Light dark states with electromagnetic form factors

New particles $\chi$ that are electrically neutral but couple to the electromagnetic current via higher-dimensional operators and that are sufficiently light, at or below the GeV-mass scale, can be produced in pairs in a number of dedicated high-intensity experiments. In this work we consider the production of $\chi$ through magnetic- and electric-dipole moments as well as through anapole moment and charge radius interactions in electron beams. We derive new constraints from BaBar, NA64 and mQ and forecast the future sensitivity on the existence of such states, from Belle-II, LDMX and BDX. We present for the first time a detailed treatment of the off-shell production of photons in electron beams with subsequent decay into a $\chi\bar\chi$ pair in a 2-to-4 process. These direct limits are then compared to the effects on SM precision observables, as well as to bounds from flavor physics and high energy colliders. Finally, we consider the possibility that $\chi$ is dark matter and study ensuing astrophysical and cosmological constraints. We find that a combination of all considered probes rule out $\chi$ particles with mass-dimension five and six photon interactions as dark matter when assuming a standard freeze-out abundance.