Vacuum stability conditions from copositivity criteria

A scalar potential of the form lambda(ab)phi(2)(a)phi(2)(b) 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 Z(2) symmetry. Copositivity criteria allow us 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 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.