# Vacuum Stability Conditions From Copositivity Criteria

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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 U1 symmetry.

Author: Kristjan Kannike

Source: https://archive.org/

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