Triplicity of Quarks and Leptons

Quarks come in three colors and have electric charges in multiples of one-third. There are also three families of quarks and leptons. Whereas the first two properties can be understood in terms of unification symmetries such as SU(5), SO(10), or E6, why there should only be three families remains a mystery. I propose how all three properties involving the number three are connected in a fivefold application of the gauge symmetry SU(3). The fundamental building blocks of particle physics are quarks and leptons. The former have electric charges of 2/3 and −1/3, and are triplets under the unbroken gauge symmetry SU(3)C of Quantum Chromodynamics. The latter have electric charges 0 and −1, and do not have SU(3)C interactions. There are also 3 families of quarks and leptons, each one transforming in the same way under the standard SU(3)C × SU(2)L × U(1)Y gauge group. The ubiquitous occurrence of the number three may be indicative of an underlying symmetry larger than that of the present observed Standard Model. Although each family of quarks and leptons may be considered as components of 5∗ + 10 under SU(5), or of 16 under SO(10), or of 27 under E6, the existence of 3 families remains unexplained in this context. A strong hint as to what the underlying symmetry could be comes from the maximal subgroup of E6, i.e. SU(3)C × SU(3)L × SU(3)R, under which quarks are contained in the representations (3, 3, 1) and (3, 1, 3), and leptons in (1, 3, 3). Here I propose the following extension. Let the gauge symmetry be SU(3)C × SU(3)L × SU(3)M × SU(3)R × SU(3)F , where SU(3)F is the family symmetry and SU(3)M is the missing link in the lepton sector which allows this scheme to work. The gauge symmetry SU(3)F is assumed to be broken at or above the unification scale already to its non-Abelian discrete subgroup ∆(12) [1]. This group is the same as A4, the symmetry group of the even permutation of four objects. It is also the symmetry group of the regular tetrahedron, one of five perfect geometric solids known to the ancient Greeks and identified by Plato with the element “fire” [2]. There are four irreducible representations of A4: 1, 1 , 1, and 3, with the multiplication rule [3] 3× 3 = 1 + 1′ + 1′′ + 3 + 3. (1) Under G = SU(3)C × SU(3)L × SU(3)M × SU(3)R × A4, (2) the quark and lepton assignments are assumed to be q ∼ (3, 3, 1, 1; 3), q ∼ (3∗, 1, 1, 3∗; 3), (3) 2 l ∼ (1, 3∗, 3∗, 1; 3), l ∼ (1, 1, 3, 3; 3), (4) with their electric charges given by Q = I3L + YL 2 + I3R + YR 2 + I3M − YM 2 . (5) A good visual summary of this scheme is Fig. 1.