Further thoughts on supersymmetric $E_8$ as a family and grand unification theory

Abstract

We continue the analysis of the possibility of supersymmetric $E_8$ as a family unification and grand unification theory, this time under the assumption that there is a vacuum gluino condensate, but that this condensate is {\it not} accompanied by dynamical generation of a mass gap in the pure $E_8$ gauge theory. Arguments supporting these assumptions are given. When the $E_8$ theory is coupled to supergravity, assuming vanishing of the cosmological constant and a supersymmetry breaking scale of around a TeV, we show that the gluino mass induced by gravitational coupling to the condensate is of order $10^{-3}$ eV or smaller, compatible with the fermion (and particularly the neutrino) mass spectrum. We suggest that composite scalar Higgs superfields can arise from a chiral glueball in the attractive 3875 channel (and possibly other channels), permitting the breaking of the original $E_8$ gauge group to a SO(10) grand unification group, times a SU(3) family symmetry group and an extra U(1) factor. A general analysis of the Higgs superpotential shows that in the absence of gravitational couplings, there is always a supersymmetric vacuum in the (unphysical) limit of infinite Higgs superfields. However, when gravitational couplings are included, dimensional analysis of the superpotential shows that the vacuum can be stabilized for finite Higgs superfields, with the occurrence of dynamical ${\cal F}$-type supersymmetry breaking. We conclude that an $E_8$ unification may be theoretically viable, providing an alternative paradigm for low energy model building, in which the supersymmetric partners of the standard model fermions are vectors, and in which the only chiral superfields are symmetry breaking Higgs fields.