What are the similarities and differences between Stueckelberg mechanism and Higgs mechanism?

Question

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?

Answer 1

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.