Generalized CP and $\Delta (3n^2)$ Family Symmetry for Semi-Direct Predictions of the PMNS Matrix

TL;DR

This paper explores how generalized CP transformations combined with $ abla(3n^2)$ family symmetry constrain lepton mixing, leading to specific trimaximal PMNS matrix patterns in a semi-direct approach.

Contribution

It provides a comprehensive analysis of automorphisms and CP transformations in $ abla(3n^2)$, identifying conditions for consistent CP definitions and deriving resulting lepton mixing patterns.

Findings

PMNS matrix is trimaximal for all CP transformations

Only two distinct PMNS forms are possible under these symmetries

Automorphisms depend on whether n is divisible by 3

Abstract

The generalized CP transformations can only be consistently defined in the context of $\Delta(3n^2)$ lepton symmetry if a certain subset of irreducible representations are present in a model. We perform a comprehensive analysis of the possible automorphisms and the corresponding CP transformations of the $\Delta(3n^2)$ group. It is sufficient to only consider three automorphisms if $n$ is not divisible by 3 while additional eight types of CP transformations could be imposed for the case of $n$ divisible by 3. We study the lepton mixing patterns which can be derived from the $\Delta(3n^2)$ family symmetry and generalized CP in the semi-direct approach. The PMNS matrix is determined to be the trimaximal pattern for all the possible CP transformations, and it can only take two distinct forms.