Proton Stability, Gauge Coupling Unification and a Light $Z^\prime$ in Heterotic-string Models

We explore the phenomenological viability of a light $Z^\prime$ in heterotic-string models, whose existence has been motivated by proton stability arguments. A class of quasi-realistic string models that produce such a viable $Z^\prime$ are the Left-Right Symmetric (LRS) heterotic-string models in the free fermionic formulation. A key feature of these models is that the matter charges under $U(1)_{Z^\prime}$ do not admit an $E_6$ embedding. The light $Z^\prime$ in the LRS heterotic-string models forbids baryon number violating operators, while allowing lepton number violating operators, hence suppressing proton decay yet allowing for sufficiently small neutrino masses via a seesaw mechanism. We show that the constraints imposed by the gauge coupling data and heterotic-string coupling unification nullify the viability of a light $Z^\prime$ in these models. We further argue that agreement with the gauge coupling data necessitates that the $U(1)_{Z^\prime}$ charges admit an $E_6$ embedding. We discuss how viable string models with this property may be constructed.