# Boundedness from below in the $U(1)\times U(1)$ three-Higgs-Doublet model

## TL;DR

This paper derives the necessary and sufficient conditions for the potential of a specific three-Higgs-doublet model with $U(1) imes U(1)$ symmetry to be bounded from below, highlighting the importance of charge-breaking directions in the analysis.

## Contribution

It provides a comprehensive method to determine BFB conditions for the 3HDM with $U(1) imes U(1)$ symmetry, including charge-breaking directions, and revises previous approaches for Weinberg's model.

## Key findings

- Derived necessary and sufficient BFB conditions for the 3HDM with $U(1) imes U(1)$ symmetry.
- Charge-breaking directions are crucial in BFB analysis, even with a neutral minimum.
- Revealed limitations of a 2009 method for Weinberg's 3HDM.

## Abstract

Establishing if multi-Higgs potentials are bounded from below (BFB) can be rather challenging, and it may impede efficient investigation of all phenomenological consequences of such models. In this paper, we find the necessary and sufficient BFB conditions for the Three-Higgs-Doublet model (3HDM) with the global symmetry group $U(1)\times U(1)$. We observed an important role played by charge-breaking directions in the Higgs space, even for situations when a good-looking neutral minimum exists. This remark is not limited to the particular model we consider but represents a rather general feature of elaborate multi-Higgs potentials which must be carefully dealt with. Also, applying this method to Weinberg's model (the $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetric 3HDM) turned out to be more challenging than was believed in the literature. In particular, we have found that the approach taken in a paper from 2009 does not lead to the necessary and sufficient BFB conditions for this case.