| chiral anomaly | |
|---|---|
| Name | Chiral anomaly |
| Field | Quantum field theory |
| Discovered | 1960s |
| Introduced by | Adler, Bell and Jackiw |
chiral anomaly
The chiral anomaly is the breakdown of a classical chiral symmetry of a field theory upon quantization, producing nonconservation of the chiral (axial) current. It is significant in Quantum field theory and particle physics because it modifies symmetry-based predictions, enforces constraints on consistent gauge theory models, and connects deep topology with observable processes like pion decay and transport in condensed matter physics.
The chiral anomaly arises when a classical symmetry under independent left- and right-handed rotations of fermions does not survive in the quantized theory. In a theory with massless Dirac fermions, the separate conservation of vector and axial currents at the classical level can be spoiled by loop effects, producing an anomalous divergence proportional to field-strength combinations such as F\tilde{F}. This phenomenon has direct physical consequences: it explains the rate of the neutral pion decay π0 → 2γ through the axial anomaly calculation by Adler and Bell–Jackiw, constrains model building in the Standard Model via anomaly cancelation requirements, and underpins effects like the chiral magnetic effect observed in heavy-ion collisions and certain Weyl semimetals.
In Quantum electrodynamics and non-Abelian Yang–Mills theory, the chiral anomaly is captured by the anomalous Ward identity for the axial current Ja5μ: ∂μ Ja5μ = (g^2/16π^2) Tr[Fμν \tilde{F}μν] for massless fermions, where F is the gauge field strength and \tilde{F} its dual. The computation typically evaluates the triangle Feynman diagram with one axial and two vector vertices, originally performed by Adler and independently by Bell and Jackiw. The anomaly coefficient is related to traces over gauge group generators and the representation of fermions, making it a group-theoretic constraint in model construction for GUTs and the Standard Model.
Topological interpretations connect the anomaly to the index theorem of Atiyah–Singer: spectral flow of the Dirac operator in background gauge fields leads to nonconservation of chiral charge equal to the topological winding number (instantons) in SU(N) gauge theories. The anomaly can also be formulated using functional determinants in the path integral approach, where Jacobians under chiral transformations produce the anomalous term, an analysis advanced by Fujikawa.
Gauge anomalies, if present, render a quantum field theory inconsistent due to loss of gauge invariance and unitarity. Thus anomaly cancelation is a stringent requirement for consistent gauge theorys. In the Standard Model, anomaly cancelation between quark and lepton representations (e.g., cancellation of the U(1), SU(2), and SU(3) mixed anomalies) is essential; this guided early particle physics model building and influenced the structure of electroweak theory developed by Glashow, Salam, and Weinberg. In proposed extensions such as GUT models or string theory, anomaly cancelation mechanisms—like the Green–Schwarz mechanism in heterotic string constructions—play a crucial role. Global anomalies, such as the Witten anomaly, further constrain allowed fermion representations in gauge groups like SU(2).
In particle physics, the chiral anomaly explains π0 → 2γ decay rate and affects processes mediated by instantons, baryon- and lepton-number violating amplitudes in electroweak theory, and properties of the axial current in deep inelastic scattering studied at facilities like CERN and SLAC. In condensed matter, emergent relativistic quasiparticles in graphene, Weyl semimetals, and topological insulators realize analogues of the chiral anomaly, producing measurable signatures such as negative longitudinal magnetoresistance associated with the chiral magnetic effect. Experimental programs at institutions including Brookhaven National Laboratory and materials research groups in universities have probed these transport anomalies, connecting high-energy anomaly concepts to table-top experiments.
The anomaly was first elucidated in the 1960s by Adler and by Bell and Jackiw independently, resolving a long-standing discrepancy in pion decay predictions. Subsequent theoretical work by Stephen Weinberg, Gerard 't Hooft, and others linked anomalies to topology and instantons, with 't Hooft demonstrating anomaly-induced processes that violate classical symmetries. Experimental confirmation of anomaly-predicted π0 decay rates helped establish the result; later observations of anomaly-like transport in Weyl semimetal experiments (e.g., works by groups at Max Planck Institute for Chemical Physics of Solids and Columbia University) have provided condensed-matter evidence. Large-scale tests of anomaly-related phenomena have been pursued at LHC and in heavy-ion collision programs at RHIC.
Calculation of the chiral anomaly requires careful regularization to preserve gauge invariance while tracking axial-current nonconservation. Methods include Pauli–Villars regularization, dimensional regularization with specific treatment of γ5, and the Fujikawa path-integral method which evaluates the Jacobian under chiral rotations. Lattice gauge theory addresses anomalies via discretizations like Wilson fermions and overlap fermions, with the Ginsparg–Wilson relation ensuring correct anomaly reproduction; key institutions advancing these techniques include CERN lattice groups and collaborations such as the USQCD consortium. Perturbative computations use triangle diagram evaluations and renormalization-group considerations to track anomaly coefficients, while nonperturbative insights employ instanton calculus and index-theorem machinery.
Category:Quantum field theory Category:Particle physics Category:Condensed matter physics