Perturbative unification of gauge couplings in supersymmetric $E_6$ models

We study gauge coupling unification in supersymmetric $E_6$ models where an additional $\mathrm{U}(1{)}^{\mathrm{\prime }}$ gauge symmetry is broken near the TeV scale and a number of exotic matter fields from the $$ representations have $O(\mathrm{T}\mathrm{e}\mathrm{V})$ mass. Solving the 2-loop renormalization group equations of gauge couplings and a kinetic mixing coupling between the $\mathrm{U}(1{)}^{\mathrm{\prime }}$ and $\mathrm{U}(1{)}_Y$ gauge fields, we find that the gauge couplings fall into the non-perturbative regime below the GUT scale. We examine threshold corrections on the running of gauge couplings from both light and heavy ($\sim$ GUT scale) particles and show constraints on the size of corrections to achieve the perturbative unification of gauge couplings.