After 40 years of searching, the Higgs boson was finally found at CERN's Large Hadron Collider in 2012 and weighed in at 125 GeV. However, this small mass gives rise to a vacuum stability problem, namely that the Standard Model Higgs quartic self-coupling becomes negative below the Planck scale, necessitating new physics beyond the Standard Model.

In this thesis, we study four minimal extensions of the Standard Model which solve the Higgs vacuum stability problem by adding a second Higgs-like scalar boson. In addition to the new scalar boson, we study the effects of adding new fermion singlets and doublets. It is shown that while new fermion generations only decrease the Higgs quartic coupling at high energies---only exacerbating the problem---the addition of a new Higgs-like scalar provides a positive contribution which is enough to overcome the vacuum stability limit.

We consider four Standard Model extensions containing different combinations of new fermions with this extended Higgs sector, and identify the allowed masses and mass mixing angles of these hypothetical particles that satisfy the vacuum stability condition. The allowed masses surround the 1 TeV range approximately, explaining why such particles have not yet been found.