Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCopositivity of Three-Dimensional Symmetric Tensors
129
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic 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.
rate research
Read More
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.
In this paper, we mainly discuss the analytic expression of exact copositivity of 4th order symmetric tensor defined by the special physical model. We first show that for the general 4th order 2-dimensional symmetric tensor, it can be transformed into solving the quadratic polynomials, and then we give a necessary and sufficient condition to test the copositivity of 4th order 2-dimensional symmetric tensor. Based on this, we consider a special 4th order 3-dimensional symmetric tensor defined by the vacuum stability for $mathbb{Z}_{3}$ scalar dark matter, and obtain the necessary and sufficient condition for its copositivity.
In this paper, we present an eleven invariant isotropic irreducible function basis of a third order three-dimensional symmetric tensor. This irreducible function basis is a proper subset of the Olive-Auffray minimal isotropic integrity basis of that tensor. The octic invariant and a sextic invariant in the Olive-Auffray integrity basis are dropped out. This result is of significance to the further research of irreducible function bases of higher order tensors.
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.
In this paper, we mainly discuss analytical expressions of positive definiteness for a special 4th order 3-dimensional symmetric tensor defined by the constructed model for a physical phenomenon. Firstly, an analytically necessary and sufficient conditions of 4th order 2-dimensional symmetric tensors are given to test its positive definiteness. Furthermore, by means of such a result, a necessary and sufficient condition of positive definiteness is obtained for a special 4th order 3-dimensional symmetric tensor. Such an analytical conditions can be used for verifying the vacuum stability of general scalar potentials of two real singlet scalar fields and the Higgs boson. The positive semi-definiteness conclusions are presented too.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
