Non-invertible Global Symmetries in the Standard Model (2205.05086v2)

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.

• The paper demonstrates that non-invertible global symmetries in massless QED establish an infinite set of symmetry operators and selection rules for scattering amplitudes.
• It extends the analysis to QCD by showing that the π0 F∧F coupling matches non-invertible symmetries, offering a fresh interpretation of the ABJ anomaly.
• The study paves the way for future research into complex fusion algebras and the broader implications of generalized symmetries in quantum field theories.

Analyzing Non-Invertible Global Symmetries in Quantum Electrodynamics and Quantum Chromodynamics

The paper by Choi, Lam, and Shao explores the presence and implications of non-invertible global symmetries within the Standard Model, particularly focusing on Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD). Despite the longstanding recognition of global symmetries in theoretical physics, this work highlights their non-invertibility, offering new insights into their roles and effects in quantum field theories.

Summary of Findings

1. Non-Invertible Global Symmetries in Massless QED:
   • The authors identify an infinite number of non-invertible generalized global symmetries in massless QED. They demonstrate that, despite the lack of a conserved Noether current for the axial $U(1)_\text{A}$ symmetry due to the Adler-Bell-Jackiw (ABJ) anomaly, conserved topological symmetry operators can be constructed for any rational angle $2\pi p/N$. These operators, composed of axial rotations and coupled fractional quantum Hall states with the electromagnetic field, form a non-invertible fusion algebra.
   • This results in selection rules for scattering amplitudes in QED, which are consistent with observed phenomena, such as helicity conservation. While these symmetries act invertibly on local operators, their action on 't Hooft lines is nontrivial, generating rich algebraic structures in the gauge theory.
2. Generalization to QCD:
   • The paper extends to QCD, focusing on the coupling $\pi^0 F\wedge F$ in the effective pion Lagrangian. The authors show that this coupling is crucial for matching the non-invertible symmetries in the ultraviolet theory, providing an alternative perspective to the conventional ABJ anomaly explanation.
   • The discussion offers an invariant characterization of the ABJ anomaly not as the absence of symmetry but rather in terms of generalized symmetries, emphasizing a shift from the usual understanding of symmetry breaking in quantum field theories.

Implications and Future Directions

The exploration of non-invertible global symmetries opens up several theoretical avenues for further research:

• Fusion Algebra and Topological Operators:

The construction and fusion of these non-invertible symmetry operators suggest a complex algebraic structure underlying these symmetries. Future work could involve a detailed mapping of this algebra and its physical implications, possibly uncovering new invariants and consistency conditions in quantum field theories.

• Extension to Non-Abelian Gauge Theories:

While the paper primarily addresses abelian theories (QED), extending these results to non-abelian gauge theories like QCD could reveal deeper insights into symmetry and anomaly interactions in more complex gauge structures.

• Impact on the Standard Model:

Recognizing these as some of the first realizable instances of non-invertible symmetries might lead to reevaluating their roles and implications within the broader Standard Model. This could include reevaluating naturalness problems or hierarchy puzzles given these symmetries' fundamental nature.

• Broader Applications of Non-Invertible Symmetries:

Beyond the Standard Model, these symmetries may have implications for other areas, such as effective field theories and condensed matter physics, particularly in systems exhibiting topologically nontrivial phases or in the context of quantum anomalies.

The identification and analysis of these non-invertible global symmetries present a paradigm shift in discussing global symmetries in quantum field theories, providing a foundation for reinterpreting classical symmetry arguments in light of modern advances in quantum physics and topology.