Mixing Matrix of Quarks Having Natural Twisting in E₆ Grand Unified Model

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Mass matrix of quarks is studied in the Supersymmetric E₆ Grand Unified Theory (GUT). The fundamental representation 27 in E₆ which corresponds to one generation contains two sets of 5*'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 5* representations may reproduce the observed quark and lepton mixing matrices. If E₆ is directly broken down to SU(3)C × SU(2)ₗ × 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 × 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 Vud, Vus, Vcd, Vcs and Vtb of the CKM matrix elements. As a by-product, one vector-like down-quark appears at the experimentally accessible TeV scale.