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

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.

arXiv comment26 pages, 4 figures

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Cited by in corpus (11)

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• A viable $A_4$ 3HDM theory of quark mass matrices
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• Scalar Lie point symmetries of the Standard Model with one or two real gauge singlets
• Vacuum Stability Conditions for New $SU(2)$ Multiplets