https://doi.org/10.1140/epjc/s10052-012-2093-z
Regular Article - Theoretical Physics
Vacuum stability conditions from copositivity criteria
1
Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, 56126, Pisa, Italy
2
National Institute of Chemical Physics and Biophysics, Rävala 10, Tallinn, Estonia
* e-mail: kannike@cern.ch
Received:
19
June
2012
Revised:
13
July
2012
Published online:
31
July
2012
A scalar potential of the form 
is bounded from below if its matrix of quartic couplings λ
ab
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 λ
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 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.
© Springer-Verlag / Società Italiana di Fisica, 2012
