Chiral anomaly

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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.

The non-conservation happens in a tunneling process from one vacuum to another. Such a process is called instanton. In the case of a symmetry related to the conservation of a fermionic particle number, one may understand the creation of such particles as follows. The definition of a particle is different in the two vacuum states between which the tunneling occurs; Therefore a state of no particles in one vacuum corresponds to a state with some particles in the other vacuum.

In particular, there is a Dirac sea of fermions and when such a tunneling happens, it causes the energy levels of the sea fermions to gradually shift upwards for the particles and downwards for the anti-particles, or vice versa. This means particles which once belonged to the Dirac sea become real (positive energy) particles and particle creation happens.

Technically, an anomalous symmetry is a symmetry of the action, but not of the measure. The measure consists of a part depending of the fermion field dψ and a part depending on its complex conjugate d\bar{\psi}. The transformations of both parts under a chiral symmetry do not cancel in general. Note that if ψ is a Dirac fermion, then the chiral symmetry can be written as \psi \rightarrow e^{i \alpha \gamma^5}\psi where α is some matrix acting on ψ.

The anomaly is proportional to the instanton number of a gauge field to which the fermions are coupled (note that the gauge symmetry is always non anomalous and is exactly respected, as is required by the consistency of the theory). This means that the non-conservation happens

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[edit] Calculation

Chiral anomaly can be calculated exactly by one-loop Feynman diagrams. Image:Triangle_diagram.svg

It can also be calculated directly from the change in the measure of the fermionic fields under the chiral transformation.

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.


[edit] 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.

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

\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

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

where C is a numerical constant,

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

and the gauge field strength Gμν is given by the expression

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: G_{\mu\nu}\tilde{G}_{\mu\nu} = \partial_\mu K_\mu where the anomalous current Kμ is

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).

The last term in this expression is non-vanishing only for non-Abelian gauge theories because the antisymmetric product of three vector potentials Aν 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).

[edit] See also

[edit] Published articles

  • S. Adler (1969). "Axial-Vector Vertex in Spinor Electrodynamics". Physical Review 177: 2426–2438. doi:10.1103/PhysRev.177.2426. 
  • J.S.Bell and R.Jackiw (1969). "A PCAC puzzle: π0→γγ in the σ-model". Il Nuovo Cimento A 60: 47. 
  • P.H. Frampton & T.W. Kephart, Anomalies in Higher Dimensions, Phys. Rev. Lett. 50, 1343, 1347 (1983); Phys. Rev. D28, 1010 (1983).

[edit] 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 0-521-55002-5. 

[edit] 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.