Gauge Kinetic Mixing and Leptophobic $Z'$ in $E_6$ and SO(10)

We examine the influence of gauge kinetic mixing on the couplings of a TeV scale $Z'$ in both $E_6$ and SO(10) models. The strength of such mixing, which arises due to the existence of incomplete matter representations at low scale, can be described by a single parameter, $\delta$. The value of this parameter can significantly influence the ability of both hadron and lepton colliders to detect a $Z'$ using conventional search techniques. In addition, $\delta \neq 0$ also adds to the complexities involved in separating $E_6$ $Z'$ models from those arising from alternative scenarios. Employing a reasonable set of assumptions we have determined the allowed range for this parameter within a wide class of models via an RGE analysis. In particular, given the requirements of Standard Model gauge coupling unification, anomaly freedom and perturbativity up to the GUT scale, we demonstrate that the necessary condition for exact leptophobia in $\eta$ type $E_6$ models, $\delta=-1/3$, is impossible to achieve in this scenario. Furthermore we show that the allowed range for $\delta$ is rather restricted for arbitrary values of the mixing between the $U(1)_\chi$ and $U(1)_\psi$ type couplings. The SO(10) $Z'$ model $\chi$ is discussed as a separate case since it requires special attention.