Relations Between Standard Model Parameters from T ′ Flavor Symmetry

Spontaneously broken flavor symmetry involving the binary tetrahedral group T ′ relates parameters in the standard model. In one example the CP violating phase δPMNS and atmospheric mixing angle θ23 of neutrinos are related to the Cabibbo mixing angle ΘC for quarks. A second formula relates a ratio of neutrino mass eigenvalues to the Cabibbo angle. [email protected] [email protected] Introduction. The standard model (SM) of particle theory based on SU(3) × SU(2) × U(1) suffers from a surfeit of parameters. With three quark-lepton families, non-zero neutrino masses and one Higgs doublet there are twenty-eight of them. Of particular interest for us here are the twenty parameters for masses and mixings of the quarks [ten: Θij(1 ≤ i, j ≤ 3), δKM , m a q(1 ≤ a ≤ 6)] and of the leptons [another ten: θij(1 ≤ i, j ≤ 3), δPMNS, mbl (1 ≤ b ≤ 6). One approach to obtain non-trivial relationships between these parameters is to postulate a flavor (or horizontal) global symmetry G which commutes with the SM gauge group: [SU(3) × SU(2) × U(1)]local × [G]global and an especially appealing choice for G which we advocate involves T ′ , the binary tetrahedral group. The present model employs G = T ′ × Z3 × Z ′ 3 × Z ′′ 3 . The group T ′ is an order g = 24 non-abelian finite group which has various alternative designations such as 24/13 in Thomas and Wood [1], or SL2(F3). T ′ has seven conjugacy classes and hence seven irreducible representations 11, 12, 13, 21, 22, 23, 3. In theoretical physics, to our knowledge, the group T ′ was first used, for quarks, in [2]. Its use for neutrino mixings, subsuming a T symmetry [3] which may underwrite tribimaximal mixing [4], was first suggested in [5] then elaborated in [6]. Between [2] and [5] other papers on T ′ symmetry appeared [7]. Paper [5] led to further articles [8] making use of T ′ although it should be emphasized that, unlike models discussed in [8], the model discussed in the present Letter is renormalizable. The classification and details of all non-abelian finite groups of order g ≤ 31 were provided, in a form most suitable to particle theory model builders, in [2, 9] which reformulated information from [1]. In particular, the multiplication table for T ′ is repeated here for the reader’s convenience: Kronecker Products for Irreducible Representations of T ′ 11 12 13 21 22 23 3 11 11 12 13 21 22 23 3 12 12 13 11 22 23 21 3 13 13 11 12 23 21 22 3 21 21 22 23 11 + 3 12 + 3 13 + 3 21 + 22 + 23 22 22 23 21 12 + 3 13 + 3 11 + 3 21 + 22 + 23 23 23 21 22 13 + 3 11 + 3 12 + 3 21 + 22 + 23 3 3 3 3 21 + 22 + 23 21 + 22 + 23 21 + 22 + 23 11 + 12 + 13 + 3 + 3