Vacuum Stability Conditions From Copositivity Criteria

TL;DR

The authors address vacuum stability for scalar potentials whose quartic part is a biquadratic form $V_4=\lambda_{ab}\varphi_a^2\varphi_b^2$, showing that stability as field values become large reduces to the copositivity of $\lambda_{ab}$. They develop and apply analytic copositivity criteria to derive necessary and sufficient conditions for stability in several models: the inert doublet (Z2-stabilised DM), a complex singlet DM, and a complex singlet plus inert doublet model with a global U(1) symmetry. The method yields explicit inequalities on the quartic couplings, often more permissive than positivity-based bounds, and they discuss practical approaches for larger matrices (e.g., CHL theorem). Overall, the work provides a rigorous, algebraic framework to guarantee vacuum stability in extended scalar sectors relevant for DM and beyond. The results offer a valuable tool for model-building and phenomenology by delivering closed-form stability constraints that practitioners can implement directly.

Abstract

A scalar potential of the form $λ_{ab} φ_a^2 φ_b^2$ 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 $\mathbb{Z}_2$ symmetry. Copositivity criteria allow 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 $\mathbb{Z}_2$ complex singlet dark matter, and for the model with both a complex singlet and an inert doublet invariant under a global U(1) symmetry.

Figures (1)

• Figure 1: Allowed parameter space for the quartic self-couplings of $S$. Area with dashed border: the parameter space allowed for $\lambda"_S = 0$. Only $\lambda'_S \geqslant 0$ is allowed by positivity (light red), while copositivity also includes the area below zero (light green). Area with solid border: the parameter space allowed by positivity (red); the constraint from copositivity (the whole area) for $\lambda"_S = 1/2$. In the transparent grey area \ref{['eq:arccos:cond']} and thus \ref{['eq:S:self-couplings:copos:min']} do not hold; if we did not take that into account, the green sliver would be erroneously excluded.