How common are grand unified theories?

TL;DR

This paper investigates how often Standard Model-like theories can be unified under grand unified theories, finding that unification is rare when considering larger gauge groups or representations, thus providing a group-theoretical perspective on GUT likelihood.

Contribution

It quantifies the frequency of grand unification compatibility among theories similar to the Standard Model using a purely group-theoretical approach.

Findings

Unification is common among small, anomaly-free fermion representations.

Unification becomes rare with larger gauge algebras or representations.

The analysis offers a bottom-up, naturalness-like argument for GUT plausibility.

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.