Vectorlike Particles, $Z'$ and Yukawa Unification in F-theory inspired $E_6$

We explore the low energy implications of an F-theory inspired $E_6$ model whose breaking yields, in addition to the MSSM gauge symmetry, a $Z'$ gauge boson associated with a $U(1)$ symmetry broken at the TeV scale. The zero mode spectrum of the effective low energy theory is derived from the decomposition of the $27$ and $\overline{27}$ representations of $E_6$ and we parametrise their multiplicities in terms of a minimum number of flux parameters. We perform a two-loop renormalisation group analysis of the gauge and Yukawa couplings of the effective theory model and estimate lower bounds on the new vectorlike particles predicted in the model. We compute the third generation Yukawa couplings in an F-theory context assuming an $E_8$ point of enhancement and express our results in terms of the local flux densities associated with the gauge symmetry breaking. We find that their values are compatible with the ones computed by the renormalisation group equations, and we identify points in the parameter space of the flux densities where the $t-b-\tau$ Yukawa couplings unify.