Copositivity for 3rd order symmetric tensors and applications

TL;DR

This paper develops analytical conditions for copositivity of third and fourth order symmetric tensors, with applications to vacuum stability in scalar dark matter models, advancing theoretical understanding in tensor analysis and particle physics.

Contribution

It introduces new analytical sufficient conditions for copositivity of 3rd and 4th order symmetric tensors, facilitating stability analysis in physics models.

Findings

Derived sufficient conditions for copositivity of 3rd order tensors in 2D and 3D.

Extended these conditions to 4th order tensors in 2D and 3D.

Applied results to establish vacuum stability criteria for $\ ext{Z}_3$ scalar dark matter.

Abstract

The strict opositivity of 4th order symmetric tensor may apply to detect vacuum stability of general scalar potential. For finding analytical expressions of (strict) opositivity of 4th order symmetric tensor, we may reduce its order to 3rd order to better deal with it. So, it is provided that several analytically sufficient conditions for the copositivity of 3th order 2 dimensional (3 dimensional) symmetric tensors. Subsequently, applying these conclusions to 4th order tensors, the analytically sufficient conditions of copositivity are proved for 4th order 2 dimensional and 3 dimensional symmetric tensors. Finally, we apply these results to present analytical vacuum stability conditions for vacuum stability for $\mathbb{Z}_3$ scalar dark matter.