A new family symmetry for SO(10) GUTs

Abstract

We argue that the projective special linear group PSL(2,7), also known as Sigma(168), has unique features which make it the most suitable discrete family symmetry for describing quark and lepton masses and mixing in the framework of SO(10) type unified models. In such models flavon fields in the sextet representation of PSL(2,7) play a crucial role both in obtaining tri-bimaximal neutrino mixing as well as in generating the third family charged fermion Yukawa couplings. In preparation for physical applications, we derive the triplet representation of PSL(2,7) in the basis S,T,U,V where S,T,U are the familiar triplet generators of S(4) in the diagonal charged lepton basis where T is diagonal. We also derive an analogous basis for the real sextet representation and identify the vacuum alignments which lead to tri-bimaximal neutrino mixing and large third family charged fermion Yukawa couplings.