Theorem on vacuum stability

We consider an extension of the Standard Model with one or more scalar multiplets beyond the Higgs doublet <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="normal">Φ</a:mi></a:math>. The additional scalar multiplets are supposed to carry arbitrary hypercharges (the precise conditions on the scalar potential are less stringent). We prove that, in such a model, if the field configuration where only <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi mathvariant="normal">Φ</d:mi></d:math> has a nonzero vacuum expectation value (VEV) is a local minimum of the potential, then it has a lower value of the potential than any extremum where both <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi mathvariant="normal">Φ</g:mi></g:math> and other scalar multiplets have nonzero VEVs.