chiral anomaly

chiral anomaly. In quantum field theory, the chiral anomaly is the quantum mechanical violation of a classically conserved axial current. This phenomenon arises when the path integral measure is not invariant under a chiral symmetry transformation that is preserved at the classical level. The discovery, formalized in the Adler–Bell–Jackiw anomaly, has profound implications for the structure of the Standard Model and our understanding of gauge theories.

Definition and mathematical formulation

The anomaly manifests in the divergence of the axial current \(J^\mu_5\) in the presence of an external gauge field, such as the electromagnetic field. While classical Noether's theorem would predict conservation, quantum corrections from triangle diagrams introduce a non-zero divergence. This is mathematically expressed through the famous relation involving the field strength tensor of the photon or gluon fields. The formal derivation relies on careful regularization of the path integral formulation, where the Fujikawa method elegantly shows the non-invariance of the fermion measure under chiral transformations. This result is deeply tied to the mathematical structure of gauge theory and the Atiyah–Singer index theorem.

Physical consequences

A primary consequence is the decay of the neutral pion into two photons, a process forbidden without the anomaly but observed experimentally with the rate predicted by the Adler–Bell–Jackiw anomaly. In quantum chromodynamics, it explains the large mass of the eta prime meson via the U(1) problem, resolving a key puzzle in strong force physics. The anomaly also underpins the baryon number non-conservation in the electroweak theory of the Standard Model, which is relevant for scenarios like sphaleron processes in the early universe. Furthermore, it is central to the axial–vector current conservation in interactions involving W and Z bosons.

Historical development

The anomaly was independently discovered in 1969 by Stephen L. Adler, and by John Stewart Bell and Roman Jackiw, while studying the decay of the pion into photons. Their work, the Adler–Bell–Jackiw anomaly, resolved a long-standing discrepancy between theory and experiment for this decay rate. Earlier, the Veltman–'t Hooft theorem had suggested such symmetries should remain exact, making the discovery a pivotal moment. Subsequent work by Kazuhiko Nishijima, Luis Alvarez-Gaumé, and Edward Witten deepened the understanding, linking it to topological concepts in gauge theory. The Fujikawa method, introduced by Kazuo Fujikawa, provided a powerful path-integral perspective.

Experimental evidence

The most direct and celebrated evidence comes from the measured decay rate of the neutral pion to two gamma rays, which agrees precisely with the anomaly's prediction. Experiments at facilities like CERN and Fermilab have consistently confirmed this. The anomaly's role in the Standard Model is also tested indirectly through precision measurements of processes involving the Z boson at the Large Hadron Collider. Furthermore, predictions related to the eta prime meson mass, explained by the anomaly via the U(1) problem, are consistent with data from detectors like CLEO and BABAR.

Role in quantum field theory

The chiral anomaly is fundamental to the consistency and structure of quantum field theory. It imposes stringent constraints on gauge theories, requiring the cancellation of anomalies among fermion representations to maintain renormalizability and unitarity—a key principle in constructing the Standard Model. It is intimately connected to topological effects, such as instantons in quantum chromodynamics and the Atiyah–Singer index theorem. The anomaly also plays a crucial role in theoretical frameworks beyond the Standard Model, including string theory and condensed matter physics phenomena like the quantum Hall effect and topological insulators.

Category:Quantum field theory Category:Particle physics Category:Theoretical physics