Copositivity of Three-Dimensional Symmetric Tensors

TL;DR

This paper develops an analytical method to determine copositivity of three-dimensional symmetric tensors, including applications to scalar dark matter models, by solving quartic and quadratic equations explicitly.

Contribution

It provides the first analytical necessary and sufficient conditions for copositivity of 3D symmetric tensors, applicable to physical models like dark matter stability.

Findings

Analytical solutions for copositivity conditions of third order tensors.

Explicit criteria for vacuum stability in a dark matter model.

Application of classical mathematical results to tensor copositivity.

Abstract

In this paper, we seek analytically checkable necessary and sufficient condition for copositivity of a three-dimensional symmetric tensor. We first show that for a general third order three-dimensional symmetric tensor, this means to solve a quartic equation and some quadratic equations. All of them can be solved analytically. Thus, we present an analytical way to check copositivity of a third order three dimensional symmetric tensor. Then, we consider a model of vacuum stability for $\mathbb{Z}_3$ scalar dark matter. This is a special fourth order three-dimensional symmetric tensor. We show that an analytically expressed necessary and sufficient condition for this model bounded from below can be given, by using a result given by Ulrich and Watson in 1994.