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In physics, a '''chiral anomaly''' is the [[anomaly (physics)|anomalous]] [[nonconservation]] of a [[chirality (physics)|chiral]] current. In some theories of [[fermion]]s with [[chiral symmetry]], the [[Quantization (physics)|quantization]] may lead to the breaking of this (global) chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.
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The non-conservation happens in a [[Quantum tunneling|tunneling]] process from one [[Vacuum#Quantum-mechanical_definition|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 level]]s 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 (physics)|action]] <math>\mathcal A</math>, but not of the [[measure (physics)|measure]] <math>\,\mu</math>, and  therefore ''not'' of the [[partition function (quantum field theory)|generating functional]] <math>\mathcal Z=\int {d \mu\,\exp (i \mathcal A/\hbar)}</math> of the quantized theory (<math>\hbar</math> is Planck's action-quantum divided by <math>2\pi</math>). The measure consists of a part depending of the fermion field <math>\! d\psi</math> and a part depending on its complex conjugate <math>d\bar{\psi}</math>. The transformations of both parts under a chiral symmetry do not cancel in general. Note that if <math>\! \psi</math> is a [[Dirac fermion]], then the  chiral symmetry can be written as <math>\psi \rightarrow e^{i \alpha \gamma^5}\psi</math> where <math>\! \alpha</math> is some matrix acting on <math>\! \psi</math>.
 
From the formula for <math>\mathcal Z</math> one also sees explicitly that in the classical limit, <math>\hbar \to 0,</math> anomalies don't come into play, since in this limit only the extrema of <math>\mathcal A</math> 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 non-anomalous and is exactly respected, as is required by the consistency of the theory).
 
==Calculation==
The chiral anomaly can be calculated exactly by [[one-loop Feynman diagram]]s, e.g. the famous "triangle diagram", contributing to the [[pion]] decays, <math>\pi^0 \to\gamma\gamma</math> and <math>\pi^0\to e^+e^-\gamma</math>.
 
The amplitute for this process can be calculated directly from the change in the [[measure (physics)|measure]] of the fermionic fields under the chiral transformation.
 
Wess and Zumino developed a set of conditions on how the [[Partition function (quantum field theory)|partition function]] ought to behave under [[gauge transformation]]s called the [[Wess-Zumino consistency conditions]].
 
Fujikawa derived this anomaly using the correspondence between [[functional determinant]]s and the [[Partition function (quantum field theory)|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]] <math>C</math> and [[CP violation]] <math>CP</math> (charge+parity), baryonic charge violation appears through the Adler-Bell-Jackiw anomaly of the  <math>U(1)</math> group.
 
Baryons are not conserved by the usual electroweak interactions due to quantum chiral anomaly. The classic electroweak [[Lagrangian]] conserves [[baryon]]ic charge. Quarks always enter in bilinear combinations <math>q\bar q</math>, so that a quark can disappear only in collision with an antiquark. In other words, the classical baryonic current <math>J_\mu^B</math> is conserved:
 
:<math>\partial^\mu J_\mu^B = \sum_j \partial^\mu(\bar q_j \gamma_\mu q_j) = 0. </math>
 
However, quantum corrections destroy this conservation law and instead of zero in the right hand side of this equation, one gets
 
:<math>\partial^\mu J_\mu^B = \frac{g^2 C}{16\pi^2} G^{\mu\nu a} \tilde{G}_{\mu\nu}^a,</math>
 
where <math>C</math> is a numerical constant,
 
:<math>\tilde{G}_{\mu\nu}^a = \frac{1}{2} \epsilon_{\mu\nu\alpha\beta} G^{\alpha\beta a},</math>
 
and the gauge field strength <math>G_{\mu\nu}^a</math> is given by the expression
 
:<math>G_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f^{a}_{bc} A_\mu^b A_\nu^c. </math>
 
An important fact is that the anomalous current non-conservation is proportional to the total derivative of a vector operator, <math>G^{\mu\nu a}\tilde{G}_{\mu\nu}^a = \partial^\mu K_\mu</math> (this is non-vanishing due to [[instanton]] configurations of the gauge field, which are pure gauge at the infinity), where the anomalous current <math>K_\mu</math> is:
 
:<math>K_\mu = 2\epsilon_{\mu\nu\alpha\beta} \left( A^{\nu a} \partial^\alpha A^{\beta a} + \frac{1}{3} f^{abc} A^{\nu a} A^{\alpha b} A^{\beta c} \right),</math>
 
which is the [[Hodge star|Hodge dual]] of the [[Chern-Simons]] 3-form.
 
==See also==
* [[Anomaly (physics)]]
* [[Global anomaly]]
* [[Gravitational anomaly]]
* [[strong CP problem#Strong_CP_problem|Strong CP problem]]
 
==Further reading==
 
===Published articles===
*{{cite journal
| author = S. Adler
| year = 1969
| title = Axial-Vector Vertex in Spinor Electrodynamics
| journal = [[Physical Review]]
| volume = 177 | pages = 2426
| doi = 10.1103/PhysRev.177.2426
|bibcode = 1969PhRv..177.2426A
| issue = 5 }}
*{{cite journal
| author = J.S. Bell and R. Jackiw
| year = 1969
| title=A PCAC puzzle: &pi;<sup>0</sup>&rarr;&gamma;&gamma; in the &sigma;-model
| journal = [[Il Nuovo Cimento A]]
| volume = 60 | pages = 47
| doi=10.1007/BF02823296
|bibcode = 1969NCimA..60...47B }}
*{{cite journal
| author = P.H. Frampton and T.W. Kephart
| year = 1983
| title = Explicit Evaluation of Anomalies in Higher Dimensions
| journal =[[Physical Review Letters]]
| volume = 50 |pages=1343
| doi =10.1103/PhysRevLett.50.1343
| bibcode=1983PhRvL..50.1343F
| issue = 18
}}
*{{cite journal
| author = P.H. Frampton and T.W. Kephart
| year = 1983
| title = Analysis of anomalies in higher space-time dimensions
| journal = Physical Review
| volume = D28 |pages=1010
| doi =10.1103/PhysRevD.28.1010
|bibcode = 1983PhRvD..28.1010F
| issue = 4 }}
*{{cite journal
| author = Alan R. White
| year = 2004
| title = Electroweak High-Energy Scattering and the Chiral Anomaly
| journal = Physical Review
| volume = D69 | pages = 096002
| doi = 10.1103/PhysRevD.69.096002
| arxiv = hep-ph/0308287
|bibcode = 2004PhRvD..69i6002W
| issue = 9 }}
*{{cite journal
| author = T. Csörgő, R. Vértesi and J. Sziklai
| year = 2010
| title = Indirect Observation of an In-Medium η′ Mass Reduction in sqrt(s_{NN})=200  GeV Au+Au Collisions
| journal = [[Physical Review Letters]]
| volume = 105 | pages = 182301
| doi = 10.1103/PhysRevLett.105.182301
| arxiv = 0912.5526
|bibcode = 2010PhRvL.105r2301C }}
 
===Textbooks===
* {{cite book
| author=K. Fujikawa and H. Suzuki
| title=Path Integrals and Quantum Anomalies
| publisher=Clarendon Press
|date=May 2004
| isbn = 0-19-852913-9
}}
* {{cite book
| author=S. Weinberg
| title=The Quantum Theory of Fields. Volume II: Modern Applications
| publisher=Cambridge University Press
| year=2001
| isbn = 0-521-55002-5
}}
 
===Preprints===
*{{cite arXiv
|eprint=hep-ph/0309311
|last1=Yang | first1=J.-F.
|title=Trace and chiral anomalies in QED and their underlying theory interpretation
|class=hep-ph
|year=2003
}}
*{{cite journal
|last1=Yang | first1=J.-F.
|title=Trace anomalies and chiral Ward identities
|year=2004
|doi=10.1088/0256-307X/21/5/008
|journal=Chinese Physics Letters
|volume=21
|issue=5
|pages=792–794
|arxiv=hep-ph/0403173
|bibcode = 2004ChPhL..21..792Y }}
*{{cite journal
|last1=Gozzi | first1=E.
|last2=Mauro | first2=D.
|last3=Silvestri | first3=A.
|title=Chiral Anomalies via Classical and Quantum Functional Methods
|year=2004
|doi=10.1142/S0217751X05025085
|journal=International Journal of Modern Physics A [Particles and Fields; Gravitation; Cosmology; Nuclear Physics]
|volume=20
|issue=20 & 21
|pages=5009
|arxiv=hep-th/0410129
|bibcode = 2005IJMPA..20.5009G }}
*{{cite arXiv
|eprint=hep-ph/9707419
|last1=Dolgov | first1=A.D.
|title=Baryogenesis, 30 Years after
|class=hep-ph
|year=1997
}}
 
[[Category:Anomalies in physics]]
[[Category:Quantum chromodynamics]]
[[Category:Standard Model]]
[[Category:Conservation laws]]

Revision as of 21:59, 20 February 2014

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