A type-I (Higgs-only VEV) local minimum is lower than every type-0 and type-III extremum in the restricted scalar potentials considered, by a proof using homogeneous-function identities.
This is a trivial consequence of SU (2) × U (1)- invariance in the case of a Ψ k that is not SU (2) × U (1)-invariant
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Theorem on vacuum stability
A type-I (Higgs-only VEV) local minimum is lower than every type-0 and type-III extremum in the restricted scalar potentials considered, by a proof using homogeneous-function identities.