Invariant approach to CP in unbroken $\Delta(27)$

TL;DR

This paper applies the invariant approach to analyze CP violation in unbroken (27) symmetric Lagrangians, classifying cases based on field representations and constructing invariants to study CP properties.

Contribution

It introduces a systematic classification of (27) invariant Lagrangians and constructs CP-odd invariants to analyze CP violation depending on field representations.

Findings

CP violation depends on the number of triplet fields

Constructed CP-odd invariants for different cases

Identified conditions for CP conservation and violation

Abstract

The invariant approach is a powerful method for studying CP violation for specific Lagrangians. The method is particularly useful for dealing with discrete family symmetries. We focus on the CP properties of unbroken $\Delta(27)$ invariant Lagrangians with Yukawa-like terms, which proves to be a rich framework, with distinct aspects of CP, making it an ideal group to investigate with the invariant approach. We classify Lagrangians depending on the number of fields transforming as irreducible triplet representations of $\Delta(27)$. For each case, we construct CP-odd weak basis invariants and use them to discuss the respective CP properties. We find that CP violation is sensitive to the number and type of $\Delta(27)$ representations.