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Vacuum stability conditions from copositivity criteria

  • Kristjan KannikeEmail author
Regular Article - Theoretical Physics

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 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.

Keywords

Dark Matter Higgs Boson Grand Unify Theory Vacuum Stability Quartic Coupling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We thank Martti Raidal for comments and suggestions and Julia Polikarpus for consultation. This work was supported by the ESF grants 8090, 8943, MTT8, MTT60, MJD140 by the recurrent financing SF0690030s09 project and by the European Union through the European Regional Development Fund.

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Copyright information

© Springer-Verlag / Società Italiana di Fisica 2012

Authors and Affiliations

  1. 1.Scuola Normale Superiore and INFNPisaItaly
  2. 2.National Institute of Chemical Physics and BiophysicsTallinnEstonia

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