Uniqueness of theSUNgauge groups for implementing absolute proton stability with a globalU1of color

Abstract

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 $\mathrm{S}{\mathrm{U}}_{3}^{c}$ of color. To guarantee that the new quantum number $\ensuremath{\chi}$ be conserved after the spontaneous symmetry breaking, we require that all color-singlet Higgs fields in the theory have $\ensuremath{\chi}=0$. This assumption has the following consequences: The gauge symmetry must be $\mathrm{S}{\mathrm{U}}_{N}$, the global symmetry must be ${\mathrm{U}}_{N}$, $\ensuremath{\chi}$ must generate the ${\mathrm{U}}_{1}^{c}$ of color, and only certain representations for the Higgs scalars are possible. Conservation of $\ensuremath{\chi}$ also restricts the possible $\mathrm{S}{\mathrm{U}}_{N}$ fermion assignments. In particular, if all fermions are either singlets, triplets, or antitriplets of color and if the only massless fermions are $\ensuremath{\chi}=0$ neutrinos, then conservation of $\ensuremath{\chi}$ 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 $\ensuremath{\chi}$. Both cases are anomaly free.