# Recognizing symmetries in 3HDM in basis-independent way

## TL;DR

This paper develops a basis-independent method to identify symmetries in the three-Higgs-doublet model (3HDM), enabling more efficient exploration of its parameter space and understanding of its phenomenology.

## Contribution

It introduces a formalism with basis-independent conditions to recognize symmetries in 3HDM, addressing a key technical challenge in the field.

## Key findings

- Derived necessary and sufficient basis-independent conditions for symmetry recognition.
- Formalism applicable to all realizable symmetry groups in 3HDM.
- Facilitates more efficient phenomenological analysis of 3HDM.

## Abstract

Higgs doublets may come in three generations. The scalar sector of the resulting three-Higgs-doublet model (3HDM) may be constrained by global symmetry groups $G$ leading to characteristic phenomenology. There exists the full list of symmetry groups $G$ realizable in the 3HDM scalar sector and the expressions for $G$-symmetric scalar potentials written in special bases where the generators of $G$ take simple form. However recognizing the presence of a symmetry in a generic basis remains a major technical challenge, which impedes efficient exploration of the 3HDM parameter space. In this paper, we solve this problem using the recently proposed approach, in which basis-independent conditions are formulated as relations among basis-covariant objects. We develop the formalism and derive basis-independent necessary and sufficient conditions for the 3HDM scalar sector to be invariant under each of the realizable symmetry group. We also comment on phenomenological consequences of these results.