# Stueckelberg action

In field theory, the Stueckelberg action (named after Ernst Stueckelberg) describes a massive spin-1 field as an R (the real numbers are the Lie algebra of U(1)) Yang–Mills theory coupled to a real scalar field $\phi$. This scalar field takes on values in a real 1D affine representation of R with $m$ as the coupling strength.
    ${\mathcal {L}}=-{\frac {1}{4}}(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu })(\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu })+{\frac {1}{2}}(\partial ^{\mu }\phi +mA^{\mu })(\partial _{\mu }\phi +mA_{\mu })$
This is a special case of the Higgs mechanism, where, in effect, λ and thus the mass of the Higgs scalar excitation has been taken to infinity, so the Higgs has decoupled and can be ignored, resulting in a nonlinear, affine representation of the field, instead of a linear representation — in contemporary terminology, a U(1) nonlinear σ-model.
Gauge-fixing $\phi =0$, yields the Proca action.
This explains why, unlike the case for non-abelian vector fields, quantum electrodynamics with a massive photon is, in fact, renormalizable, even though it is not manifestly gauge invariant (after the Stückelberg scalar has been eliminated in the Proca action).

## Stueckelberg extension of the Standard Model

The Stueckelberg extension of the Standard Model (StSM) 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
    ${\mathcal {L}}_{\rm {St}}=-{\frac {1}{4}}C_{\mu \nu }C^{\mu \nu }+g_{X}C_{\mu }{\mathcal {J}}_{X}^{\mu }-{\frac {1}{2}}\left(\partial _{\mu }\sigma +M_{1}C_{\mu }+M_{2}B_{\mu }\right)^{2}.$
The first term above is the Stueckelberg field strength, $M_{1}$ and $M_{2}$ are topological mass parameters and $\sigma$ is the axion. After symmetry breaking in the electroweak sector the photon remains massless. The model predicts a new type of gauge boson dubbed $Z'_{\rm {St}}$ 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 U(N) 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.