High scale perturbative gauge coupling in R-parity conserving SUSY SO(10) with larger proton lifetime

It is well known that in single step breaking of R-parity conserving SUSY SO(10) that needs the Higgs representations $126+\bar 126$ the GUT gauge coupling violates the perturbative constraint at mass scales few times larger than the GUT scale. Imposing the perturbative constraint up to the Planck scale we obtain the smallest lower bound on the unification scale $M_U \ge 1.5X10^{17}$ GeV and the lower bound increases with the size of other Higgs representations. Using similar SO(10) Higgs representations as in single step breaking case we show that when the spontaneous symmetry breaking to MSSM takes place through $SU(2)_LXSU(2)_RXU(1)_{BL}XSU(3)_C$ intermediate gauge symmetry, investigated recently, the resulting threshold and gravitational corrections permit such large values of the unification scale ensuring the validity of the perturbative constraint at least up to the Planck scale. We also show that when $SO(10) \to G_{2213}$ by $210 + 54$ gravitational corrections with negligible threshold effects may guarantee such perturbative gauge coupling. The lifetime of the proton is found to increase by nearly 6 orders over the current experimental limit for $p \to e^+\pi^0$. The possibility of significant increase in the lifetime for the proton decay mediated by $dim.5$ operator is also pointed out