Adler–Bell–Jackiw anomaly

Adler–Bell–Jackiw anomaly
Name	Adler–Bell–Jackiw anomaly
Field	Quantum field theory
Discovered	1969
Discoverers	Stephen Adler; John Bell; Roman Jackiw

Adler–Bell–Jackiw anomaly is a quantum anomaly discovered in 1969 that reveals the nonconservation of the axial vector current in certain quantum field theories despite classical symmetries. It plays a central role in the understanding of symmetries in particle physics and connects with phenomena in condensed matter, topology, and mathematics. The anomaly has influenced developments at institutions such as Princeton University, Harvard University, and CERN and has implications for experimental programs at facilities like SLAC National Accelerator Laboratory and Fermilab.

History and discovery

The anomaly was identified independently by Stephen L. Adler and by John S. Bell with Roman Jackiw in the late 1960s during studies connected to the Veltman-era investigations of renormalization and current algebra at CERN and Harvard University. Historical context includes earlier work on the axial current by Noether-inspired analyses and applications to processes studied at Brookhaven National Laboratory and CERN SPS. The discovery resolved puzzles arising in analyses by groups at University of Cambridge and Princeton University concerning the decay rates of the neutral pion deduced from current algebra and the PCAC program pursued by theorists at Cornell University and MIT. Following publication, the anomaly influenced research trajectories at Institute for Advanced Study and in collaborations involving researchers from Caltech and Columbia University.

Theoretical formulation

The anomaly appears in theories coupling fermions to gauge fields such as quantum electrodynamics and quantum chromodynamics. Its formulation uses the framework developed by Richard Feynman and Julian Schwinger for perturbative calculations and ties to symmetry principles associated with Emmy Noether. In the formulation of the anomaly, currents studied by Murray Gell-Mann and Sidney Coleman—notably the axial vector current—fail to be preserved after quantization in the presence of gauge interactions familiar from Electroweak interaction and Strong interaction models. Theoretical frameworks at Princeton University and University of Cambridge utilized path integral methods popularized by Paul Dirac and formalized in modern language by Ludwig Faddeev and Edward Witten.

Mathematical derivation

Derivations exploit regularization schemes introduced by Kenneth Wilson and the path integral jacobian analysis pioneered by Fujikawa. The standard perturbative derivation uses triangle Feynman diagrams first evaluated by practitioners trained under Richard Feynman and Freeman Dyson, demonstrating a nonvanishing divergence of the axial current via loop integrals regulated using methods of Gerard 't Hooft and Martinus Veltman. Alternative derivations draw on index theorems developed by Atiyah–Singer and topological arguments associated with instanton solutions studied by Alexander Belavin and Gabriele Veneziano. The anomaly is captured by a local operator proportional to the product of field strength tensors appearing in formulations by Steven Weinberg and relates to cohomological classifications familiar from work at Institute for Advanced Study by Edward Witten and Nathan Seiberg.

Physical implications and applications

The anomaly explains the observed decay rate of the neutral pion into two photons measured at facilities such as CERN and SLAC National Accelerator Laboratory, resolving discrepancies noted by experimental groups at Brookhaven National Laboratory. It constrains model building in Standard Model extensions pursued at Fermilab and guides anomaly cancellation conditions that determined fermion representations in constructions by Howard Georgi and Sheldon Glashow. In condensed matter, analogues of the anomaly arise in Weyl semimetals investigated at Bell Labs and Max Planck Institute for Solid State Research, impacting transport phenomena explored in collaborations with researchers at University of Cambridge and University of Illinois Urbana–Champaign. Cosmological consequences have been discussed by theorists at Harvard University and Perimeter Institute in contexts influenced by Andrei Linde and Alan Guth.

Experimental observations and tests

Empirical confirmation came from measurements of the pi0 decay rate and photon spectra at experiments conducted at CERN and SLAC National Accelerator Laboratory, with analyses by collaborations including personnel from Caltech and MIT. High-energy collider data from Large Electron–Positron Collider and subsequent programs at Large Hadron Collider collaborations provided further tests of anomaly-related predictions constraining theories by groups at Fermilab and Brookhaven National Laboratory. Condensed matter experiments probing the chiral magnetic effect in Weyl semimetal samples were carried out by teams at Max Planck Institute for Solid State Research and Stanford University, offering tabletop analogues of the anomaly comparable to particle experiments at CERN.

Generalizations and related anomalies

The Adler–Bell–Jackiw result led to recognition of other quantum anomalies such as the gravitational anomaly studied by researchers at Princeton University and Institute for Advanced Study, the trace anomaly examined by Callan and Coleman at Harvard University, and global anomalies analyzed by Edward Witten. Generalizations include anomaly inflow mechanisms developed by Juan Maldacena-era groups and applications in topological phases investigated by scientists at Microsoft Research and Max Planck Institute for Physics. Cancellation conditions inspired model building in grand unified theories proposed by Georgi–Glashow-style groups and constrained constructions in string theory developed at Institute for Advanced Study and CERN.

Category:Quantum anomalies