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The most general scalar potential of two real scalar fields and a Higgs bosonis a quartic homogeneous polynomial about 3 variables, which defines a 4thorder 3 dimensional symmetric tensor. Hence, the boundedness from below of sucha scalar potential involves the positive (semi-)definiteness of thecorresponding tensor. So, we mainly discuss analytical expressions of positive(semi-)definiteness for such a special 4th order 3-dimension symmetric tensorin this paper. Firstly, an analytically necessary and sufficient condition isgiven to test the positive (semi-)definiteness of a 4th order 2 dimensionalsymmetric tensor. Furthermore, by means of such a result, the necessary andsufficient conditions of the boundedness from below are obtained for a generalscalar potential of two real scalar fields and the Higgs boson.

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