  Dark matter particles, even if they are electrically neutral, could interact
with the Standard Model particles via their electromagnetic multipole moments.
In this paper, we focus on the electromagnetic properties of the complex vector
dark matter candidate, which can be described by means of seven form factors.
We calculate the differential scattering cross-section with nuclei due to the
interactions of the dark matter and nuclear multipole moments, and we derive
upper limits on the former from the non-observation of dark matter signals in
direct detection experiments. We also present a model where the dark matter
particle is a gauge boson of a dark $SU(2)$ symmetry, and which contains heavy
new fermions, charged both under the dark $SU(2)$ symmetry and under the
electromagnetic $U(1)$ symmetry. The new fermions induce at the one loop level
electromagnetic multipole moments, which could lead to detectable signals in
direct detection experiments.
