Effective field theory of Stückelberg vector bosons

We explore the effective field theory of a vector ${X}^{\ensuremath{\mu}}$ that has St\"uckelberg mass. The absence gauge symmetry for implies Lorentz-invariant operators are constructed directly from ${X}^{\ensuremath{\mu}}$. Beyond kinetic and mass terms, allowed interactions at renormalizable level include ${X}_{\ensuremath{\mu}}{X}^{\ensuremath{\mu}}{H}^{\ifmmode\dagger\else\textdagger\fi{}}H$, $({X}_{\ensuremath{\mu}}{X}^{\ensuremath{\mu}}{)}^{2}$, ${X}_{\ensuremath{\mu}}{j}^{\ensuremath{\mu}}$, where ${j}^{\ensuremath{\mu}}$ is global current SM or hidden sector. show all these lead to scattering amplitudes grow with powers $\sqrt{s}/{m}_{X}$, except case ${X}_{\ensuremath{\mu}}{j}^{\ensuremath{\mu}}$ nonanomalous current. latter well known when $X$ identified as dark photon coupled electromagnetic current, often written equivalently mixing between photon. power counting energy growth facilitated by isolating longitudinal enhancement. examine in detail interaction an anomalous ${X}_{\ensuremath{\mu}}{j}_{\mathrm{anom}}^{\ensuremath{\mu}}$, carefully finite contribution fermion triangle diagram. calculate longitudinally-enhanced observables $Z\ensuremath{\rightarrow}X\ensuremath{\gamma}$ (when ${m}_{X}<{m}_{Z}$), $f\overline{f}\ensuremath{\rightarrow}X\ensuremath{\gamma}$, $Z\ensuremath{\gamma}\ensuremath{\rightarrow}Z\ensuremath{\gamma}$ couples baryon number Introducing ``fake'' gauge-invariance writing ${X}^{\ensuremath{\mu}}={A}^{\ensuremath{\mu}}\ensuremath{-}{\ensuremath{\partial}}^{\ensuremath{\mu}}\ensuremath{\pi}/{m}_{X}$, would-be anomaly associated ${A}_{\ensuremath{\mu}}{j}_{\mathrm{anom}}^{\ensuremath{\mu}}$ canceled ${j}_{\mathrm{anom}}^{\ensuremath{\mu}}{\ensuremath{\partial}}_{\ensuremath{\mu}}\ensuremath{\pi}/{m}_{X}$; this four-dimensional Green-Schwarz anomaly-cancellation mechanism work. Our analysis demonstrates there much larger set possible EFT can have, revealing energy. be tamed Higgs sector, but requires boson (and reintroduces fine-tuning sector) separated only limit $g\ensuremath{\ll}1$.