Yet another lesson on the stability conditions in multi-Higgs potentials

TL;DR

This paper examines the impact of soft symmetry breaking on the bounded-from-below conditions in multi-Higgs models, revealing that previous conditions may be insufficient when such breaking is introduced, and provides exact criteria for specific models.

Contribution

It introduces a geometric formalism to derive exact BFB conditions for 3HDMs with $S_4$ symmetry, considering both exact and softly broken cases, highlighting limitations of prior results.

Findings

Existing BFB conditions are invalidated by soft symmetry breaking.

Derived exact BFB conditions for $S_4$-symmetric 3HDM.

Reviewed BFB conditions for $A_4$-symmetric models.

Abstract

We discuss a rather common but often unnoticed pitfall which arises when deriving the bounded-from-below (BFB) conditions in multi-Higgs models with softly broken global symmetries. Namely, necessary and sufficient BFB conditions derived for the case with an exact symmetry can be ruined by introducing soft symmetry breaking terms. Using $S_4$ and $A_4$-symmetric three-Higgs-doublet models as an example, we argue that all published necessary and sufficient BFB conditions, even those which are correct for the exactly symmetric case, are no longer sufficient if soft symmetry breaking is added. Using the geometric formalism, we derive the exact necessary and sufficient BFB conditions for the 3HDM with the symmetry group $S_4$, either exact or softly broken, and review the situation for the $A_4$-symmetric case.