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Copositivity for a class of fourth order symmetric tensors given by scalar dark matter

Yisheng Song, Xudong Li

Abstract

The mathematical model of multiple microscopic particles potentials corresponds to a fourth order symmetric tensor with a particular structure in particle physics. In this paper, we mainly dedicate to the study of copositivity for a class of tensors defined by the scalar dark matter with the standard model Higgs and an inert doublet and a complex singlet. With the help of its structure, we obtain the necessary and sufficient conditions, which attains the analytic conditions required by the physical problems. At the same time, this analytic expression provides how to determine a unique solution of the corresponding tensor complementarity problem with a parameter.

Copositivity for a class of fourth order symmetric tensors given by scalar dark matter

Abstract

The mathematical model of multiple microscopic particles potentials corresponds to a fourth order symmetric tensor with a particular structure in particle physics. In this paper, we mainly dedicate to the study of copositivity for a class of tensors defined by the scalar dark matter with the standard model Higgs and an inert doublet and a complex singlet. With the help of its structure, we obtain the necessary and sufficient conditions, which attains the analytic conditions required by the physical problems. At the same time, this analytic expression provides how to determine a unique solution of the corresponding tensor complementarity problem with a parameter.

Paper Structure

This paper contains 4 sections, 8 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Basic facts
  3. Main Results
  4. Conclusions

Key Result

lemma 1

A symmetric $2\times 2$ matrix $M=(m_{ij})$ is (strictly) copositive if and only if A symmetric $3\times 3$ matrix $M=(m_{ij})$ is (strictly) copositive if and only if

Theorems & Definitions (15)

  • lemma 1
  • lemma 2
  • lemma 3
  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • proof
  • theorem 4
  • ...and 5 more