Depending on the representation of the scalar field, not every gauge field acquires a mass. A simple example is in the renormalizable version of an early electroweak model due to Julian Schwinger. In this model, the gauge group is SO (3) (or SU (2) − there are no spinor representations in the model), and the gauge invariance is broken down to U (1) or SO (2) at long distances. To make a consistent renormalizable version using the Higgs mechanism, introduce a scalar field ϕ ϕ a which transforms as a vector (a triplet) of SO (3). If this field has a vacuum expectation value, it points in some direction in field space. Without loss of generality, one can choose the z -axis in field space to be the direction that ϕ ϕ is pointing, and then the vacuum expectation value of ϕ ϕ is (0, 0, Ã), where Ã is a constant with dimensions of mass ( c = ℏ ℏ = 1 ).