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 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.

Bounded-from-below conditions for $A_4$-symmetric 3HDM

“…just as stated in [16]. In this work we proved that these conditions are necessary and sufficient BFB conditions in the neutral Higgs subspace.…”

“…In summary, we proved in this paper that the inequalities (43) conjectured in [16] indeed represent the necessary and sufficient bounded-from-below conditions for the A 4 symmetric 3HDM, at least within the neutral Higgs space. We could achieve this result with the aid of a novel technique, in which we used an auxiliary function ϕ defined in (25) and related to the Higgs potential.…”

“…Our task is to find the necessary and sufficient conditions on the parameters a, b, d which guarantee that V 4 ≥ 0 everywhere in C 3 (the BFB conditions). For real d, the full set of BFB conditions was recently established [16,27]:…”

“…However, in particular versions of the 3HDM equipped with additional global symmetries, the structure of the Higgs potential simplifies, and in certain cases the necessary and sufficient BFB conditions are also known. These examples include the 3HDM with the U(1) × U(1) symmetry group [13], where the copositivity methods are at work [14,15], and the S 4 -symmetric 3HDM [16].…”

Boundedness from below of SU(n) potentials

“…Even so, deriving the BFB conditions quickly becomes a very complicated problem as the number of scalar fields increases, so much so that in the literature one can find the derivation of these conditions for just a few models. Among the cases which were considered there is the two Higgs doublet model [1], the type-II seesaw potential with the Higgs doublet plus an SU (2) triplet [2,3], special three Higgs doublet models [4][5][6] and also an SU (3) invariant potential with three triplets [7]. Several other works have analyzed the vacuum stability of specific models or discussed general techniques for doing so [8][9][10][11].…”