High scale perturbative gauge coupling in R-parity conserving SUSY   SO(10) with longer 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. Therefore, if the SO(10) gauge coupling is to remain perturbative up the Planck scale(=2x10^{18} GeV), the scale $M_U$ of GUT symmetry breaking is to be bounded from below. The bound depends upon specific Higgs representations used for SO(10) symmetry breaking but, as we find, can not be lower than $1.5x10^{17}$ GeV. In order to obtain such high unification scale we propose a two-step SO(10) breaking through $SU(2)_LXSU(2)_RXU(1)_{B-L}XSU(3)_C(g_{2L} \neq g_{2R})$ intermediate gauge symmetry. We estimate potential threshold and gravitational corrections to the running of gauge couplings and show that they can make the picture of perturbative GUT- gauge coupling running consistent at least up to the Planck scale. We also show that when $SO(10) \to G_{2213}$ by $210 + 54$, gravitational corrections alone 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 $p \to e^+\pi^0$. For the proton decay mediated by dim.5 operator a wide range of lifetimes is possible extending from the current experimental limit up to values 2-3 orders longer. 
M. K. Parida and B. D. Cajee