Vacuum Stability Conditions From Copositivity Criteria

Abstract

A scalar potential of the form ${\lambda }_{ab}{\varphi }_a^2{\varphi }_b^2$ 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.

Comments: 15 pages, 1 figure, complex singlet in polar form clarified