 Chiral anomaly

A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with 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 nonconservation happens in a tunneling process from one vacuum to another. Such a process is called an 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 antiparticles, 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 , and therefore not of the generating functional of the quantized theory ( is Planck's actionquantum divided by 2π). The measure consists of a part depending of the fermion field and a part depending on its complex conjugate . The transformations of both parts under a chiral symmetry do not cancel in general. Note that if is a Dirac fermion, then the chiral symmetry can be written as where is some matrix acting on .
From the formula for one also sees explicitly that in the classical limit, anomalies don't come into play, since in this limit only the extrema of are relevant.
The anomaly is in fact proportional to the instanton number of a gauge field to which the fermions are coupled (note that the gauge symmetry is always nonanomalous and is exactly respected, as is required by the consistency of the theory).
Contents
Calculation
The chiral anomaly can be calculated exactly by oneloop Feynman diagrams, e.g. the famous "triangle diagram", contributing to the pion decays, .
Here, a wavy line signifies a photon, and the rightgoing (leftgoing) bold lines signify an electron and a positron, alternatingly. The fact that by encircling the triangle an odd number of times, a line which originally may be seen as a positron, returns as an electron, is actually already a signature for the anomalous behavior of the diagram.
This 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 WessZumino consistency conditions.
Fujikawa derived this anomaly using the correspondence between functional determinants and the partition function using the AtiyahSinger index theorem. See Fujikawa's method.
An example: baryonic charge nonconservation
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 (charge+parity), baryonic charge violation appears through the AdlerBellJackiw 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 , so that a quark can disappear only in collision with an antiquark. In other words, the classical baryonic current is conserved:
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 strength is given by the expression
An important fact is that the anomalous current nonconservation is proportional to the total derivative of a vector operator, (this is nonvanishing due to instanton configurations of the gauge field, which are pure gauge at the infinity), where the anomalous current K_{μ} is:
which is the Hodge dual of the ChernSimons 3form.
See also
 Anomaly (physics)
 Global anomaly
 Gravitational anomaly
 Strong CP problem
Further reading
Published articles
 S. Adler (1969). "AxialVector Vertex in Spinor Electrodynamics". Physical Review 177 (5): 2426. Bibcode 1969PhRv..177.2426A. doi:10.1103/PhysRev.177.2426.
 J.S. Bell and R. Jackiw (1969). "A PCAC puzzle: π^{0}→γγ in the σmodel". Il Nuovo Cimento A 60: 47. Bibcode 1969NCimA..60...47B. doi:10.1007/BF02823296.
 P.H. Frampton and T.W. Kephart (1983). "Explicit Evaluation of Anomalies in Higher Dimensions". Physical Review Letters 50 (18): 1343. Bibcode 1983PhRvL..50.1343F. doi:10.1103/PhysRevLett.50.1343.
 P.H. Frampton and T.W. Kephart (1983). "Analysis of anomalies in higher spacetime dimensions". Physical Review D28 (4): 1010. Bibcode 1983PhRvD..28.1010F. doi:10.1103/PhysRevD.28.1010.
 Alan R. White (2004). "Electroweak HighEnergy Scattering and the Chiral Anomaly". Physical Review D69 (9): 096002. arXiv:hepph/0308287. Bibcode 2004PhRvD..69i6002W. doi:10.1103/PhysRevD.69.096002.
 T. Csörgő, R. Vértesi and J. Sziklai (2010). "Indirect Observation of an InMedium η′ Mass Reduction in sqrt(s_{NN})=200 GeV Au+Au Collisions". Physical Review Letters 105: 182301. arXiv:0912.5526. Bibcode 2010PhRvL.105r2301C. doi:10.1103/PhysRevLett.105.182301.
Textbooks
 K. Fujikawa and H. Suzuki (May 2004). Path Integrals and Quantum Anomalies. Clarendon Press. ISBN 0198529139.
 S. Weinberg (2001). The Quantum Theory of Fields. Volume II: Modern Applications. Cambridge University Press. ISBN 0521550025.
Preprints
 Yang, J.F. (2003). "Trace and chiral anomalies in QED and their underlying theory interpretation". arXiv:hepph/0309311 [hepph].
 Yang, J.F. (2004). "Trace anomalies and chiral Ward identities". Chinese Physics Letters 21 (5): 792–794. arXiv:hepph/0403173. Bibcode 2004ChPhL..21..792Y. doi:10.1088/0256307X/21/5/008.
 Gozzi, E.; Mauro, D.; Silvestri, A. (2004). "Chiral Anomalies via Classical and Quantum Functional Methods". International Journal of Modern Physics A [Particles and Fields; Gravitation; Cosmology; Nuclear Physics] 20 (20 & 21): 5009. arXiv:hepth/0410129. Bibcode 2005IJMPA..20.5009G. doi:10.1142/S0217751X05025085.
 Dolgov, A.D. (1997). "Baryogenesis, 30 Years after". arXiv:hepph/9707419 [hepph].
Categories: Anomalies in physics
 Quantum chromodynamics
 Standard Model
 Conservation laws
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