Stueckelberg action

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In field theory, the Stueckelberg action (named after Ernst Stueckelberg) describes a massive spin-1 field as a R (the real numbers are the Lie algebra of U(1)) Yang-Mills theory coupled to a real scalar field φ which takes on values in a real 1D affine representation of R with m as the coupling strength.

$\mathcal{L}=-\frac{1}{4}(\partial^\mu A^\nu-\partial^\nu A^\mu)(\partial_\mu A_\nu-\partial_\nu A_\mu)+\frac{1}{2}(\partial^\mu \phi+m A^\mu)(\partial_\mu \phi+m A_\mu)$

The usual Higgs mechanism of spontaneous symmetry breaking applies here. The only difference being we have an affine representation instead of a linear representation.

By gauge-fixing φ=0, we get the Proca action.

This explains why, unlike the case for non-abelian vector fields, quantum electrodynamics with a massive photon is renormalizable even though it's not manifestly gauge invariant (after the Stückelberg scalar has been eliminated in the Proca action).

[edit] The Stueckelberg Extension of the Standard Model

The Stueckelberg Lagrangian of the StSM (Stueckelberg Extension of the Standard Model) consists of a gauge invariant kinetic term for a massive U(1) gauge field. Such a term can be implemented into the Lagrangian of the Standard Model without destroying the renormalizability of the theory and further provides a mechanism for mass generation that is distinct from the Higgs mechanism in the context of abelian gauge theories.

The model involves a non-trivial mixing of the Stueckelberg and the Standard Model sectors by including an additional term in the effective Lagrangian of the Standard Model given by

$\mathcal{L}_{St}=-\frac{1}{4}C_{\mu \nu }C^{\mu\nu }+g_XC_{\mu }\mathcal{J}_X^{\mu }-\frac{1}{2}\left(\partial _{\mu }\sigma +M_1C_{\mu}+M_2B_{\mu }\right)^2.$

The first term above is the Stueckelberg field strength, $M 1$ and $M 2$ are topological mass parameters and $σ$ is the axion. After symmetry breaking in the electroweak sector the photon remains massless. The model predicts a new type of gauge boson dubbed $Z' S t$ which inherits a very distinct narrow decay width in this model. The St sector of the StSM decouples from the SM in limit $M_2/M_1 \to 0$ .

Stueckelberg type couplings arise quite naturally in theories involving compactifications of higher dimensional string theory, in particular, these couplings appear in the dimensional reduction of the ten dimensional N = 1 supergravity coupled to supersymmetric Yang-Mills gauge fields in the presence of internal gauge fluxes. In the context of intersecting D brane model building, products of abelian gauge groups are broken to their SU(N) subgroups via the Stueckelberg couplings and thus the abelian gauge fields become massive. Further, in a much simpler fashion one may consider a model with only one extra dimension (a type of Kaluza-Klein model) and compactify down to a four dimensional theory. The resulting Lagrangian will contain massive vector gauge bosons that acquire masses through the Stueckelberg mechanism.