Effective Field Theory of Stückelberg Vector Bosons

Abstract

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 µ X µ H † H, (X µ X µ ) 2 , and X µ j 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 √ s/m X , except for the case of X µ j 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 µ j µ anom , carefully isolating the finite contribution to the fermion triangle diagram. We calculate the longitudinally-enhanced observables Z → Xγ (when m X < m Z ), f f → Xγ, and Zγ → Zγ when X couples to the baryon number current. Introducing a "fake" gauge-invariance by writing X µ = A µ − ∂ µ π/m X , the would-be gauge anomaly associated with A µ j µ anom is canceled by j µ anom ∂ µ π/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 (with its fine tunings and additional dark Higgs boson interactions) is replaced by the non-generic set of interactions that the Stückelberg vector field must have to avoid amplitudes that grow with energy.

The problem with Proca: ghost instabilities in self-interacting vector fields

“…The key cause of the ghost instability appears to be the loss of the longitudinal degree of freedom when the effective mass becomes (instantaneously, and locally at any event) zero [57]. This degree of freedom decouples from the Lagrangian and, even though a mass is recovered, it no longer possesses a continuous solution.…”

Dark Photon Dark Matter without Stueckelberg Mass

“…This is due to the symmetry breaking condition of eq. (10). To emphasize the effect of the coupling to curvature we mention, for example, that the conformal coupling ξ = 1/6 always restores the symmetry in the regime H I > M φ .…”

“…In string theory realizations no light dark photon has been found compatible with a large cut-off of the effective theory [8]. The Stueckelberg action describes three degrees freedom and any addition of interactions would spoil the nice features of the theory, see [9][10][11]. While none of these arguments is conclusive it seems natural to explore the possibility that the mass of a weakly coupled spin-1 field is generated through the standard Higgs mechanism.…”