Effective field theory of Stückelberg vector bosons

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

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