Uniqueness of the SUN gauge groups for implementing absolute proton stability with a global U1 of color

The problem of ensuring proton stability in grand unified models of the strong, weak, and electromagnetic interactions by means of a new conserved quantum number is investigated. It is assumed that spontaneous symmetry breaking will preserve an SU3c of color. To guarantee that the new quantum number χ be conserved after the spontaneous symmetry breaking, we require that all color-singlet Higgs fields in the theory have χ=0. This assumption has the following consequences: The gauge symmetry must be SUN, the global symmetry must be UN, χ must generate the U1c of color, and only certain representations for the Higgs scalars are possible. Conservation of χ also restricts the possible SUN fermion assignments. In particular, if all fermions are either singlets, triplets, or antitriplets of color and if the only massless fermions are χ=0 neutrinos, then conservation of χ requires that either the model is vectorlike or that there are many fermion representations, some of which include families of particles with unusual values of χ. Both cases are anomaly free.