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; this choice of gauge gluon and preon multiplets is motivated by the possibility of embedding them in an N=6 supergravity multiplet. Hypercolor singlets are forbidden in the fermionic sector of this theory; we propose that SU(4) symmetry spontaneously breaks to $SU(3) \times U(1)$, with the binding of triality nonzero preons and gluons into composites, and with the formation of a color singlet condensate that breaks the initial $Z_{12}$ vacuum symmetry to $Z_{6}$. The spin 1/2 fermionic composites have the triality structure of a quark lepton family, and the initial $Z_{12}$ 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 $Z_{12}$ 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 $SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ 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