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

TL;DR

This paper derives analytical conditions to determine when a general scalar potential involving two real scalars and the Higgs boson is bounded from below, using tensor positivity criteria.

Contribution

It provides necessary and sufficient conditions for the positive definiteness of a specific 4th order 3-dimensional symmetric tensor relevant to scalar potentials.

Findings

Analytical expression for positive definiteness of 4th order 2D symmetric tensors.

Necessary and sufficient conditions for scalar potential boundedness.

Application to Higgs-related scalar potentials.

Abstract

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