# Chiral anomaly

Chiral anomaly is the anomalous nonconservation of charge in a quantized theory of chiral fermions coupled to a background gauge field.

It may be a bit surprising, but charges simply are not conserved in such a theory. 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 conspicious particles and what looks like a particle creation happens. This isn't a very satisfactory explanation, however.

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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.

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

[itex]\partial_\mu J_\mu^B = \sum_j \partial_\mu(\bar q_j \gamma_\mu q_j) = 0. [itex]

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

[itex]\partial_\mu J_\mu^B = \frac{g^2 C}{16\pi^2} G_{\mu\nu} \tilde{G}_{\mu\nu},[itex]

where [itex]C[itex] is a numerical constant,

[itex]\tilde{G}_{\mu\nu} = \frac{1}{2} G_{\alpha\beta} \epsilon_{\mu\nu\alpha\beta}[itex]

and the gauge field strenth [itex]G_{\mu\nu}[itex] is given by the expression

[itex]G_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + g[A_\mu A_\nu]. [itex]

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

[itex]K_\mu = 2\epsilon_{\mu\nu\alpha\beta} \left( A_\nu \partial_\alpha A_\beta + \frac{2}{3} i g A_\nu A_\alpha A_\beta \right).[itex]

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

### Published articles

• S. Adler, Physical Review volume 177, page 2426, 1969.
• J.S.Bell and R.Jackiw, Il Nuovo Cimento A volume 60, page 47, 1969.

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

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