Vacuum Stability of a General Scalar Potential of a Few Fields

TL;DR

This paper derives analytical vacuum stability conditions for general scalar potentials involving multiple fields, including specific models like two Higgs doublets and scalar dark matter, with practical tools provided.

Contribution

It extends copositivity methods to more general scalar potentials and provides explicit stability conditions for several models, including tensor positivity and computational tools.

Findings

Derived analytical stability conditions for multi-field scalar potentials.

Provided explicit conditions for two Higgs doublet and scalar dark matter models.

Included a Mathematica notebook with stability criteria and tensor positivity overview.

Abstract

We find analytical vacuum stability or bounded below conditions for general scalar potentials of a few fields. After a brief review of copositivity we go beyond it. We discuss the vacuum stability conditions of the general potential of two real scalars, without and with the Higgs boson included in the potential. As further examples, we give explicit vacuum stability conditions for the two Higgs doublet model with no explicit CP breaking, and for the $\mathbb{Z}_{3}$ scalar dark matter with an inert doublet and a complex singlet. We give a short overview of positivity conditions for tensors of quartic couplings via tensor eigenvalues. A Mathematica notebook with the conditions is included with the source files.