# Spontaneous CP violation in multi-Higgs potentials with triplets of $\Delta(3n^2)$ and $\Delta(6n^2)$

## TL;DR

This paper investigates spontaneous CP violation in multi-Higgs models with triplet representations under discrete symmetries like $ ext{A}_4$, $ ext{S}_4$, $ ext{Δ}(27)$, and $ ext{Δ}(54)$, identifying conditions for CP violation and geometrical phases.

## Contribution

It provides a systematic analysis of SCPV in potentials with triplet Higgs fields under various discrete symmetries, including the identification of geometrical CP violation cases.

## Key findings

- Classified global minima for different parameter regions.
- Separated CP-violating and CP-conserving VEVs using CP-odd invariants.
- Identified cases of geometrical CP violation with calculable phases.

## Abstract

Motivated by discrete flavour symmetry models, we analyse Spontaneous CP Violation (SCPV) for potentials involving three or six Higgs fields (both electroweak doublets and singlets) which fall into irreducible triplet representations of discrete symmetries belonging to the $\Delta(3n^2)$ and $\Delta(6n^2)$ series, including $A_4$, $S_4$, $\Delta(27)$ and $\Delta(54)$. For each case, we give the potential and find various global mimima for different regions of the parameter space of the potential. Using CP-odd basis Invariants that indicate the presence of Spontaneous CP Violation we separate the VEVs into those that do or do not violate CP. In cases where CP is preserved we reveal a CP symmetry of the potential that is preserved by those VEVs, otherwise we display a non-zero CP-odd Invariant. Finally we identify interesting cases where there is Spontaneous Geometrical CP Violation in which the VEVs have calculable phases.