How common are grand unified theories?

TL;DR

This work quantifies how frequently the Standard Model fermions can be embedded into representations of simple Grand Unified Theory algebras, using a bottom-up, UV-agnostic notion of unifiability that excludes additional fermions and does not rely on gauge coupling unification. By constructing SM-like theory bases under strict and loose definitions and counting anomaly-free, completely chiral representations up to a maximum dimension $D_{ m max}$ with charge bounds $|Q|<Q_{ m max}$ and a limit on identical irreps $ ilde{S}$, the authors compute the unifiable fraction $r$. They find that near the single-generation SM, unifiability is relatively common (order one in some tightly restricted cases), but becomes exceedingly rare ($<10^{-2}$) as the theory space is broadened to include larger algebras or less restricted chirality; allowing vector-like fermions up to total dimension 30 can raise the fraction by up to a factor of 2. Overall, the results provide a conservative, group-theoretic indication supporting Grand Unification while highlighting the sensitivity to the chosen framework and the absence of a principled probability measure over possible 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.

Figures (3)

• Figure 1: Fraction $r$ of SM-like, completely chiral, anomaly-free fermion representations that are unifiable into a representation of a simple GUT gauge algebra, as a function of the maximal considered fermion dimension $D_\mathrm{max}$. Representations are restricted to $U(1)$ charges of $|Q|<10$ and at most $\tilde{S}=4$ identical irreducible representations under the semi-simple part of the algebra. Green shows the result considering only representations of $SU(3)\times SU(2)\times U(1)$ (strictly SM-like theories), whereas red includes representations of all semi-simple gauge algebras with a $U(1)$ factor and rank smaller than three (loosely SM-like theories). For computational reasons, the number of anomaly-free fermion representations of $SU(2)^2\times U(1)$ and $SU(2)^3\times U(1)$ has been computed only up to $D_{\rm max} = 20$ and $D_{\rm max} = 18$, respectively. The dashed curve hence constitutes only an upper limit.
• Figure 2: Dependence of the unifiable fraction on $Q_{\rm max}$ and $\tilde{S}$ (strictly SM-like case). Left: Cut on maximal considered integer charge $Q_{\rm max}$. Right: Cut on the number $\tilde{S}$ of equal irreps of the semi-simple part of the gauge algebra.
• Figure 3: Number of anomaly-free fermion representations up to dimension $D_\mathrm{max}$. Solid lines correspond to all anomaly-free representations, while dashed lines only count those that unify. $Q_\mathrm{max}$ is fixed to $10$ in this figure. Left: Strictly SM-like theories (ie. SM gauge algebra representations), showing relaxed chirality restrictions. The red lines show completely chiral representations (fiducial case). The green lines show results also including partially chiral representations, while the blue lines additionally include completely VL representations. The jumps in the number of unifiable theories occur when unification into $SO(10)$ ($SU(6)$ and $E_6$) become possible at $D_{\rm max}=16$ ($D_{\rm max} =27$). Right: Contribution from the different gauge algebras to the number of loosely SM-like theories. Due to computational reasons we only determine the number of representations for $SU(2)^2\times U(1)$ and $SU(2)^3\times U(1)$ up to $D_{\rm max}=20$ and $D_{\rm max} = 18$, respectively.