o-positivity of tensors and boundedness from below conditions of CP conserving two-Higgs-doublet potential

In this paper, the analytic sufficient and necessary conditions are obtained for the CP conserving two-Higgs-doublet  potential to be bounded from below  by using the co-positivity of  tensors. This is achieved by treating the potential as a  quartic homogeneous polynomial about the moduli of the two Higgs doublet fields,  where the ‘angles’ is described as the misalignment of the two doublets, then solving three minimum problems with respect to the misalignment.   Finally,  the analytic conditions are established with the help of the corresponding theory and methods of higher order tensors.