Electroweak couplings and LHC constraints on alternative $Z^{\prime}$ models in $E_6$

We report the most general expression for the chiral charges of a $Z'$ gauge boson coming from an $E_6$ unification model, as a function of the electroweak parameters and the charges of the $U(1)$ factors in the chain of subgroups. These charges are valid for an arbitrary Higgs sector and only depend on the branching rules of the $E_6$ fundamental representation and the corresponding rules for the fermionic representations of their subgroups. By assuming $E_6$ unification, the renormalization group equations~(RGE) allow us to calculate the electroweak parameters at low energies for most of the chains of subgroups in $E_6$. From RGE and unitary conditions, we show that at low energies there must be a mixing between the gauge boson of the standard model hypercharge and the $Z'$. From this, it is possible to delimit the preferred region in the parameter space for a breaking pattern in $E_6$. In general, without unification, it is not viable to determine this region; however, for some models and under certain assumptions, it is possible to limit the corresponding parameter space. By using the most recent upper limits on the cross-section of extra gauge vector bosons $Z'$ decaying into dileptons from the ATLAS data at 13~TeV with accumulated luminosities of 36.1~fb$^{-1}$ and 13.3~fb$^{-1}$, we report the 95$\%$ C.L. lower limits on the $Z'$ mass for the typical $E_6$ benchmark models. We also show the contours in the 95\% C.L. of the $Z'$ mass bounds for the entire parameter space of $E_6$.