Tree Level Vacuum Stability in Multi Higgs Doublet Models: A Cumbersome Analysis?

We introduce a simple approach to obtain the constraint for multi Higgs doublet potentials to be stable, i. e. to have tree-level vacuum stability. We use the methodology for some special cases with three doublets. One of the most important open problems in the annals of particle physics is the nature of electroweak symmetry breaking (EWSB). This challenging question has received either theoretical or experimental attentions during the past decades and Many scenarios such as the Higgs bosons of low-energy supersymmetry, little Higgs models, extra-dimensional theories of EWSB and strongly-coupled EWSB sectors have been proposed to answer this controversial topic. An other elegant idea is the introduction of the so-called multi- or N-Higgs-doublet models (NHDMs). It is also quite well-known that the standard model (SM) of particle physics postulates the existence of one Higgs boson through the introduction of one Higgs doublet and thereby explains the nature of EWSB. However, having only one Higgs doublet is not much appealing since it can not account for several phenomena such as the hierarchy problem of Yukawa couplings in the third generation of quarks, neutrino oscillations, some flavor changing neutral currents etc. Apart from the fact that there is no deep theoretical reason for the existence of a minimal Higgs sector and also some failures of SM mentioned above, the CP violation problem has caused great attention toward the NHDMs since unlike the situation existing in SM, where the only source of CP violation comes from the complex phases in the Yukawa couplings that are transferred to the Cabibbo-Kobayashi-Maskawa (CKM) matrix after spontaneous EWSB, there are new sources of CP violation within NHDMs. Using basic mathematical properties, we have formulated a methodology to study the positivity constraint with NHDM potentials. Perhaps this is the simplest approach ever formulated. (author)