Effects of Dimension-5 Operators in $E_6$ Grand Unified Theories

The effective dimension-5 operators can be induced by quantum gravity or inspired by string and M theories. They have important impacts on grand unified theories. We investigate the group theoretic nature of them for the well known E(6) model. Considering the breaking chains $E_{6}\mapsto H=SO(10)\times U_{V'}(1)\mapsto SU(5)\times U_{V}(1)\times U_{V'}(1)\mapsto SU(3)\times SU(2)\times U_{Z}(1)\times U_{V}(1)\times U_{V'}(1)$ and $E_{6}\mapsto H=SO(10)\times U_{V'}(1)\mapsto SU(4)\times SU_{L}(2)\times SU_{R}(2)\times U_{V'}(1)\mapsto SU(3)\times SU_{L}(2)\times SU_{R}(2)\times U_{S}(1)\times U_{V'}(1)$, we derive and give all of the Clebsch-Gordan coefficients $\Phi^{(r)}_{s,z}$ associated with $E_6$ breaking to the standard model. Some applications of the results are discussed shortly.