**44**: 447-457

https://doi.org/10.1140/epjc/s2005-02374-x

Theoretical Physics

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

Physics Department, North-Eastern Hill University, 793022, Shillong, India

^{*} e-mail: mparida@sancharnet.in

It is well known that in single step breaking of R-parity conserving SUSY *SO*(10) that needs the Higgs representations , the GUT gauge coupling violates the perturbative constraint at mass scales a few times larger than the GUT scale. Therefore, if the *SO*(10) gauge coupling is to remain perturbative up to the Planck scale ( GeV), the scale *M*
_{
U
} 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, cannot be lower than $1.5 \times 10$^{17} GeV. In order to obtain such a high unification scale we propose a two-step *SO*(10) breaking through *SU*(2)_{
L
} $\times$ *SU*(2)_{
R
} $\times$ *U*(1)_{
B-L
} $\times$*SU*(3)_{
C
} () intermediate gauge symmetry. We estimate the 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 by the Higgs representations , 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 . For the proton decay mediated by a dim = 5 operator a wide range of lifetimes is possible, extending from the current experimental limit up to values 2-3 orders longer.

*© Springer-Verlag, 2005*