The double cover of the icosahedral symmetry group and quark mass textures

Lisa L Everett,Alexander J Stuart

doi:10.1016/j.physletb.2011.02.054

Abstract

We investigate the idea that the double cover of the rotational icosahedral symmetry group is the family symmetry group in the quark sector. The icosahedral (A5) group was previously proposed as a viable family symmetry group for the leptons. To incorporate the quarks, it is highly advantageous to extend the group to its double cover, as in the case of tetrahedral (A4) symmetry. We provide the basic group theoretical tools for flavor model-building based on the binary icosahedral group I′ and construct a model of the quark masses and mixings that yields many of the successful predictions of the well-known U(2) quark texture models.

Highlights

With the measurement of neutrino oscillations [1,2,3,4,5,6], an intriguing pattern of lepton mixing has emerged
Of the possible discrete non-Abelian symmetry groups to consider as a basis for flavor model building, we have argued in this paper that the double cover of the rotational icosahedral symmetry group, the binary icosahedral group I′, is well-suited for this purpose due to its nontrivial triplet and doublet representations
In comparison to other discrete groups based on the Platonic solids, there has been a relative paucity of appearances of I′ within the physics literature, which is due at least in part to the fact that the icosahedron does not obey translational symmetry

Summary

INTRODUCTION

With the measurement of neutrino oscillations [1,2,3,4,5,6], an intriguing pattern of lepton mixing has emerged. By considering its double cover, the binary tetrahedral group T , and assigning the lighter generation quarks to doublets and the third generation quarks to singlets, the successful U (2) quark textures [32] can be obtained together with the lepton sector prediction of Harrison-Perkins-Scott [HPS] “tri-bimaximal” mixing [33], as shown in [12,13,14]. We explored the interesting hypothesis that the solar angle is given by tan θsol = 1/φ, where φ = (1 + 5)/2 is the golden ratio This hypothesis was first suggested in [26] and later explored within a Z2 × Z2 framework in [17], who provided a prediction for the Cabibbo angle based on the golden ratio that is correct to three digits and suggested A5 as a useful setting for lepton flavor model building since the golden ratio appears in the geometry of the icosahedron.

THEORETICAL BACKGROUND

GROUP PRESENTATION AND GROUP INVARIANTS

CONCLUSIONS