Family symmetries and CP

TL;DR

This paper explores how CP-odd invariants can be used to analyze CP properties in models with family symmetries, focusing on $ abla(27)$, and investigates explicit and automatic CP conservation or violation.

Contribution

It systematically applies the invariant approach to Yukawa-like Lagrangians with $ abla(27)$ symmetry, including models with explicit geometrical CP violation and automatic CP conservation.

Findings

Identified conditions for explicit geometrical CP violation.

Demonstrated models with automatic CP conservation despite $ abla(27)$ symmetry.

Provided a systematic framework for analyzing CP in family symmetry models.

Abstract

CP-odd invariants, independent of basis and valid for any choice of CP transformation are a powerful tool in the study of CP. They are particularly convenient to study the CP properties of models with family symmetries. After interpreting the consequences of adding specific CP symmetries to a Lagrangian invariant under $\Delta(27)$, I use the invariant approach to systematically study Yukawa-like Lagrangians with an increasing field content in terms of $\Delta(27)$ representations. Included in the Lagrangians studied are models featuring explicit CP violation with calculable phases (referred to as explicit geometrical CP violation) and models that automatically conserve CP, despite having all the $\Delta(27)$ representations.