SU(8) family unification with boson-fermion balance

Abstract: We formulate an $SU(8)$ family unification model motivated by requiring that the theory should incorporate the graviton, gravitinos, and the fermions and gauge fields of the standard model, with boson--fermion balance. Gauge field $SU(8)$ anomalies cancel between the gravitinos and spin $\frac {1}{2}$ fermions. The 56 of scalars breaks $SU(8)$ to $SU(3)_{family} \times SU(5)\times U(1)/Z_5$, with the fermion representation content needed for "flipped" $SU(5)$ with three families, and with residual scalars in the $10$ and $\overline{10}$ representations that break flipped $SU(5)$ to the standard model. Dynamical symmetry breaking can account for the generation of $5$ representation scalars needed to break the electroweak group. Yukawa couplings of the 56 scalars to the fermions are forbidden by chiral and gauge symmetries, so in the first stage of $SU(8)$ breaking fermions remain massless. In the limit of vanishing gauge coupling, there are $N=1$ and $N=8$ supersymmetries relating the scalars to the fermions, which restrict the form of scalar self-couplings and should improve the convergence of perturbation theory, if not making the theory finite and "calculable". In an Appendix we give an analysis of symmetry breaking by a Higgs component, such as the $(1,1)(-15)$ of the $SU(8)$ 56 under $SU(8) \supset SU(3) \times SU(5) \times U(1)$, which has nonzero $U(1)$ generator.