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.
