Connect discreteness to continuousness in leptonic flavor symmetries

TL;DR

This paper introduces a novel mathematical framework using group-algebra to unify discrete and continuous leptonic flavor symmetries, classifies elements for S4, and explores their phenomenological implications.

Contribution

It presents a new approach employing group-algebra to connect discrete and continuous symmetries in leptonic flavor models, with a comprehensive classification of S4 GA elements.

Findings

Classified 198 nontrivial S4 group-algebra elements.

Established equivalence between GA elements and various symmetries.

Discussed phenomenological consequences of S4 GA elements.

Abstract

Discrete groups are widely used in the expression of flavor symmetries of leptons. In this paper, we employ a novel mathematical object called group-algebra (GA) to describe symmetries of the leptonic mass matrices. A GA element is constructed by a discrete group with continuous parameters. For a GA element, there is an equivalent symmetry which can be continuous, discrete, and hybrid. According to the equivalence between a GA element and other symmetries, we perform a classification of 198 nontrivial elements of the GA generated by the group $S_{4}$. Based on the results of the classification, the phenomenological consequences of the $S_{4}$ GA are illustrated.