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| Adler–Bardeen theorem | |
|---|---|
| Name | Adler–Bardeen theorem |
| Field | Quantum field theory |
| Proved | 1969 |
| Proposer | Stephen L. Adler; William A. Bardeen |
| Related | Chiral anomaly; Renormalization; Gauge theory |
Adler–Bardeen theorem The Adler–Bardeen theorem is a result in quantum field theory asserting that certain quantum anomalies are exact at one-loop order and receive no further perturbative corrections. The theorem connects work in particle physics involving Stephen L. Adler, William A. Bardeen, John S. Bell, Roman Jackiw, and groups at institutions such as Princeton University and Fermi National Accelerator Laboratory, and influences analyses in models studied by Sheldon Glashow, Steven Weinberg, Abdus Salam, and Murray Gell-Mann.
The theorem addresses the behavior of the chiral or axial anomaly discovered in calculations by Stephen L. Adler and John S. Bell with Roman Jackiw and formalized in later work by William A. Bardeen. In contexts involving gauge symmetries studied in theories by Sheldon Glashow, Steven Weinberg, Abdus Salam, and Gerard 't Hooft, anomalies can spoil consistency unless cancelled as in constructions by Michael Green and John H. Schwarz. The Adler–Bardeen result constrains perturbative corrections in frameworks developed at places such as Harvard University, Yale University, and University of Cambridge and has ramifications for model building used at CERN, SLAC National Accelerator Laboratory, and Brookhaven National Laboratory.
In its common formulation the theorem states that the coefficient of the anomalous divergence of an axial current computed at one-loop order is exact and unaffected by higher-loop diagrams in renormalizable gauge theories. The assertion was proven in formulations influenced by techniques from Kenneth G. Wilson's renormalization ideas, the regularization schemes used by Gerard 't Hooft and Martinus Veltman, and algebraic methods related to work by Claude Itzykson and Jean-Bernard Zuber. The statement is often invoked in analyses of anomaly cancellation in models proposed by Howard Georgi and David Gross and in discussions connecting to the axial anomaly earlier described by Lev Landau-era investigations and later by Yoichiro Nambu.
The discovery of the axial anomaly originated with calculations by Stephen L. Adler and by John S. Bell with Roman Jackiw in the late 1960s, followed by the formal nonrenormalization statement by William A. Bardeen in 1969. Subsequent clarifications and alternative derivations involved contributors from institutions such as Princeton University, Massachusetts Institute of Technology, and Institute for Advanced Study including Gerard 't Hooft, Martinus Veltman, Kenneth G. Wilson, Alexander Polyakov, and Sidney Coleman. Later developments tying anomalies to topology drew on mathematics associated with Michael Atiyah and Isadore Singer and influenced string theory results by Michael Green and John H. Schwarz.
Proofs employ perturbative diagrammatic analysis pioneered in the schools of Richard Feynman and formal operator methods influenced by Julian Schwinger and Erwin Schrödinger-era operator formalisms. Methods include regularization schemes such as dimensional regularization developed by G. 't Hooft and Martin Veltman, Pauli–Villars regularization introduced by Wolfgang Pauli and Felix Villars, and algebraic renormalization techniques linked to work by Olivier Piguet and Silvio Sorella. Nonperturbative and topological proofs connect to the index theorem of Michael Atiyah and Isadore Singer, and to instanton analyses by Alexander Belavin, Alexander Polyakov, Albert Schwarz, and Giorgio 't Hooft.
The theorem constrains anomaly cancellation conditions crucial for consistency of the Standard Model developed by Sheldon Glashow, Steven Weinberg, and Abdus Salam and for grand unified theories proposed by Howard Georgi, Sheldon Glashow, and Hugh Georgi. It plays a central role in computations of processes like the decay π0 → γγ analyzed by Nambu-influenced chiral models and in electroweak precision tests performed at LEP and Tevatron. The Adler–Bardeen result informs model building in supersymmetric frameworks introduced by Pierre Fayet and Sergio Ferrara, in string constructions by Michael Green and John H. Schwarz, and in anomaly-mediated supersymmetry breaking scenarios discussed in literature from Stanford University and Princeton University research groups.
Generalizations extend the nonrenormalization property to supersymmetric gauge theories studied by Edward Witten and Nathan Seiberg and to algebraic renormalization frameworks explored by Olivier Piguet. Limitations appear when nonperturbative effects, global anomalies identified by Edward Witten in SU(2) contexts, or gravitational anomalies considered by Alvarez-Gaumé and Erik Witten modify naive conclusions. The interplay with topology via the Atiyah–Singer index theorem imposes constraints in contexts studied at Institute for Advanced Study and in string theory by Michael Green and John H. Schwarz.
Canonical examples include the computation of the axial anomaly in quantum electrodynamics as performed by Stephen L. Adler and in quantum chromodynamics studied by Murray Gell-Mann and David Gross. In grand unified models such as those proposed by Howard Georgi and Sheldon Glashow, the Adler–Bardeen nonrenormalization ensures anomaly coefficients computed at one loop determine gauge consistency. Supersymmetric examples analyzed by Edward Witten and Nathan Seiberg illustrate extensions where holomorphy and nonrenormalization theorems combine, while instanton computations by Alexander Belavin and Giorgio 't Hooft show how nonperturbative sectors interact with the perturbative Adler–Bardeen result.