Global anomalies in the Standard Model(s) and Beyond

TL;DR

This work provides a topology-based audit of anomaly freedom for the Standard Model and a broad class of Beyond-Standard-Model theories by computing five-dimensional spin bordism groups of classifying spaces for various gauge groups using the Atiyah–Hirzebruch spectral sequence and the Dai–Freed framework. It shows that, across four SM variants and multiple BSM constructions (including U(1) extensions, Pati-Salam, and trinification), there are no new global anomalies beyond the familiar SU(2) Witten anomaly in some cases, and none at all in others; hypercharge constraints and representation content play a crucial role in these conclusions. The results hold for arbitrary fermion content and extend to spin_c formulations (e.g., gauged B−L) where applicable, thereby providing robust, topology-based consistency checks for a wide landscape of gauge theories. Methodologically, the paper furnishes a systematic approach to anomaly analysis in four dimensions that can guide model-building and illuminate when global topological obstructions arise or are absent.

Abstract

We analyse global anomalies and related constraints in the Standard Model (SM) and various Beyond the Standard Model (BSM) theories. We begin by considering four distinct, but equally valid, versions of the SM, in which the gauge group is taken to be $G=G_{\text{SM}}/Γ_n$, with $G_{\text{SM}}=SU(3)\times SU(2) \times U(1)$ and $Γ_n$ isomorphic to $\mathbb{Z}/n$ where $n\in\left\{1,2,3,6\right\}$. In addition to deriving constraints on the hypercharges of fields transforming in arbitrary representations of the $SU(3)\times SU(2)$ factor, we study the possibility of global anomalies in theories with these gauge groups by computing the bordism groups $Ω^{\text{Spin}}_5(BG)$ using the Atiyah-Hirzebruch spectral sequence. In two cases we show that there are no global anomalies beyond the Witten anomaly, while in the other cases we show that there are no global anomalies at all, illustrating the subtle interplay between local and global anomalies. While freedom from global anomalies has been previously shown for the specific fermion content of the SM by embedding the SM in an anomaly-free $SU(5)$ GUT, our results here remain true when the SM fermion content is extended arbitrarily. Going beyond the SM gauge groups, we show that there are no new global anomalies in extensions of the (usual) SM gauge group by $U(1)^m$ for any integer $m$, which correspond to phenomenologically well-motivated BSM theories featuring multiple $Z^\prime$ bosons. Nor do we find any new global anomalies in various grand unified theories, including Pati-Salam and trinification models. We also consider global anomalies in a family of theories with gauge group $SU(N)\times Sp(M)\times U(1)$, which share the phase structure of the SM for certain $(N, M)$.

Figures (11)

• Figure 1: The results of Dai and Freed give a prescription for writing down a fermionic partition function $Z_\psi$ when spacetime $\Sigma$ is the boundary of a five-manifold $X$.
• Figure 2: Gluing of two manifolds $Y_1$ and $Y_2$ with a shared boundary component $\Sigma$, under which the exponentiated $\eta$-invariant factorizes.
• Figure 3: Gluing of two manifolds $X$ and $X^\prime$ with a shared boundary $\Sigma$ into a closed manifold $\bar{X} = X \cup (-X^\prime)$.
• Figure 4: Schematic illustration of a spectral sequence
• Figure 5: The $E^2$ page of the Atiyah-Hirzebruch spectral sequence for $G=G_{\text{SM}}$. We see that there is only a single entry relevant to the computation of $\Omega_5^{\text{Spin}}(BG_{\text{SM}})$, with a map ($\gamma$) going in and a map ($\beta$) going out.