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 ⊕ 126 the GUT-gauge coupling violates perturbative constraint at mass scales few times larger than the GUT scale. Therefore, if the SO(10) gauge coupling is to remain perturbative up to the Planck scale(≡ 2× 10 GeV), the scale MU of the 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 1.5× 10 GeV. In order to obtain such a high unification scale we propose a two-step SO(10) breaking through SU(2)L × SU(2)R × U(1)B−L × SU(3)C(g2L 6= g2R) intermediate gauge symmetry. We estimate potential threshold and gravitational corrections to the gauge coupling running and show that they can make the picture of perturbative gauge coupling running consistent at least up to the Planck scale. We also show that when SO(10) → G2213 by Higgs representations 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 present experimental limit for p → eπ. 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.