A novel mathematical construct for the family of leptonic mixing patterns

TL;DR

This paper introduces a new mathematical construct based on finite group theory to generate leptonic mixing patterns consistent with experimental data, particularly for Dirac neutrinos, using a hybrid element from the group algebra.

Contribution

It proposes a novel group-algebra based construct combining two group elements with a parameter, providing a new approach to model leptonic mixing patterns.

Findings

Construct can reduce to cyclic group generators when the parameter is rational.

Example based on group S4 demonstrates viability for Dirac neutrino mixing patterns.

Infinite cyclic groups can produce realistic leptonic mixing patterns.

Abstract

In order to induce a family of mixing patterns of leptons which accommodate the experimental data with a simple mathematical construct, we construct a novel object from the hybrid of two elements of a finite group with a parameter $\theta$. This construct is an element of a mathematical structure called group-algebra. It could reduce to a generator of a cyclic group if $\theta/2\pi$ is a rational number. We discuss a specific example on the base of the group $S_{4}$. This example shows that infinite cyclic groups could give the viable mixing patterns for Dirac neutrinos.