# Spontaneous symmetry breaking in three-Higgs-doublet $S_3$-symmetric models

## TL;DR

This paper investigates the vacuum structure of three-Higgs-doublet models with $S_3$ symmetry, analyzing conditions for spontaneous symmetry breaking and CP violation, with implications for particle physics model building.

## Contribution

It provides a detailed analysis of vacua in $S_3$-symmetric three-Higgs-doublet models, identifying conditions for spontaneous CP violation and CP-conserving vacua.

## Key findings

- Spontaneous CP violation is possible in the model.
- Certain vacua preserve CP despite complex phases.
- Constraints from vacuum stability are established.

## Abstract

The talk summarises work done by the authors consisting of a detailed study of the possible vacua in models with three Higgs doublets with $S_3$ symmetry and without explicit CP violation. Different vacua require special regions of the parameter space which were analysed in our work. We establish the possibility of spontaneous CP violation in this framework and we also show which complex vacua conserve CP. In our work we discussed constraints from vacuum stability. The results presented here are relevant for model building.