Magnitude of R-parity violation in String Inspired GUTs

The nature of R-parity violating interactions in two classes of string inspired supersymmetric grand unified theories (SISUSY GUT), based on the gauge groups $SO(10)$ (and its subgroup $SU(2)_L\times SU(2)_R\times SU(4)_c\equiv G_{224}$) as well as $[SU(3)]^3$, are discussed and their strengths are related to the ratio of symmetry breaking scales present in the model. We first argue that for the R-parity violating couplings $\lambda_{R\!\!\!/}$ to be suppressed to the desired level, the $B-L$ local symmetry must break at an intermediate scale $M_{B-L}$ since $\lambda_{R\!\!\!/} =M_{B-L}/M_{GUT}$. We then construct scenarios where such intermediate scales arise being consistent with gauge coupling unification in a two-loop renormalization group study. In the resulting $SO(10)$ models, higher-dimensional-operator-induced R-parity violating couplings are potentially large (except in one case), and are therefore inconsistent with limits on the proton life time unless the couplings associated with the higher dimensional terms are fine tuned to very small values. However, the $[SU(3)]^3$ and $G_{224}$ models can be consistent if a certain class of quark-lepton coupling in the superpotential is forbidden by a discrete summetry (unrelated to R-parity). An interesting prediction of these models is neutron-anti-neutron oscillation with observable strength.