Mixing Matrix of Quarks Having Natural Twisting in E_6 Grand Unified Model

Abstract

Mass matrix of quarks is studied in the Supersymmetric $E_6$ Grand Unified Theory (GUT). The fundamental representation $\text{27}$ in $E_6$ which corresponds to one generation contains two sets of ${\text{5}}^{\ast }$'s in SU(5), so that there are six flavors of lepton doublets and right-handed down-quark triplets. It is known that the twisting (interchange) among the ${\text{5}}^{\ast }$ representations may reproduce the observed quark and lepton mixing matrices. If $E_6$ is directly broken down to $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$ and two extra U(1)'s, the extra 3 sets of down-type quarks do not necessarily decouple from the quark mass matrix, under the appropriate choice of the U(1) flavor charges on the quark and Higgs fields. Then, by diagonalizing the 6 $\times$ 6 down-quark mass matrix, we find a certain set of parameters, in which the twisting between a pair of the right-handed down-quarks occurs naturally, reproducing the reasonable values for the $d$-, $s$- and $b$-quark masses, and for $V_{ud}$, $V_{us}$, $V_{cd}$, $V_{cs}$ and $V_{tb}$ of the CKM matrix elements. As a by-product, one vector-like down-quark appears at the experimentally accessible TeV scale.

Comments: 16 pages, latex, no figures, some discussions added, results unchanged