How common are grand unified theories?

Abstract

The individual fermion generations of the Standard Model fit neatly into a representation of a simple Grand Unified Theory gauge algebra. If Grand Unification is not realized in nature, this would appear to be a coincidence. We attempt to quantify how frequently this coincidence occurs among theories with group structure and fermion content similar to the Standard Model. While many of the completely chiral, anomaly-free fermion representations of the Standard Model gauge algebra that are no larger than the single generation Standard Model are unifiable, we find that unifiability quickly becomes rare when the analysis is extended to include other gauge algebras or larger representations. This purely group-theoretical analysis may be taken as a bottom-up indication for Grand Unification, conceptually similar to a naturalness argument.