Simple $E_6$ Unification with Anomalous $U(1)_A$ Symmetry

We propose simpler Higgs sectors in $E_6$ grand unified theory with anomalous $U(1)_A$ gauge symmetry than the previous model in Ref.1). As in the previous model, the doublet-triplet splitting can be realized in a natural way, while proton decay via dimension 5 operators is suppressed and gauge coupling unification is also realized without fine-tuning. Combining the matter sector, simple complete GUTs can be obtained. Since the Higgs sector is simpler, the gauge coupling constant at the cutoff scale can be in perturbative region, and therefore, the estimated value of the lifetime of a nucleon in this model, $\tau_p(p\to e^+\pi^0)\sim 5\times 10^{33}$ years, becomes more reliable.