# Vacuum Stability Conditions for Higgs Potentials with $SU(2)_L$ Triplets

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 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 but much simpler than the ones used in the literature.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03087658
Contributor : Gilbert Moultaka <>
Submitted on : Thursday, December 24, 2020 - 9:06:16 AM
Last modification on : Thursday, January 7, 2021 - 3:07:06 AM

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triplet-vacuum-stability.pdf
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### Identifiers

• HAL Id : hal-03087658, version 1
• ARXIV : 2012.13947

### Citation

Gilbert Moultaka, Michel Peyranère. Vacuum Stability Conditions for Higgs Potentials with $SU(2)_L$ Triplets. 2020. ⟨hal-03087658⟩

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