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.

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.

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.

Fujikawa derived this anomaly using the correspondence between functional determinants and the partition function using the Atiyah-Singer index theorem. See Fujikawa's method.

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 ^* of the $U(1)$ group.

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 $q\backslash bar\; q$, so that a quark can disappear only in collision with an antiquark. In other words, the classical baryonic current $J\_\backslash mu^B$ is conserved:

$\backslash partial\_\backslash mu\; J\_\backslash mu^B\; =\; \backslash sum\_j\; \backslash partial\_\backslash mu(\backslash bar\; q\_j\; \backslash gamma\_\backslash mu\; q\_j)\; =\; 0.$

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

$\backslash partial\_\backslash mu\; J\_\backslash mu^B\; =\; \backslash frac\{g^2\; C\}\{16\backslash pi^2\}\; G\_\{\backslash mu\backslash nu\}\; \backslash tilde\{G\}\_\{\backslash mu\backslash nu\},$

where $C$ is a numerical constant,

$\backslash tilde\{G\}\_\{\backslash mu\backslash nu\}\; =\; \backslash frac\{1\}\{2\}\; G\_\{\backslash alpha\backslash beta\}\; \backslash epsilon\_\{\backslash mu\backslash nu\backslash alpha\backslash beta\}$

and the gauge field strenth $G\_\{\backslash mu\backslash nu\}$ is given by the expression

$G\_\{\backslash mu\backslash nu\}\; =\; \backslash partial\_\backslash mu\; A\_\backslash nu\; -\; \backslash partial\_\backslash nu\; A\_\backslash mu\; +\; gA\_\backslash nu.$

An important fact is that the anomalous current nonconservation is proportional to the total derivative of a vector operator: $G\_\{\backslash mu\backslash nu\}\backslash tilde\{G\}\_\{\backslash mu\backslash nu\}\; =\; \backslash partial\_\backslash mu\; K\_\backslash mu$ where the anomalous current $K\_\backslash mu$ is

$K\_\backslash mu\; =\; 2\backslash epsilon\_\{\backslash mu\backslash nu\backslash alpha\backslash beta\}\; \backslash left(\; A\_\backslash nu\; \backslash partial\_\backslash alpha\; A\_\backslash beta\; +\; \backslash frac\{2\}\{3\}\; i\; g\; A\_\backslash nu\; A\_\backslash alpha\; A\_\backslash beta\; \backslash right).$

The last term in this expression is non-vanishing only for non-Abelian gauge theories because the antisymmetric product of three vector potentials $A\_\backslash nu$ 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 $Z^0$).

See also

• Anomaly (physics)
• Global anomaly
• Gravitational anomaly

Published articles

Textbooks

• .

External links

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

Quantum field theory | Quantum chromodynamics | Standard Model | Conservation laws