Stueckelberg action

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In

${\displaystyle \phi }$. This scalar field takes on values in a real 1D affine representation of R with ${\displaystyle m}$ as the coupling strength.

${\displaystyle {\mathcal {L}}=-{\frac {1}{4}}(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu })(\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu })+{\frac {1}{2}}(\partial ^{\mu }\phi +mA^{\mu })(\partial _{\mu }\phi +mA_{\mu })}$

This is a special case of the

— in contemporary terminology, a U(1) nonlinear σ-model.

Gauge-fixing ${\displaystyle \phi =0}$, yields the Proca action.

This explains why, unlike the case for non-abelian vector fields,

(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

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

${\displaystyle {\mathcal {L}}_{\rm {St}}=-{\frac {1}{4}}C_{\mu \nu }C^{\mu \nu }+g_{X}C_{\mu }{\mathcal {J}}_{X}^{\mu }-{\frac {1}{2}}\left(\partial _{\mu }\sigma +M_{1}C_{\mu }+M_{2}B_{\mu }\right)^{2}.}$

The first term above is the Stueckelberg field strength, ${\displaystyle M_{1}}$ and ${\displaystyle M_{2}}$ are topological mass parameters and ${\displaystyle \sigma }$ is the axion. After symmetry breaking in the electroweak sector the photon remains massless. The model predicts a new type of gauge boson dubbed ${\displaystyle Z'_{\rm {St}}}$ which inherits a very distinct narrow

in this model. The St sector of the StSM decouples from the SM in limit ${\displaystyle M_{2}/M_{1}\to 0}$.

Stueckelberg type couplings arise quite naturally in theories involving

) 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

• Higgs mechanism#Affine Higgs mechanism

References

• The edited PDF files of the physics course of Professor Stueckelberg, openly accessible, with commentary and complete biographical documents.
• Review:Stueckelberg Extension of the Standard Model and the MSSM
• Boris Kors, Pran Nath: [1], [2], [3]
• Daniel Feldman, Zuowei Liu, Pran Nath: [4], [5]