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

Abstract

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 $E_6\mapsto H=SO(10)\times U_{V^{\prime }}(1)\mapsto SU(5)\times U_V(1)\times U_{V^{\prime }}(1)\mapsto SU(3)\times SU(2)\times U_Z(1)\times U_V(1)\times U_{V^{\prime }}(1)$ and $E_6\mapsto H=SO(10)\times U_{V^{\prime }}(1)\mapsto SU(4)\times S{U}_L(2)\times S{U}_R(2)\times U_{V^{\prime }}(1)\mapsto SU(3)\times S{U}_L(2)\times S{U}_R(2)\times U_S(1)\times U_{V^{\prime }}(1)$, we derive and give all of the Clebsch-Gordan coefficients ${\Phi }_{s,z}^{(r)}$ associated with $E_6$ breaking to the standard model. Some applications of the results are discussed shortly.

Comments: 8 pages, latex, minor changes, accepted for the publication in MPLA