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 S4 and A4-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 S4, either exact or softly broken, and review the situation for the A4-symmetric case.
Highlights
Theoretical search for New Physics beyond the Standard Model (SM) is driven, in the absence of direct experimental indications, by “educated guesses”
Using S4 and A4-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
The quadratic part of this model is extremely simple due to the group-theoretic arguments: if all Higgs doublets transform as an irreducible representation of the chosen symmetry group, one can construct only one quadratic group invariant respecting gauge symmetries
Summary
Theoretical search for New Physics beyond the Standard Model (SM) is driven, in the absence of direct experimental indications, by “educated guesses”. There is strong evidence that in the absence of New Physics all the way up to the Planck scale, the Standard Model effective Higgs potential goes deeper than the electroweak vacuum around the scale of Λ ∼ 1010 GeV [16], either developing a second, deeper minimum at transplanckian scales or even remaining unbounded from below. Such cases are still based on tree level potentials which are completely stable; metastability occurs at higher values of the scalar fields in effective potentials. Several appendices provide details of the calculations in support of statements made in the main text
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