 High-energy obstruction for a hard vector mass

A free Proca field is consistent and has three physical particle polarizations. The difficulty for generic interactions is the ultraviolet behavior of its Proca propagator: the longitudinal numerator makes part of it approach order $1/m^2$, rather than decay as $1/k^2$. Longitudinal external polarization vectors also behave as ${\epsilon }_L^{\mu }=k^{\mu }/m+O(m/E)$. Without suitable identities and cancellations, amplitudes grow with energy and perturbative counterterms of increasing dimension are needed. Conserved-current Abelian models and the Stueckelberg mechanism are important exceptions; a massive interacting vector is not intrinsically inconsistent as an effective field theory.