# Stitching an Asymmetric Texture with $\mathcal{T}_{13} \times \mathcal{Z}_5$ Family Symmetry

## TL;DR

This paper introduces a novel non-Abelian discrete family symmetry, $ ext{T}_{13}$, to explain asymmetric textures in fermion mass matrices, linking down-quark and charged-lepton masses without fine-tuning.

## Contribution

It proposes the $ ext{T}_{13}$ symmetry as the basis for asymmetric textures, providing a new framework for fermion mass relations in GUT-like models.

## Key findings

- $ ext{T}_{13}$ symmetry distinguishes Yukawa matrix elements.
- Model naturally relates down-quark and charged-lepton masses.
- No fine-tuning required for mass relations.

## 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.