# Chiral anomaly

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 ${\displaystyle C}$ and CP violation ${\displaystyle CP}$, baryonic charge violation appears through the Adler-Bell-Jackiw anomaly [5] of the ${\displaystyle U(1)}$ 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 ${\displaystyle q{\bar {q}}}$, so that a quark can disappear only in collision with an antiquark. In other words, the classical baryonic current ${\displaystyle J_{\mu }^{B}}$ is conserved:

${\displaystyle \partial _{\mu }J_{\mu }^{B}=\sum _{j}\partial _{\mu }({\bar {q}}_{j}\gamma _{\mu }q_{j})=0.}$

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

${\displaystyle \partial _{\mu }J_{\mu }^{B}={\frac {g^{2}C}{16\pi ^{2}}}G_{\mu \nu }{\tilde {G}}_{\mu \nu },}$

where ${\displaystyle C}$ is a numerical constant,

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

and the gauge field strenth ${\displaystyle G_{\mu \nu }}$ is given by the expression

${\displaystyle G_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }+g[A_{\mu }A_{\nu }].}$

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

${\displaystyle K_{\mu }=2\epsilon _{\mu \nu \alpha \beta }\left(A_{\nu }\partial _{\alpha }A_{\beta }+{\frac {2}{3}}igA_{\nu }A_{\alpha }A_{\beta }\right).}$

The last term in this expression is non-vanishing only for non-Abelian gauge theories because the antisymmetric product of three vector potentials ${\displaystyle A_{\nu }}$ can be nonzero due to different group indices (e.g. for the electroweak group it should contain the product of ${\displaystyle W^{+}}$, ${\displaystyle W^{-}}$ and the isospin part of ${\displaystyle Z^{0}}$).