| chiral anomaly | |
|---|---|
| Name | Chiral anomaly |
| Field | Quantum field theory |
| Discovered | 1960s |
| Discoverer | John Bell and Roman Jackiw; related work by Stephen Adler |
| Related | Axial current, Anomaly (physics), Adler–Bell–Jackiw anomaly |
chiral anomaly
The chiral anomaly, also called the axial anomaly or Adler–Bell–Jackiw anomaly, is the nonconservation of a classically conserved chiral or axial current at the quantum level. It arises when regularization and renormalization of a quantum field theory break a classical symmetry, with profound consequences for particle physics, quantum electrodynamics, and topological aspects of condensed matter physics. The anomaly explains otherwise puzzling decay rates and constrains consistent gauge theory constructions.
The chiral anomaly occurs when a symmetry of the classical action, typically a chiral symmetry rotating left- and right-handed components of fermions, fails to survive quantization. Seminal analyses by Stephen Adler, John Bell and Roman Jackiw in the late 1960s demonstrated that the divergence of the axial current acquires an extra term proportional to F ∧ F, where F is the electromagnetic field tensor in Quantum electrodynamics (QED). Physically, this explains the observed rate of the neutral pion decay π0 → γγ and constrains the particle content of Standard Model extensions. In condensed matter, analogues of the chiral anomaly produce measurable transport phenomena in Weyl semimetals via the chiral magnetic effect.
In relativistic quantum field theory, consider a massless Dirac fermion ψ coupled to a gauge field Aμ. Classically the vector current Jμ = ψ̄γμψ and the axial current Jμ5 = ψ̄γμγ5ψ are conserved. Quantum mechanically, triangle Feynman diagrams with three external currents produce a nonzero divergence: ∂μ Jμ5 = (e^2/16π^2) εμνρσ Fμν Fρσ, known as the Adler–Bell–Jackiw anomaly expression in QED. The precise coefficient is fixed by one-loop computation and gauge invariance of the vector current. In nonabelian Yang–Mills theory the anomaly involves tr(F ∧ F) and is related to the Chern–Simons form and topological charge (instanton number). Anomaly cancellation conditions in the Standard Model of particle physics enforce specific charge assignments and family structure to preserve gauge consistency.
The path integral formulation provides a transparent derivation attributed to Fujikawa. Under an infinitesimal chiral rotation of fermion fields, the fermionic measure in the functional integral acquires a Jacobian. Regularizing the eigenmode sum (e.g., with a heat kernel or Pauli–Villars regulators) yields the same F ∧ F term. The Fujikawa method connects anomalies to the spectral properties of the Dirac operator and to index theorems: the integrated anomaly equals twice the Atiyah–Singer index theorem index for the Dirac operator on a compact manifold, linking anomalies to topology and instantons in nonperturbative sectors.
Anomalies that break global symmetries can have physical consequences, but anomalies that break gauge symmetries render a theory inconsistent due to loss of unitarity or renormalizability. In constructing grand unified theorys and extensions like supersymmetry or string theory, anomaly cancellation is a central constraint. Famous mechanisms include the Green–Schwarz mechanism in string theory and the requirement that chiral fermion representations yield vanishing gauge and mixed anomalies in the Standard Model. Global anomalies, such as the Witten anomaly for SU(2), impose further topological consistency requirements.
In particle physics, the chiral anomaly explains the decay π0 → γγ via the axial current breaking and sets selection rules for processes involving gauge bosons and pseudo-Goldstone bosons from spontaneous symmetry breaking. Anomaly matching conditions by Gerard 't Hooft constrain strongly coupled dynamics and confinement scenarios. In condensed matter, analogues appear in Weyl semimetals and graphene, where low-energy quasiparticles behave as Weyl fermions and show signatures like the chiral magnetic and chiral separation effects; these link to experimental platforms studied at institutions such as Max Planck Institute for Solid State Research and collaborations across ETH Zurich and Harvard University.
Direct particle physics evidence comes from measured branching ratios of π0 decay and from processes constrained by anomaly-cancellation-based charge assignments, verified at accelerators including CERN and SLAC National Accelerator Laboratory. In condensed matter, characteristic transport signatures—negative longitudinal magnetoresistance and nonlocal responses under parallel electric and magnetic fields—have been reported in materials like TaAs and NbAs, interpreted as manifestations of the chiral anomaly in Weyl nodes. Neutron electric dipole moment searches, precision electroweak tests at LEP, and rare decay searches further probe related anomaly-driven effects.
Beyond gauge anomalies, gravitational and mixed anomalies arise when chiral currents couple to spacetime curvature. The gravitational anomaly leads to nonconservation proportional to R ∧ R (with R the Riemann curvature tensor two-form) and affects chiral fermions in curved backgrounds and in theories with quantized gravity. Mixed gauge–gravitational anomalies and their cancellation are crucial in consistent string compactifications and in the study of black hole entropy via anomaly inflow. These extensions connect to modern topics such as holography in the AdS/CFT correspondence, topological phases of matter, and the role of anomalies in transport coefficients calculated using hydrodynamics and Kubo formulas.
Category:Quantum field theory Category:Anomalies (physics)