Complexification of The Vacuum and The Electroweak Gauge Symmetry

In this paper we investigate the consequences of imposing an SO(4) symmetry both on the scalar field and on spacetime prior to any spontaneous symmetry breakdown in the electroweak model. First we argue that the local electroweak gauge symmetry is implied by the initial SO(4) symmetry of the scalar field. Then we parameterize the vacuum, in a slightly different way with respect to the conventional assignment in the electroweak model, around the ground state mainly in two different ways: Once over the local electroweak gauge transformations and once over the local $\gamma$ basis which generates the SO(4) symmetry of the scalar field. Through comparing the two, we conclude that a consistent parametrization, requires the complexification of the scalar field, which simultaneously requires spacetime to obey local Lorentz invariance in the broken phase of the electroweak vacuum. After spontaneous symmetry breakdown and complexification, the SO(3,1) invariant quantity of the scalar field is identified to be the Higgs mass, which yields $m_H=v/2 \cong 123$ GeV. Also the vacuum state is normalized to a unit state by rescaling the local $\gamma$ basis, so that the metric tensor inherently contains the speed of light in the Higgs vacuum. We shortly consider the validity of the SO(3,1) invariance property of the vacuum state in terms of the Goldstone modes, in case the Higgs field departs from the ground state. The main goal of the paper is to treat spacetime and vacuum as the two faces of the same medallion, and to relate the local spacetime symmetry with the electroweak gauge symmetry, both in the broken and unbroken phases of the vacuum via complexification and spontaneous symmetry breaking.