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.
Categories: hep-ph