Tri-bimaximal neutrino mixing, A4A4 and the modular symmetry

We formulate and discuss a 4-dimensional SUSY version of an A4A4 model for tri-bimaximal neutrino mixing which is completely natural. We also study the next-to-the-leading corrections and show that they are small, once the ratios of A4A4 breaking VEVs to the cutoff are fixed in a specified interval. We also point out an interesting way of presenting the A4A4 group starting from the modular group. In this approach, which could be interesting in itself as an indication on a possible origin of A4A4, the Lagrangian basis where the symmetry is formulated coincides with the basis where the charged leptons are diagonal. If the same classification structure in A4A4 is extended from leptons to quarks, the CKM matrix coincides with the unit matrix in leading order and a study of non-leading corrections shows that the departures from unity of the CKM matrix are far too small to accomodate the observed mixing angles.