Adding CP to flavour symmetries

TL;DR

This paper introduces CP-odd invariants as a basis-independent method to analyze CP violation in models with flavour symmetries, exemplified through $ ext{Delta}(27)$ invariant Lagrangians and a toy model.

Contribution

It proposes a novel approach using CP-odd invariants for studying CP in flavour symmetric theories, including explicit geometrical CP violation examples.

Findings

CP-odd invariants are basis-independent tools for CP analysis

Adding specific CP symmetries affects $ ext{Delta}(27)$ singlet couplings

A toy model demonstrates explicit geometrical CP violation with calculable phases

Abstract

I propose the use of CP-odd invariants, which are independent of basis and valid for any choice of CP transformation, as a powerful approach to study CP in the presence of flavour symmetries. As examples of the approach I focus on Lagrangians invariant under $\Delta(27)$. I comment on the consequences of adding a specific CP symmetry to a Lagrangian and distinguish cases where several $\Delta(27)$ singlets are present depending on how they couple to the triplets. One of the examples included is a very simple toy model with explicit CP violation with calculable phases, which is referred to as explicit geometrical CP violation by comparison with previously known cases of (spontaneous) geometrical CP violation.