学术文献库

We explore the effective field theory of a vector field $X^μ$ that has a Stückelberg mass. The absence of a gauge symmetry for $X^μ$ implies Lorentz-invariant operators are constructed directly from $X^μ$. Beyond the kinetic and mass terms, allowed interactions at the renormalizable level include $X_μX^μH^\dagger H$, $(X_μX^μ)^2$, and $X_μj^μ$, where $j^μ$ 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 $X_μj^μ$ where $j^μ$ 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 $X_μj_{anom}^μ$, carefully isolating the finite contribution to the fermion triangle diagram. We calculate the longitudinally-enhanced observables $Z \rightarrow Xγ$ (when $m_X < m_Z$), $f\bar{f} \rightarrow X γ$, and $Zγ\to Zγ$ when $X$ couples to the baryon number current. Introducing a fake gauge-invariance by writing $X^μ= A^μ- \partial^μπ/m_X$, the would-be gauge anomaly associated with $A_μj_{anom}^μ$ is canceled by $j_{anom}^μ\partial_μπ/m_X$; this is the four-dimensional Green-Schwarz anomaly-cancellation mechanism at work. Our analysis suggests there is no free lunch by appealing to Stückelberg for the mass of a vector field: the price paid for avoiding a dark Higgs sector is replaced by the non-generic set of interactions that the Stückelberg vector field must have to avoid amplitudes that grow with energy.