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What is the difference and similarities between the Stueckelberg mechanism and the Higgs mechanism? They both make the gauge field massive. Is the Stueckelberg mechanism a special case about $U(1)$ gauge fields of the Higgs mechanism? Does there exist Spontaneously symmetry breaking in the Stueckelberg mechanism?

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There are no deep differences! The Stueckelberg action is, in fact, a weird limit of the standard Goldstone sombrero potential, where, in effect, the self-coupling $\lambda$ is taken to infinity, while keeping the v.e.v. $\rm v$ fixed.

As a result, the $\sigma$ (analog of the Higgs) becomes infinitely massive and "decouples" (disappears from the spectrum), so it is dropped off the action, and only the massless goldston survives, Stueckelberg's celebrated scalar; this is the so-called $U(1)$ non-linear $\sigma$-model.

This could extend to nonabelian settings, cf. Affine Higgs mechanism ; Stueckelberg action, but the surviving scalar sector is substantially messier, on account of the interactions among the goldstone modes that have not decoupled: the nonabelian $\sigma$-model is non-renormalizable.

In the unitary gauge, the resulting action is called the Proca action, and represents a pure massive "would be" gauge field, but, of course it is not manifestly gauge invariant. Unlike the case for non-abelian vector fields, quantum electrodynamics with a massive photon is, nevertheless, in fact, renormalizable!! How can this be? Well, it is actually a covert gauge invariant Higgs/Stueckelberg action, only the fact was not recognized! (Well, with the possible exception of Ernst S in 1938...)

Englert and Higgs et al simply recognized the generality and non-Abelian extensions of the fact, which made electroweak gauge theories a reality.

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