This article is cited in 1 paper

Boundedness-below conditions for a general scalar potential of two real scalar fields and the Higgs boson

Yisheng Song^a, Liqun Qi<sup>bc

a</sup> School of Mathematical Sciences, Chongqing Normal University, Chongqing, China
^b Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, China
^c Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Abstract: The most general scalar potential of two real scalar fields and a Higgs boson is a quartic homogeneous polynomial in three variables, which defines a $4$th-order three-dimensional symmetric tensor. Hence, the boundedness of such a scalar potential from below involves the positive (semi-)definiteness of the corresponding tensor. In this paper, we therefore mainly discuss analytic expressions of positive (semi-)definiteness for such a special tensor. First, an analytically necessary and sufficient condition is given to test the positive (semi-)definiteness of a $4$th-order two-dimensional symmetric tensor. Furthermore, by means of such a result, the analytic necessary and sufficient conditions of the boundedness from below are obtained for a general scalar potential of two real scalar fields and the Higgs boson.

Keywords: scalar potentials, boundedness from below, 4th-order tensors, positive definiteness, homogeneous polynomial, analytic expression.

Received: 16.11.2023
Revised: 18.12.2023

DOI: 10.4213/tmf10645

 English version: Theoretical and Mathematical Physics, 2024, 220:3, 1567–1579

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