On the Grand Unified Model of the Nambu-Jona-Lasino Type

It is widely believed that the world of elementary particles may be governed by some local gauge symmetry principle with the use of Higgs mechanism. It is an old attempt ' ) to explain a gauge field (photon) as a Nambu-Goldstone composite state of self-interacting fermion-antifermion pairs. And some years ago it was demonstrated') that gauge field theories with Higgs scalar fields are also effectively derived from some Nambu-Jona-Lasino type Lagrangeans. Subsequently a simple method to derive the effective gauge field theory, by making use of the path-integral technique, from a Nambu-Jona-Lasino type Lagrangean was proposed ) and applied ).6) to unified gauge field theories of elementary particles. In the 5U(2hx UO) model with leptons, the Weinberg angle is fixed to be sin 2 8w= 1/4. And in the 5U(3)c x 5U(2h x UO),model with quarks and leptons, it is obtained that sin Bw=3/S and gc=a/sin Bw where gc and a are the colour coupling and fine structure constants, respectively. This predicted value of the Weinberg angle, sin Bw = 3/8, is the same as that of some grand unified gauge field models 7 ) and should be extrapolated down from the super heavy mass scale of order 10 Ge V to the ordinary one of order 10 GeV. 8 ) This suggests that the 5U(3)cx5U(2hx UO) gauge group is to be extended to larger ones. It is the purpose of this paper to examine this possibility. We postulate the following four conditions on our model in order to select appropriate gauge groups and fermion assignments. Condition l. All basic fermions are left-handed spinors and belong to one small irreducible representation of the gauge group (at least in the case of one generation). Condition 2. The composite Higgs scalars Uermion-antifermion pair excitations) should have appropriate transformation properties to give necessary hierarchical patterns of symmetry breakings. Condition 3. Although all Higgs scalars couple to the basic fermion multiplets, the fermions corresponding to the observed leptons and quarks should not become super heavy through the super heavy symmetry breakings. Condition 4. The effective grand unified gauge field theory should be anomaly free. The exceptional group E 6 9 ) satisfies all the above conditions and this choice of the gauge group is almost unique as far as various simple classical and exceptional Lie groups are concerned. For example, the 5 U (5), 50(10) and E7 are excluded by Conditions 1, 2, 3; 2, 3, and 2, 3, respectively. In the following we discuss the grand unified model of the Nambu-Jona-Lasino type based on E 6 • The fundamental and adjoint representations of E6 are 27 and 78, respectively. According to Condition 1, bet us assign all the basic laft-handed fermions to 27 of E6 and denote it as XI. Then XR=(XC)L also trans-