Conditions for boundedness from below of a $\Delta(54)$-symmetric three-Higgs-doublet model

TL;DR

This paper analyzes the geometric structure of the scalar potential in a $ abla(54)$-symmetric three-Higgs-doublet model, proposing conditions for its boundedness from below based on orbit space properties.

Contribution

It introduces a geometric approach to determine boundedness conditions of the potential, including a conjecture supported by brute-force minimization.

Findings

Orbit space is a polytope if $CP$ invariance holds.

Non-$CP$ symmetric potential has a non-convex boundary in its orbit space.

Conjectured necessary and sufficient conditions for boundedness from below.

Abstract

We investigate the orbit space of the scalar potential of a $\Delta(54)$-symmetric three-Higgs-doublet model. We find that, if the potential enjoys $CP$ invariance, then its three-dimensional orbit space is a polytope; if the potential has no $CP$ symmetry, then its four-dimensional orbit space has a boundary that is sometimes slightly concave, but seems never to be convex. Consequently, we conjecture necessary and sufficient conditions for the potential to be bounded from below; brute-force minimization of a large number of potentials affirms the accuracy of our conjecture. We list all possible charge-conserving and charge-breaking minima of the potential.