Two-Loop Analysis of Gauge Coupling Unification with Anomalous U(1) Symmetry and Proton Decay

Recently, a new mechanism, which explains why the three gauge coupling constants meet at a certain scale in the minimal supersymmetric standard model, has been proposed in a scenario of grand unified theories with anomalous $U(1)_A$ gauge symmetry. It is a non-trivial result that although there are many superheavy fields whose mass scales are below the unification scale, this mechanism explains this fact by means of one-loop renormalization group equations. Since the unification scale generically becomes below the usual GUT scale, $2\times 10^{16}$ GeV, and proton decay via dimension 5 operators is suppressed, the scenario predicts that proton decay via dimension 6 operators, $p\to e \pi^0$, will be observed in the near future. In this paper, we attempt to estimate a reasonable range of values of the lifetime of the proton predicted within this scenario by using a two-loop renormalization group calculation and the ambiguities of \order1 coefficients.