Frustrated SU(4) as the Preonic Precursor of the Standard Model

We give a model for composite quarks and leptons based on the semisimple gauge group SU(4), with the preons in the 10 representation. Hypercolor singlets are forbidden in the fermionic sector of this theory; we propose that SU(4) symmetry spontaneously breaks to SU(3)× U(1), with the binding of triality nonzero preons and gluons into composites, and with the formation of a color singlet condensate that breaks the initial Z12 vacuum symmetry to Z6. The spin 1/2 fermionic composites have the triality structure of a quark lepton family, and the initial Z12 symmetry implies that there are six massless families, which mix to give three distinct families, two massless with massive partners and one with both states massive, at the scale of the condensate. The spin 1 triality zero composites of the color triplet SU(4) gluons, when coupled to the condensate and with the color singlet representation of the 10 acting as a doorway state, lead to weak interactions of the fermionic composites through an exact SU(2) gauge algebra. The initial Z12 symmetry implies that this SU(2) gauge algebra structure is doubled, which in turn requires that the corresponding independent gauge bosons must couple to chiral components of the composite fermions. Since the U(1) couples to the 10 representation as B −L, an effective SU(2)L × SU(2)R ×U(1)B−L electroweak theory arises at the condensate scale, with all composites having the correct electric charge structure. A renormalization group analysis shows that the conversion by binding of one 10 of SU(4) to 12 triplets of SU(3) can give a very large, calculable hierarchy ratio between the SU(4) and the hadronic mass scales. Finally, we suggest that the representations used in the model may be embeddable in a single N = 6 supergraviton or conformal supergraviton multiplet. The structure of the scalar sector of the model, which we have not addressed, will be determined by the scalar content of this embedding.