The vacuum structure of the Higgs complex singlet-doublet model
  The complex singlet-doublet model is a popular theory to account for dark matter and electroweak baryogenesis, wherein the Standard Model particle content is supplemented by a complex scalar gauge singlet, with certain discrete symmetries imposed. The scalar potential which results thereof can have seven different types of minima at tree-level, which may coexist for specific choices of parameters. There is therefore the possibility that a given minimum is not global but rather a local one, and may tunnel to a deeper extremum, thus causing vacuum instability. This rich vacuum structure is explained and discussed in detail. 
P.M. Ferreira
hep-ph