Non-universal Z' from SO(10) GUTs with vector-like family and the origin of neutrino masses

A $Z'$ gauge boson with mass around the (few) TeV scale is a popular example of physics beyond the Standard Model (SM) and can be a fascinating remnant of a Grand Unified Theory (GUT). Recently, $Z'$ models with non-universal couplings to the SM fermions due to extra vector-like states have received attention as potential explanations of the present $R_K$, $R_{K^{\ast}}$ anomalies; this includes GUT model proposals based on the $\mathrm{SO}(10)$ group. In this paper we further develop GUT models with a flavour non-universal low scale $Z'$ and clarify several outstanding issues within them. First, we successfully incorporate a realistic neutrino sector (with linear and/or inverse low scale seesaw mechanism), which was so far a missing ingredient. Second, we investigate in detail their compatibility with the $R_K$, $R_{K^{\ast}}$ anomalies; we find that the anomalies do not have a consistent explanation within such models. Third, we demonstrate that these models have other compelling phenomenological features; we study the correlations between the flavour violating processes of $μ\to 3e$ and $μ$-$e$ conversion in a muonic atom, showing how a GUT imprint could manifest itself in experiments