Quarks and Leptons in a Hexagonal Chain

The seemingly disparate notions of chiral color and quark-lepton nonuniversality are combined, and shown to be essential to each other as part of an underlying (and unifying) larger symmetry, i.e. supersymmetric SU(3)^6. Both phenomena are accessible experimentally at the TeV energy scale.

In the Standard Model of quarks and leptons, the electric charge has two components, i.e.
where I 3L comes from SU(2) L and Y from U(1) Y . If the gauge group is extended to include SU(2) R , then there are two possible decompositions of the electric charge. One is based on The other is based on SU(3) L → SU(2) L × U(1) Y L and SU(3) R → SU(2) R × U(1) Y R , i.e.
[The minus signs in the above expression are due to a convention which will become clear They are different in their electric charges because they have different B − L values. Using ).
At this point, a curious fact must have already been noticed, i.e. the electric charge has two components in Eq. (1), three in Eq. (2), and four in Eq. (3). How about five or more? Using the idea of a separate SU(3) l for leptons [3], which results in a successful nonsupersymmetric SU(3) 4 model [4], the electric charge may indeed have five components, i.e.
where Y l comes from SU(3) l → SU(2) l × U(1) Y l . In this case, Going back to Eq. (3), it is also clear that quarks and leptons may belong to different SU(3) L 's and SU(3) R 's, so that the electric charge has eight components, i.e.
Combining this notion of quark-lepton nonuniversality [5,6,7] with that of chiral color [8], the group SU(3) 6 is then obtained. Note that this is very different from the previously proposed [9] nonsupersymmetric SU(4) 6 model which predicted a value of sin 2 θ W very far away from the present data. A good way of displaying the structure of this symmetry is again a hexagonal "moose" diagram [10] ( Fig. 1) with the assignments x ∼ (1, 3, 3 * , 1, 1, 1),  (the x, x c links).
The particle content of this model is given by also allows the six gauge couplings to unify with sin 2 θ W equal to the canonical 3/8 at the unification scale. To check this, consider the contributions of q, x, λ, x c , q c , η to I 2 3L and Q 2 : α s would be wrong by at least a factor of two. As it is, SU(3) 6 allows the intriguing possibility that both chiral color and quark-lepton nonuniversality may exist at experimentally accessible energies, as shown below.
Let the neutral scalar components of the supermultiplets corresponding to x c 11 , x c 22 , x c 13 , x c 31 , x c 33 , x 33 , and λ 33 acquire large vacuum expectation values, then SU(3) 6 is broken down to This symmetry embodies both the notions of chiral color and quark-lepton nonuniversality and is assumed to be valid down to M S , the supersymmetry breaking scale. The particles which remain massless between M S and M U , the unification scale, are assumed to be three copies of q, q c , and η, three copies of all the components of x except x 33 , three copies of (ν x , e x ) ∼ (1, 1, 2, 1, −1/2), (e c x , ν c x ) ∼ (1, 1, 1, 2, 1/2), (25) Above M S , the theory is supersymmetric and all these supermultiplets contribute to the running of the five gauge coupings of Eq. (21). The one-loop renormalization-group equations are given by 1 where where N f = 3 is the number of families.
At M S , in addition to the breaking of supersymmetry, assume as well that SU (3) Below M S , the particle content becomes that of the Standard Model, but with two Higgs doublets, i.e.

SU(3)
At M U , all six gauge couplings are assumed equal. Using sin 2 θ W (M U ) = 3/8, this means value of M U is of order 10 16 GeV, in good agreement with the usual theoretical expectation.
Thus a new and remarkably successful model of grand unification is obtained. It is also experimentally verifiable because it predicts specific new particles necessarily below the TeV energy scale.  They are however necessary because they render the SU(2) qL × SU(2) lL gauge extension [7] anomaly-free. This is also the purpose of the new SU ( ln This would allow M S to be somewhat larger, say of order 1 TeV. In conclusion, a new model of grand unification has been proposed based on SU (3)