Upper Bound of Proton Life-Time in Product-Group Unification

Models of supersymmetric grand unified theories based on SU(5)_GUT \times U(N)_H gauge group (N = 2,3) have a symmetry that guarantees light two Higgs doublets and absence of dimension-five proton decay operators. We analysed the proton decay induced by gauge-boson exchange in the models. Upper bounds of proton life-time are obtained; \tau(p\to \pi^0e^+) \lsim 6.0 \times 10^{33} yrs. in the SU(5)_GUT \times U(2)_H model and \tau(p\to \pi^0e^+) \lsim 5.3 \times 10^{35} yrs. in the SU(5)_GUT \times U(3)_H model. Various uncertainties in the predictions are also discussed.