Vacuum stability conditions from copositivity criteria

Abstract

A scalar potential of the form λab φa2φa2 is bounded from below if its matrix of quartic couplings λab is copositive-positive on non-negative vectors. Scalar potentials of this form occur naturally for scalar dark matter stabilised by a ℤ2 symmetry. Copositivity criteria allow us to derive analytic necessary and sufficient vacuum stability conditions for the matrix λab. 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 ℤ2 complex singlet dark matter, and for the model with both a complex singlet and an inert doublet invariant under a global U(1) symmetry. © 2012 Springer-Verlag / Società Italiana di Fisica.