Vacuum Stability Conditions From Copositivity Criteria

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

• The paper introduces an analytic framework using copositivity criteria to derive necessary and sufficient vacuum stability conditions for scalar potentials.
• It applies the method to models like the inert doublet and Z2-symmetric complex singlet, confirming and extending known stability conditions.
• The analysis broadens the parameter space for BSM models, offering practical insights for designing and testing dark matter scenarios.

Vacuum Stability Conditions From Copositivity Criteria

The paper "Vacuum Stability Conditions From Copositivity Criteria" by Kristjan Kannike explores the application of copositivity criteria to establish vacuum stability conditions for scalar potentials. The scalar potential under consideration is of a specific type, $\lambda_{ab} \varphi_{a}^{2} \varphi_{b}^{2}$, which is relevant in models involving scalar dark matter stabilized by a $Z_{2}$ symmetry. The study underscores the relevance of copositivity in determining the conditions under which a scalar potential remains positive, thereby ensuring vacuum stability in particle physics models.

Fundamental Insights and Methodology

The work hinges on the concept of copositivity of matrices. A matrix $\lambda_{ab}$ representing quartic couplings in a scalar potential is defined to be copositive if the associated quadratic form is non-negative for all non-negative vectors. Recognizing vacuum stability as a quintessential requirement in the formulation of particle physics models, especially those extending beyond the Standard Model (BSM), the paper provides an analytic framework to derive necessary and sufficient conditions for a scalar potential to be bounded from below.

The core methodology involves leveraging the copositivity criteria, which provides a more extensive parameter space than the classical positive definiteness approach. Copositivity conditions for matrices are thoroughly reviewed, including those for $2 \times 2$ and $3 \times 3$ matrices, as well as general criteria applicable to larger matrices, leveraging the Cottle-Habetler-Lemke theorem.

The paper applies these mathematical formulations to re-derive known vacuum stability conditions for the inert doublet model and extends the analysis to models including complex singlet dark matter and combinations of singlets and inert doublets under specific symmetry conditions.

Numerical Results and Theoretical Implications

The paper explicitly derives the vacuum stability conditions for several models. For instance, in the inert doublet model, the author confirms known conditions by copositively analyzing a matrix of quartic couplings. Additionally, more complex scenarios involving a $Z_{2}$-symmetric complex singlet are considered, with the copositivity criteria providing conditions that are both necessary and sufficient for the boundedness of the scalar potential.

These results are not trivial, as demonstrated by conditions that vary based on the existence of symmetries in the potential, such as a global $U(1)$ symmetry. Copositivity, in these models, yields results that potentially expand the parameter space previously considered viable under positivity constraints alone.

Implications and Future Prospects

The analysis in this paper provides a crucial stepping stone for future research in both theoretical and phenomenological particle physics. By showcasing the utility of copositivity criteria in deriving vacuum stability conditions, this work paves the way for more nuanced investigations into BSM physics, particularly in scenarios involving scalar fields such as those anticipated in dark matter models.

From a theoretical perspective, this framework could be further extended to other classes of potentials or models which include additional fields and symmetries. Practically, this could imply more relaxed conditions for model parameters, possibly simplifying experimental searches by broadening the spectrum of permissible model configurations.

Conclusion

Kristjan Kannike's paper presents a significant contribution to the field of particle physics by applying copositivity criteria to the problem of vacuum stability, ensuring analytic determination of potential stability requirements. This approach not only reaffirms existing models under new criteria but also furnishes the research community with an expanded toolkit for exploring the parameter landscapes of more intricate BSM theories. As the field progresses towards understanding the dark sector and other particle dynamics, the insights provided here will likely inform both theoretical inquiry and experimental design in the arena of high-energy physics.