Classes of complete dark photon models constrained by Z-Physics

Dark Matter models that employ a vector portal to a dark sector are usually treated as an effective theory that incorporates kinetic mixing of the photon with a new U(1) gauge boson, with the $Z$ boson integrated out. However, a more complete theory must employ the full SU(2)$_L\times$U(1)$_Y \times$U(1)$_{Y^\prime}$ gauge group, in which kinetic mixing of the $Z$ boson with the new U(1) gauge boson is taken into account. The importance of the more complete analysis is demonstrated by an example where the parameter space of the effective theory that yields the observed dark matter relic density is in conflict with a suitably defined electroweak $\rho$-parameter that is deduced from a global fit to $Z$ physics data.