  We construct an effective field theory for complex Stueckelberg dark photon
dark matter. Such an effective construction can be realized by writing down a
complete set of operators up to dimension six built with the complex dark
photon and Standard Model fields. Classifying the effective operators, we find
that in order to properly take into account the non-renormalizable nature of an
interacting massive vector, the size of the Wilson coefficients should be
naturally smaller than naively expected. This can be consistently taken into
account by a proper power counting, that we suggest. First we apply this to
collider bounds on light dark matter, then to direct detection searches by
extending the list of non-relativistic operators to include the case of complex
vectors. In the former we correctly find scaling limits for small masses, while
in the latter we mostly focus on electric dipole interactions, that are the
smoking gun of this type of dark matter. Simple UV completions that effectively
realize the above scenarios are also outlined.
