Tree-level Unitarity in SU(2)$_L\times$U(1)$_Y \times$U(1)$_{Y'}$ Models

In models with a U(1) gauge extension beyond the Standard Model, one can derive sum rules for the couplings of the theory that are a consequence of tree-level unitarity. In this paper, we provide a comprehensive list of coupling sum rules for a general SU(2)$_L\times$U(1)$_Y \times$U(1)$_{Y'}$ gauge theory coupled to an arbitrary set of fermion and scalar multiplets. These results are of particular interest for models of dark matter that employ an extended gauge sector mediated by a new (dark) $Z^\prime$ gauge boson. For the case of a minimal extension of the Standard Model with a U(1)$_{Y'}$ gauge boson, we clarify the definitions of the weak mixing angle and the electroweak $\rho$ parameter. We demonstrate the utility of a generalized $\rho$ parameter (denoted by $\rho^\prime$) whose definition naturally follows from the unitarity sum rules developed in this paper.