  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.
