Vacuum stability conditions of the general two-Higgs-doublet potential

In this paper, we present the novel analytical expressions for the bounded-from-below or the vacuum stability conditions of scalar potential for a general CP violating two-Higgs-doublet model by using the concepts of co-positivity and the gauge orbit spaces. More precisely, several analyticial sufficient conditions and necessary conditions are established for the vacuum stability of the general 2HDM potential, respectively. We also give an equivalent condition of the vacuum stability of the general 2HDM potential in theory, and then, apply it to derive the analyticial necessary conditions of the general 2HDM potential. Meanwhile, the positive semi-definiteness is proved for a class of 4th-order 2-dimensional complex tensor.