A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with a chiral symmetry the quantization may lead to the breaking of this (global) chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved. An anomaly is a deviation from the common rule. ... A phenomenon is said to be chiral if it is not identical to its mirror image (see Chirality (mathematics)). The spin of a particle may be used to define a handedness for that particle. ... Spinor ...

A heuristic handwaving way of explaining this is to suppose there is a Dirac sea of fermions and a large (and therefore adiabatic) instanton suddenly appears, and suddenly, the energy levels gradually shift upwards or downwards. This means particles which once belonged to the Dirac sea suddenly become conspicuous particles and what looks like a particle creation happens. This isn't a very satisfactory explanation, however. The term handwaving is used in mathematics and physics to describe arguments that are not mathematically rigorous. ... The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles possessing negative energy. ... In thermodynamics, an adiabatic process is a process in which no heat is transferred to or from working fluid. ... This article is in need of attention from an expert on the subject. ... A quantum mechanical system can only be in certain states, so that only certain energy levels are possible. ...

triangle feynman diagram File links The following pages link to this file: Chiral anomaly One-loop Feynman diagram Gravitational anomaly Gauge anomaly Mixed anomaly Categories: User-created public domain images ...

Wess and Zumino developed a set of conditions on how the partition function ought to behave under gauge transformations called the Wess-Zumino consistency conditions. In quantum field theory, we have a generating functional, Z[J] of correlation functions and this value, called the partition function is usually expressed by something like the following functional integral: where S is the action functional. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

Fujikawa derived this anomaly using the correspondence between functional determinants and the partition function using the Atiyah-Singer index theorem. See Fujikawa's method. If S is a linear operator mapping a space of functionals to itself, it is possible to define an infinite-dimensional generalization of the determinant in some cases. ... In quantum field theory, we have a generating functional, Z[J] of correlation functions and this value, called the partition function is usually expressed by something like the following functional integral: where S is the action functional. ... In the mathematics of manifolds and differential operators, the Atiyah-Singer index theorem is an important unifying result that connects topology and analysis. ... Fujikawas method is a way of deriving the chiral anomaly in quantum field theory. ...

An example: baryonic charge non-conservation

The Standard Model of electroweak interactions has all the necessary ingredients for successful baryogenesis. Beyond the violation of charge conjugation C and CP violation CP, baryonic charge violation appears through the Adler-Bell-Jackiw anomaly [5] of the U(1) group. In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... Baryogenesis is the generic designation for the physical processes that generate matter (more specifically, a class of fundamental particle called baryon) from an otherwise matter-empty state (such as it is generally believed to be the state of the Universe at its onset, the so-called Big Bang). ... C-symmetry means the symmetry of physical laws over a charge-inversion transformation. ... John Bell (left) and Martinus Veltman (right) discussing Physics at CERN John S. Bell (June 28, 1928 – October 1, 1990) was a physicist who became well known as the originator of Bells Theorem, regarded by some in the quantum physics community as one of the most important theorems of... Roman Jackiw is a theoretical physicist and Dirac Medallist. ...

Baryons are not conserved by the usual electroweak interactions due to quantum chiral anomaly. The classic electroweak Lagrangian conserves baryonic charge. Quarks always enter in bilinear combinations , so that a quark can disappear only in collision with an antiquark. In other words, the classical baryonic current is conserved: A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... In particle physics, the baryons are a family of subatomic particles including the proton and the neutron (collectively called nucleons), as well as a number of unstable, heavier particles (called hyperons). ...

However, quantum corrections destroy this conservation law and instead of zero in the right hand side of this equation, one gets

where C is a numerical constant,

and the gauge field strenth G_μν is given by the expression

An important fact is that the anomalous current nonconservation is proportional to the total derivative of a vector operator: where the anomalous current K_μ is

The last term in this expression is non-vanishing only for non-Abelian gauge theories because the antisymmetric product of three vector potentials A_ν can be nonzero due to different group indices (e.g. for the electroweak group it should contain the product of W ⁺ , W ⁻ and the isospin part of Z⁰). In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ...

See also

• Anomaly (physics)
• Global anomaly
• Gravitational anomaly

In physics, an anomaly is a classical symmetry — a symmetry of the Lagrangian — that is broken in quantum field theories. ... In theoretical physics, a global anomaly is a type of anomaly: in this particular case, it is a quantum effect that invalidates a discrete symmetry that would otherwise be preserved in the classical theory. ... Anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams In theoretical physics, a gravitational anomaly is an example of an anomaly: it is an effect of quantum mechanics - usually a one-loop diagram - that invalidates the general covariance of a theory of general relativity combined with some...

Published articles

• S. Adler (1969). "Axial-Vector Vertex in Spinor Electrodynamics". Physical Review 177: 2426–2438.
• J.S.Bell and R.Jackiw (1969). "A PCAC puzzle: π⁰→γγ in the σ-model". Il Nuovo Cimento A 60: 47.

Textbooks

• K. Fujikawa and H. Suzuki (May 2004). Path Integrals and Quantum Anomalies, Clarendon Press. ISBN 0-19-852913-9.
• S. Weinberg (2001). The Quantum Theory of Fields. Volume II: Modern Applications, Cambridge University Press. ISBN 0-52-155002-5..

External links

• [1] A. R. White, Electroweak High-Energy Scattering and the Chiral Anomaly. hep-ph/0308287.
• [2] J.-F. Yang, Trace and chiral anomalies in QED and their underlying theory interpretation. hep-ph/0309311.
• [3] J.-F. Yang, Trace anomalies and chiral Ward identities. hep-ph/0403173.
• [4] E. Gozzi, D. Mauro, A. Silvestri, Chiral Anomalies via Classical and Quantum Functional Methods. hep-th/0410129.
• [5] A. D. Dolgov, Baryogenesis, 30 years after. hep-ph/9707419.