  Massive vector fields feature in several areas of particle physics, e.g., as
carriers of weak interactions, dark matter candidates, or as an effective
description of photons in a plasma. Here we investigate vector fields with
self-interactions by replacing the mass term in the Proca equation with a
general potential. We show that this seemingly benign modification inevitably
introduces ghost instabilities of the same kind as those recently identified
for vector-tensor theories of modified gravity (but in this simpler, minimally
coupled theory). It has been suggested that nonperturbative dynamics may drive
systems away from such instabilities. We demonstrate that this is not the case
by evolving a self-interacting Proca field on a Kerr background, where it grows
due to the superradiant instability. The system initially evolves as in the
massive case, but instabilities are triggered in a finite time once the
self-interaction becomes significant. These instabilities have implications for
the formation of condensates of massive, self-interacting vector bosons, the
possibility of spin-one bosenovae, vector dark matter models, and effective
models for interacting photons in a plasma.
