Non-invertible Global Symmetries in the Standard Model

TL;DR

This paper uncovers infinitely many non-invertible generalized global symmetries in the Standard Model's QED and QCD, revealing new selection rules and deepening the understanding of anomalies and pion decay.

Contribution

It constructs and analyzes non-invertible symmetries in QED and QCD, linking them to anomalies and pion decay, a novel perspective in gauge theories.

Findings

Identifies non-invertible symmetries in QED and QCD.

Shows these symmetries lead to new selection rules.

Connects symmetries to pion decay via anomaly matching.

Abstract

We identify infinitely many non-invertible generalized global symmetries in QED and QCD for the real world in the massless limit. In QED, while there is no conserved Noether current for the $U(1)_\text{A}$ axial symmetry because of the ABJ anomaly, for every rational angle $2\pi p/N$, we construct a conserved and gauge-invariant topological symmetry operator. Intuitively, it is a composition of the axial rotation and a fractional quantum Hall state coupled to the electromagnetic $U(1)$ gauge field. These conserved symmetry operators do not obey a group multiplication law, but a non-invertible fusion algebra over TQFT coefficients. They act invertibly on all local operators as axial rotations, but non-invertibly on the 't Hooft lines. These non-invertible symmetries lead to selection rules, which are consistent with the scattering amplitudes in QED. We further generalize our construction to QCD, and show that the coupling $\pi^0 F\wedge F$ in the effective pion Lagrangian is necessary to match these non-invertible symmetries in the UV. Therefore, the conventional argument for the neutral pion decay using the ABJ anomaly is now rephrased as a matching condition of a generalized global symmetry.