Algorithmic Boundedness-From-Below Conditions for Generic Scalar Potentials

Abstract

Checking that a scalar potential is bounded from below (BFB) is an ubiquitous and notoriously difficult task in many models with extended scalar sectors. Exact analytic BFB conditions are known only in simple cases. In this work, we present a novel approach to algorithmically establish the BFB conditions for any polynomial scalar potential. The method relies on elements of multivariate algebra, in particular, on resultants and on the spectral theory of tensors, which is being developed by the mathematical community. We give first a pedagogical introduction to this approach, illustrate it with elementary examples, and then present the working Mathematica implementation publicly available at GitHub. Due to the rapidly increasing complexity of the problem, we have not yet produced ready-to-use analytical BFB conditions for new multi-scalar cases. But we are confident that the present implementation can be dramatically improved and may eventually lead to such results.

arXiv comment27 pages, 2 figures; v2: added references

References in corpus (21)

• Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC
• Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC
• Vacuum Stability Conditions From Copositivity Criteria
• Stability and Symmetry Breaking in the General Two-Higgs-Doublet Model
• Strong First Order Electroweak Phase Transition in the CP-Conserving 2HDM Revisited
• Building and testing models with extended Higgs sectors
• Vacuum Stability of a General Scalar Potential of a Few Fields
• The CP-Violating 2HDM in Light of a Strong First Order Electroweak Phase Transition and Implications for Higgs Pair Production
• BSMPT - Beyond the Standard Model Phase Transitions -A Tool for the Electroweak Phase Transition in Extended Higgs Sectors
• Two-Higgs-doublet model from the group-theoretic perspective
• Copositive Criteria and Boundedness of the Scalar Potential
• Constraining Flavour Symmetries At The EW Scale I: The A4 Higgs Potential
• Structure of potentials with $N$ Higgs doublets
• Geometric minimization of highly symmetric potentials
• Properties of the general NHDM. II. Higgs potential and its symmetries
• CP4 miracle: shaping Yukawa sector with CP symmetry of order four
• Single-scale Renormalisation Group Improvement of Multi-scale Effective Potentials
• Stability and symmetry breaking in the general n-Higgs-doublet model
• The Multivariate Resultant is NP-hard in any Characteristic
• Vacuum Stability and Perturbativity of SU(3) Scalars
• Stability and symmetry breaking in a three Higgs-boson doublet model with lepton family symmetry O(2)xZ2

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• Analytical expressions of copositivity for 4th order symmetric tensors and applications
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• Theoretical constraints on the Higgs potential of the general $331$ model
• Resonant top pair searches at the LHC: a window to electroweak phase transition
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