Anomaly-Free Dark Matter Models are not so Simple

We explore the anomaly-cancellation constraints on simplified dark matter (DM) models with an extra U(1)$^\prime$ gauge boson $Z'$. We show that, if the Standard Model (SM) fermions are supplemented by a single DM fermion $\chi$ that is a singlet of the SM gauge group, and the SM quarks have non-zero U(1)$^\prime$ charges, the SM leptons must also have non-zero U(1)$^\prime$ charges, in which case LHC searches impose strong constraints on the $Z'$ mass. Moreover, the DM fermion $\chi$ must have a vector-like U(1)$^\prime$ coupling. If one requires the DM particle to have a purely axial U(1)$^\prime$ coupling, which would be the case if $\chi$ were a Majorana fermion and would reduce the impact of direct DM searches, the simplest possibility is that it is accompanied by one other new singlet fermion, but in this case the U(1)$^\prime$ charges of the SM leptons still do not vanish. This is also true in a range of models with multiple new singlet fermions with identical charges. Searching for a leptophobic model, we then introduce extra fermions that transform non-trivially under the SM gauge group. We find several such models if the DM fermion is accompanied by two or more other new fermions with non-identical charges, which may have interesting experimental signatures. We present benchmark representatives of the various model classes we discuss.