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We furnish a supersymmetric extension of the Standard Model with a flavour discrete symmetry $A_5$ under which the lepton fields transform as an irreducible triplet. Additional (`flavon) superfields are used to break $A_5$ into a $mathbb{Z}_2 times mathbb{Z}_2$ subgroup in the charged-lepton sector and another $mathbb{Z}_2$ subgroup in the neutrino sector. The first column of the resulting lepton mixing matrix is predicted and has entries which are related to the golden ratio. Using the observed $theta_{13}$ as input, our model predicts a solar mixing angle $theta_{12}$ in very good agreement with experiment; it also predicts a correlation between the atmospheric mixing angle $theta_{23}$ and the $CP$-violating Dirac phase $delta$.
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After the successful determination of the reactor neutrino mixing angle theta_13 ~ 0.16 eq 0, a new feature suggested by the current neutrino oscillation data is a sizeable deviation of the atmospheric neutrino mixing angle theta_23 from pi/4. Using the fact that the neutrino mixing matrix U = U_e^dagger U_ u, where U_e and U_ u result from the diagonalisation of the charged lepton and neutrino mass matrices, and assuming that U_ u has a i) bimaximal (BM), ii) tri-bimaximal (TBM) form, or else iii) corresponds to the conservation of the lepton charge L = L_e - L_mu - L_tau (LC), we investigate quantitatively what are the minimal forms of U_e, in terms of angles and phases it contains, that can provide the requisite corrections to U_ u so that theta_13, theta_23 and the solar neutrino mixing angle theta_12 have values compatible with the current data. Two possible orderings of the 12 and the 23 rotations in U_e, standard and inverse, are considered. The results we obtain depend strongly on the type of ordering. In the case of standard ordering, in particular, the Dirac CP violation phase delta, present in U, is predicted to have a value in a narrow interval around i) delta ~ pi in the BM (or LC) case, ii) delta ~ 3pi/2 or pi/2 in the TBM case, the CP conserving values delta = 0, pi, 2pi being excluded in the TBM case at more than 4sigma. In the addendum we discuss the implications of the latest 2013 data.
In terms of its eigenvector decomposition, the neutrino mass matrix (in the basis where the charged lepton mass matrix is diagonal) can be understood as originating from a tribimaximal dominant structure with small deviations, as demanded by data. If neutrino masses originate from at least two different mechanisms, referred to as hybrid neutrino masses, the experimentally observed structure naturally emerges provided one mechanism accounts for the dominant tribimaximal structure while the other is responsible for the deviations. We demonstrate the feasibility of this picture in a fairly model-independent way by using lepton-number-violating effective operators, whose structure we assume becomes dictated by an underlying $A_4$ flavor symmetry. We show that if a second mechanism is at work, the requirement of generating a reactor angle within its experimental range always fixes the solar and atmospheric angles in agreement with data, in contrast to the case where the deviations are induced by next-to-leading order effective operators. We prove this idea is viable by constructing an $A_4$-based ultraviolet completion, where the dominant tribimaximal structure arises from the type-I seesaw while the subleading contribution is determined by either type-II or type-III seesaw driven by a non-trivial $A_4$ singlet (minimal hybrid model). After finding general criteria, we identify all the $mathbb{Z}_N$ symmetries capable of producing such $A_4$-based minimal hybrid models.
We investigate the possibility of using icosahedral symmetry as a family symmetry group in the lepton sector. The rotational icosahedral group, which is isomorphic to A5, the alternating group of five elements, provides a natural context in which to explore (among other possibilities) the intriguing hypothesis that the solar neutrino mixing angle is governed by the golden ratio. We present a basic toolbox for model-building using icosahedral symmetry, including explicit representation matrices and tensor product rules. As a simple application, we construct a minimal model at tree level in which the solar angle is related to the golden ratio, the atmospheric angle is maximal, and the reactor angle vanishes to leading order. The approach provides a rich setting in which to investigate the flavor puzzle of the Standard Model.
We present the correlation of low energy CP phases, both Dirac and Majorana, and the lepton asymmetry for the baryon asymmetry in the universe, with a certain class of Yukawa matrices that consist of two right-handed neutrinos and include one texture zero in themselves. For cases in which the amount of the lepton asymmetry $Y_L$ turns out to be proportional to $theta_{13}^2$, we consider the relation between two types of CP phases and the relation of $Y_L$ versus the Jarlskog invariant or the amplitude of neutrinoless double beta decay as $theta_{13}$ varies.
In light of the latest neutrino oscillation data, we examine whether the leptonic flavor mixing matrix can take on an exact form of tri-bimaximal (TBM), golden-ratio (GR) or bimaximal (BM) mixing pattern at a superhigh-energy scale, where such a mixing pattern could be realized by a flavor symmetry, and become compatible with experimental data at the low-energy scale. Within the framework of the Minimal Supersymmetric Standard Model (MSSM), the only hope for realizing such a possibility is to count on the corrections from the renomalization-group (RG) running. In this work we focus on these radiative corrections, and fully explore the allowed parameter space for each of these mixing patterns. We find that when the upper bound on the sum of neutrino masses $Sigma^{}_ u equiv m^{}_1 + m^{}_2 + m^{}_3 < 0.23~text{eV}$ at the $95%$ confidence level from Planck 2015 is taken into account, none of these mixing patterns can be identified as the leptonic mixing matrix below the seesaw threshold. If this cosmological upper bound on the sum of neutrino masses were relaxed, the TBM and GR mixing patterns would still be compatible with the latest neutrino oscillation data at the $3sigma$ level, but not at the $1sigma$ level. Even in this case, no such a possibility exists for the BM mixing.
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