Abstract
Tree-level dynamical stability of scalar field potentials in renormalizable theories can in principle be expressed in terms of positivity conditions on quartic polynomial structures. However, these conditions cannot always be cast in a fully analytical resolved form, involving only the couplings and being valid for all field directions. In this paper we consider such forms in three physically motivated models involving $SU(2)$ triplet scalar fields: the Type-II seesaw model, the Georgi-Machacek model, and a generalized two-triplet model. A detailed analysis of the latter model allows one to establish the full set of necessary and sufficient boundedness-from-below conditions. These can serve as a guide, together with unitarity and vacuum structure constraints, for consistent phenomenological (tree-level) studies. They also provide a seed for improved loop-level conditions and encompass in particular the leading ones for the more specific Georgi-Machacek case. Incidentally, we present complete proofs of various properties and also derive general positivity conditions on quartic polynomials that are equivalent to but much simpler than the ones used in the literature.
Highlights
Since the experimental discovery of a Standard Model (SM)–like Higgs particle at the LHC [1,2] and the lack so far of any direct evidence for physics beyond the standard model (BSM),1 one might ask whether the properties of the discovered 125 GeV scalar particle so close to the SM predictions leave any room for BSM physics to reside below the tera-electron-volt or at the nearby few tera-electron-volts scale
These constraints should eventually be studied for the general precustodial model and be combined with the necessary and sufficient (NAS) bounded from below (BFB) conditions derived in this paper
In the course of the study we were led to review some of the known results for the Type-II seesaw and Georgi-Machacek models providing complete proofs that were missing in the literature for key properties
Summary
Since the experimental discovery of a Standard Model (SM)–like Higgs particle at the LHC [1,2] and the lack so far of any direct evidence for physics beyond the standard model (BSM), one might ask whether the properties of the discovered 125 GeV scalar particle so close to the SM predictions (see, e.g., [4]) leave any room for BSM physics to reside below the tera-electron-volt or at the nearby few tera-electron-volts scale. To the best of our knowledge a model with one triplet has been treated using copositivity [41] but for which only specific directions in field space were considered in agreement with [31], while the authors of [45] obtained with this method the all-directions conditions for the Type-II seesaw model in a form different from that of [31] It should, be stressed that the copositivity method cannot always be applied to potentials with extended Higgs multiplets when all four-dimensional operators allowed by the gauge symmetries and renormalizability are considered. In Appendixes G and H, to the mathematical issue of deriving simple forms for the NAS positivity conditions of quartic polynomials
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