Rotations around the z -axis form a U (1) subgroup of SO (3) which preserves the vacuum expectation value of ϕ ϕ , and this is the unbroken gauge group. Rotations around the x and y -axis do not preserve the vacuum, and the components of the SO (3) gauge field which generate these rotations become massive vector mesons. There are two massive W mesons in the Schwinger model, with a mass set by the mass scale Ã, and one massless U (1) gauge boson, similar to the photon.