Assessing boundedness from below in the $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetric three-Higgs-doublet model: algorithm and machine learning

TL;DR

This paper introduces a new algorithm and machine learning approach to determine if scalar potentials in a specific three-Higgs-doublet model are bounded from below, enhancing precision and efficiency in stability analysis.

Contribution

It presents a Mathematica code for systematically applying necessary conditions for boundedness and a machine learning model achieving over 99% accuracy in classifying potentials as BFB.

Findings

The code can be configured for different accuracy levels, balancing precision and computational time.

The machine learning model reliably identifies BFB potentials with over 99% accuracy.

The approach improves the precision and speed of stability analysis in complex scalar potentials.

Abstract

The scalar potential of any particle-physics model must be bounded from below (BFB). We consider the extension of the Standard electroweak Model with three $SU(2)$ doublets of scalars and a symmetry under which each of those doublets changes sign. In the absence of necessary and sufficient conditions for boundedness from below (BnessFB) for this specific model, we argue that one may use ever more necessary conditions. We introduce a Mathematica code, StableWein, that implements this idea. The user is allowed to choose the level of accuracy that they want in the determination of BnessFB; more precision means the use of more necessary conditions, and usually entails a longer running time for the code. Our investigation suggests that our procedure and code can be extremely precise in the determination of the potentials that are BFB. In addition, we introduce a machine-learning code that identifies, with more than 99% accuracy, which potentials are BFB.