Abstract
We propose $\mathcal{T}_{13} = \mathcal{Z}_{13} \rtimes \mathcal{Z}_3$ as the underlying non-Abelian discrete family symmetry of the asymmetric texture presented in arXiv:1805.10684 [hep-ph]. Its mod 13 arithmetic distinguishes each Yukawa matrix element of the texture. We construct a model of effective interactions that singles out the asymmetry and equates, without fine-tuning, the products of down-quark and charged-lepton masses at a GUT-like scale.
Highlights
In this work we propose an answer to the second question with a family symmetry based on the discrete group T 13 1⁄4 Z13⋊Z3 [15]
As a first step in this direction, in this work we focus solely on the asymmetric matrices for the down quarks and charged leptons [9], and show how they can naturally arise from the discrete family symmetry T 13 × Z5
We assume that such mixing arises in the context of the seesaw mechanism but do not further specify the dynamics of the Majorana sector; the origin of the phase δ and the implications of our chosen family symmetry on Majorana physics will be the focus of a future publication [16]. (ii) A diagonal up-quark Yukawa matrix Yð23Þ 1⁄4
Summary
The observable Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix is the overlap of the unitary matrix Uð−1Þ that mixes the charged-lepton Yukawa matrix Yð−1Þ and a quasiunitary matrix USeesaw that diagonalizes the 3 × 3 Majorana matrix of the light neutrinos, Mð0Þ, i.e., UPMNS 1⁄4 Uð−1Þ†USeesaw: ð1Þ. In SUð5Þ, a symmetric downquark Yukawa matrix leads to small left-handed mixings of the charged leptons. Models based on an underlying family symmetry, where the SUð5Þ quintets and decuplets transform as the same representations of the group, can only single out symmetric and antisymmetric Yukawa matrices. In this work we propose an answer to the second question with a family symmetry (see [14] and the references therein) based on the discrete group T 13 1⁄4 Z13⋊Z3 [15] It explains the asymmetric term of the texture and yields the equality of the determinants of the matrices Yð−13Þ and Yð−1Þ, conforming to the down-quark to charged-lepton mass ratios at the GUT scale.
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