Algorithmic Boundedness-From-Below Conditions for Generic Scalar Potentials

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 condi...

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Bibliographic Details
Published inarXiv.org
Main Authors Ivanov, Igor P, Köpke, Marcel, Muhlleitner, Margarete
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 04.03.2018
Subjects
Algorithms
Mathematical analysis
Physics - High Energy Physics - Phenomenology
Physics - High Energy Physics - Theory
Polynomials
Resultants
Spectral theory
Tensors
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1802.07976