Abstract
We find analytical vacuum stability or bounded below conditions for general scalar potentials of a few fields. After a brief review of copositivity we go beyond it. We discuss the vacuum stability conditions of the general potential of two real scalars, without and with the Higgs boson included in the potential. As further examples, we give explicit vacuum stability conditions for the two Higgs doublet model with no explicit CP breaking, and for the $\mathbb{Z}_{3}$ scalar dark matter with an inert doublet and a complex singlet. We give a short overview of positivity conditions for tensors of quartic couplings via tensor eigenvalues. A Mathematica notebook with the conditions is included with the source files.
Highlights
A scalar potential has to be bounded from below to make physical sense
We discuss the vacuum stability conditions of the general potential of two real scalars, without and with the Higgs boson included in the potential
Scalar potentials have to be bounded from below in order for the physics to make sense. Finding such conditions is a hard problem of algebraic geometry
Summary
A scalar potential has to be bounded from below to make physical sense. In the Standard Model (SM), it means that the self-coupling of the Higgs boson has to be positive. The new addition to the scalar sector consists of just a couple of real scalar singlets, often in the guise of a complex singlet In this case the vacuum stability reduces to the problem of positivity of a general quartic polynomial. Similar conditions can be derived e.g. for the 2HDM, where the potential can be considered to be a quartic polynomial in magnitudes of fields, or for more complicated models, such as the Z3 scalar dark matter [22,23] with an inert doublet and a complex singlet. 7 we introduce tensor eigenvalues as a way to determine the vacuum stability conditions for a most general scalar potential.
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