Invariant approach to CP in family symmetry models

TL;DR

This paper introduces a basis invariant method to analyze CP violation in models with discrete family symmetries, providing a unified and elegant framework applicable to various symmetry groups.

Contribution

It develops a basis invariant approach for studying CP violation, applicable to any CP transformation, and demonstrates its effectiveness with $A_4$ and $ Delta(27)$ symmetry models.

Findings

Reproduces known results for $A_4$ models.

Identifies geometrical CP violation in $ Delta(27)$ models.

Shows explicit CP violation persists regardless of couplings.

Abstract

We propose the use of basis invariants, valid for any choice of CP transformation, as a powerful approach to studying specific models of CP violation in the presence of discrete family symmetries. We illustrate the virtues of this approach for examples based on $A_4$ and $\Delta(27)$ family symmetries. For $A_4$, we show how to elegantly obtain several known results in the literature. In $\Delta(27)$ we use the invariant approach to identify how explicit (rather than spontaneous) CP violation arises, which is geometrical in nature, i.e. persisting for arbitrary couplings in the Lagrangian.