As we have seen in Chap. 2, the gauge-invariant theories include massless gauge fields. From the viewpoint of physical applications both massless and massive gauge fields are relevant. Just adding a mass term to the Lagrangian for the gauge field is not allowed since it would lead to the violation of the gauge-invariance of the Lagrangian. Therefore, a different approach has been proposed in which gauge fields acquire a mass by breaking of the gauge-invariance of the vacuum, while the Lagrangian of the gauge field remains gauge invariant (spontaneous symmetry-breaking). Symmetry-breaking of the vacuum may be incomplete. A part of the gauge fields then remains massless. This makes it possible to build theories including both massive and massless gauge fields, a circumstance used in unifying the short-range and the long-range interactions (e.g., the weak and the electromagnetic interactions), whose mediators are massive intermediate bosons and massless photons, respectively.
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