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_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 \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. ==See also==