Stueckelberg action

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In field theory, the Stueckelberg action (named after Ernst Stueckelberg^[1]) 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 ${\displaystyle \phi }$. 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 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 ${\displaystyle \phi =0}$, 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).