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

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 increasingly stringent sets of 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.