$Z^{\prime }$, new fermions, and flavor-changing processes: Constraints on ${\mathrm{E}}_6$ models from $\mu \to \mathrm{eee}$

We study a new class of flavor-changing interactions, which can arise in models based on extended gauge groups (rank >4) when new charged fermions are present together with a new neutral gauge boson. We discuss the cases in which the flavor-changing couplings in the new neutral current coupled to the $Z^{\prime }$ are theoretically expected to be large, implying that the observed suppression of neutral flavor-changing transitions must be provided by heavy $Z^{\prime }$ masses together with small $Z-Z^{\prime }$ mixing angles. Concentrating on ${\mathrm{E}}_6$ models, we show how the tight experimental limit on $\mu \to \mathrm{eee}$ implies serious constraints on the $Z^{\prime }$ mass and mixing angle. We conclude that if the value of the flavor-changing parameters is assumed to lie in a theoretically natural range, in most cases the presence of a $Z^{\prime }$ much lighter than 1 TeV is unlikely.