  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.
