Vacuum Stability Conditions From Copositivity Criteria

TL;DR

This paper reviews copositivity criteria to derive analytic vacuum stability conditions for scalar potentials with quartic couplings, applying them to models with dark matter candidates and symmetries.

Contribution

It provides a systematic review of copositivity criteria and applies them to derive vacuum stability conditions for various dark matter models.

Findings

Derived analytic stability conditions for the inert doublet model.

Established stability criteria for the $ ext{Z}_2$ complex singlet dark matter.

Extended criteria to models with a complex singlet and inert doublet with U(1) symmetry.

Abstract

A scalar potential of the form $\lambda_{ab} \phi_a^2 \phi_b^2$ is bounded from below if its matrix of quartic couplings $\lambda_{ab}$ is copositive -- positive on non-negative vectors. Scalar potentials of this form occur naturally for scalar dark matter stabilised by a $\mathbb{Z}_2$ symmetry. Copositivity criteria allow to derive analytic necessary and sufficient vacuum stability conditions for the matrix $\lambda_{ab}$. We review the basic properties of copositive matrices and analytic criteria for copositivity. To illustrate these, we re-derive the vacuum stability conditions for the inert doublet model in a simple way, and derive the vacuum stability conditions for the $\mathbb{Z}_2$ complex singlet dark matter, and for the model with both a complex singlet and an inert doublet invariant under a global U(1) symmetry.