry of the quark sector.
In this talk I give an overview over continuous and discrete groups and how these are used in the field of model building as flavor symmetries. The latter act on the space of the three generations of elementary particles. I mainly concentrate on discussing generic mathematical properties of these groups relevant for understanding their possible predictive power when applied to explain fermion mass and mixing patterns. I also put emphasis on the classification of discrete groups.13 . Indeed, in all models the leading order result, in this case TB mixing, receives corrections from various sources, e.g. higher-dimensional operators. Thus, in all models a non-zero value of θ l 13 is aspected. Concerning its size one can roughly say: let us assume the size of the corrections to be δ and that the latter contribute in the same manner to all three mixing angles, then the request to not perturb too much the result for the solar mixing angle implies that δ 0.05. Thus, one might expect sin θ l 13 ∼ δ 0.05 which is too small to explain the recent experimental indication. Since in many models several operators give rise to a correction to θ l 13 , we get in general sin θ l 13 ≈ |c|δ with c complex. If c is largish, i.e. the corrections add up, larger values of θ l 13 can be explained. Alternatively, one can consider models in which the corrections to the
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