Analytical expressions of copositivity for 4th order symmetric tensors and applications

Abstract

In particle physics, scalar potentials have to be bounded from below in order for the physics to make sense. The precise expressions of checking lower bound of scalar potentials are essential, which is an analytical expression of checking copositivity and positive definiteness of tensors given by such scalar potentials. Because the tensors given by general scalar potential are 4th order and symmetric, our work mainly focuses on finding precise expressions to test copositivity and positive definiteness of 4th order tensors in this paper. First of all, an analytically sufficient and necessary condition of positive definiteness is provided for 4th order 2 dimensional symmetric tensors. For 4th order 3 dimensional symmetric tensors, we give two analytically sufficient conditions of (strictly) cpositivity by using proof technique of reducing orders or dimensions of such a tensor. Furthermore, an analytically sufficient and necessary condition of copositivity is showed for 4th order 2 dimensional symmetric tensors. We also give several distinctly analytically sufficient conditions of (strict) copositivity for 4th order 2 dimensional symmetric tensors. Finally, we apply these results to check lower bound of scalar potentials, and to present analytical vacuum stability conditions for potentials of two real scalar fields and the Higgs boson.

30 pages

References in corpus (3)

• Minimal semi-annihilating ${\mathbb{Z}}_N$ scalar dark matter
• Boundedness from below in the $U(1)\times U(1)$ three-Higgs-Doublet model
• Polynomial complementarity problems

Cited by in corpus (7)

• Copositivity for 3rd order symmetric tensors and applications
• Vacuum stability conditions of the general two-Higgs-doublet potential
• Positivity Conditions for Cubic, Quartic and Quintic Polynomials
• Copositivity for a class of fourth order symmetric tensors given by scalar dark matter
• Co-positivity of tensors and bounded from below conditions of CP conserving two-Higgs-doublet potential
• Copositivity of Three-Dimensional Symmetric Tensors
• Boundedness from below of SU(n) potentials