Effective Field Theory of St\"uckelberg Vector Bosons

We explore the effective field theory of a vector field $X^\mu$ that has a St\"uckelberg mass. The absence of a gauge symmetry for $X^\mu$ implies Lorentz-invariant operators are constructed directly from $X^\mu$. Beyond the kinetic and mass terms, allowed interactions at the renormalizable level include $X_\mu X^\mu H^\dagger H$, $(X_\mu X^\mu)^2$, and $X_\mu j^\mu$, where $j^\mu$ 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 $X_\mu j^\mu$ where $j^\mu$ is a \emph{nonanomalous} global current. The latter is well-known when $X$ is a dark photon coupled to the electromagnetic current, often written as kinetic mixing with the photon. 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 \emph{anomalous} global vector current $X_\mu j_{\rm anom}^\mu$, carefully isolating the finite contribution to the fermion triangle diagram. We calculate the longitudinally-enhanced observables $Z \rightarrow X\gamma$ (when $m_X < m_Z$), $f\bar{f} \rightarrow X \gamma$, and $Z\gamma \to Z\gamma$ when $X$ couples to the baryon number current. Introducing a ``fake'' gauge-invariance by writing $X^\mu = A^\mu - \partial^\mu \pi/m_X$, the would-be gauge anomaly associated with $A_\mu j_{\rm anom}^\mu$ is canceled by $j_{\rm anom}^\mu \partial_\mu \pi /m_X$; this is the four-dimensional Green--Schwarz anomaly-cancellation mechanism at work. Our analysis demonstrates a larger set of interactions that an EFT with a St\"uckelberg vector field can have, revealing scattering amplitudes that grow with energy. This growth can be tamed by a dark Higgs sector, but this requires additional Higgs interactions that can be separated from $X$ only in the limit $g \ll 1$.