# CP Violation from Finite Groups

## TL;DR

This paper classifies finite groups based on their automorphisms to understand how they can give rise to CP violation in models with discrete flavor symmetries, providing a group-theoretic framework for CP properties.

## Contribution

It introduces a classification scheme for finite groups using class-inverting automorphisms and the twisted Frobenius-Schur indicator to analyze CP violation origins in flavor models.

## Key findings

- Finite groups are categorized into three types based on their automorphisms and Clebsch-Gordan coefficients.
- Explicit examples of each group type are provided using elta(27), T', and 2.
- Certain generalized CP transformations do not correspond to physical CP conservation.

## Abstract

We discuss the origin of CP violation in settings with a discrete (flavor) symmetry $G$. We show that physical CP transformations always have to be class-inverting automorphisms of $G$. This allows us to categorize finite groups into three types: (i) Groups that do not exhibit such an automorphism and, therefore, in generic settings, explicitly violate CP. In settings based on such groups, CP violation can have pure group-theoretic origin and can be related to the complexity of some Clebsch-Gordan coefficients. (ii) Groups for which one can find a CP basis in which all the Clebsch-Gordan coefficients are real. For such groups, imposing CP invariance restricts the phases of coupling coefficients. (iii) Groups that do not admit real Clebsch-Gordan coefficients but possess a class-inverting automorphism that can be used to define a proper (generalized) CP transformation. For such groups, imposing CP invariance can lead to an additional symmetry that forbids certain couplings. We make use of the so-called twisted Frobenius-Schur indicator to distinguish between the three types of discrete groups. With $\Delta(27)$, $T^{\prime}$, and $\Sigma(72)$ we present one explicit example for each type of group, thereby illustrating the CP properties of models based on them. We also show that certain operations that have been dubbed generalized CP transformations in the recent literature do not lead to physical CP conservation.