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). 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. Q = I3L + 1 2 Y, (1) where I3L 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 [1] SU(4)C → SU(3)C × U(1)B−L, i.e. Q = I3L + I3R + 1 2 (B − L). (2) The other is based on SU(3)L → SU(2)L × U(1)YL and SU(3)R → SU(2)R × U(1)YR, i.e. Q = I3L + I3R − 1 2 YL − 1 2 YR. (3) [The minus signs in the above expression are due to a convention which will become clear later.] Whereas Eq. (2) is indicative of SO(10) as the unification group, Eq. (3) is indicative of [2] SU(3)C × SU(3)L × SU(3)R. However, the two are in fact equivalent if considered as subgroups of E6. Using Eq. (2), under SU(2)L × SU(2)R × U(1)B−L, the quarks and leptons transform as q = (u, d)L ∼ (2, 1, 1 6 ), q = (d, u)L ∼ (1, 2,− 1 6 ), (4) l = (ν, e)L ∼ (2, 1,− 1 2 ), l = (e, ν)L ∼ (1, 2, 1 2 ). (5) They are different in their electric charges because they have different B − L values. Using Eq. (3), under SU(2)L × SU(2)R × U(1)YL × U(1)YR , q ∼ (2, 1, 1 6 , 0), q ∼ (1, 2, 0,− 1 6 ), (6) l ∼ (2, 1,− 1 6 ,− 1 3 ), l ∼ (1, 2, 1 3 , 1 6 ). (7)