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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]].
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| 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.
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| 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.
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| 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>.
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| 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.
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| 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).
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| ==Calculation==
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| 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>.
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| 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.
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| 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]].
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| 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]].
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| ==An example: baryonic charge non-conservation==
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| 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.
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| 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:
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| :<math>\partial^\mu J_\mu^B = \sum_j \partial^\mu(\bar q_j \gamma_\mu q_j) = 0. </math>
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| However, quantum corrections destroy this conservation law and instead of zero in the right hand side of this equation, one gets
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| :<math>\partial^\mu J_\mu^B = \frac{g^2 C}{16\pi^2} G^{\mu\nu a} \tilde{G}_{\mu\nu}^a,</math>
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| where <math>C</math> is a numerical constant,
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| :<math>\tilde{G}_{\mu\nu}^a = \frac{1}{2} \epsilon_{\mu\nu\alpha\beta} G^{\alpha\beta a},</math>
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| and the gauge field strength <math>G_{\mu\nu}^a</math> is given by the expression | |
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| :<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>
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| 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:
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| :<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>
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| which is the [[Hodge star|Hodge dual]] of the [[Chern-Simons]] 3-form.
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| ==See also==
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| * [[Anomaly (physics)]]
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| * [[Global anomaly]]
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| * [[Gravitational anomaly]]
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| * [[strong CP problem#Strong_CP_problem|Strong CP problem]]
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| ==Further reading==
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| ===Published articles===
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| *{{cite journal
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| | author = S. Adler
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| | year = 1969
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| | title = Axial-Vector Vertex in Spinor Electrodynamics
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| | journal = [[Physical Review]]
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| | volume = 177 | pages = 2426
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| | doi = 10.1103/PhysRev.177.2426
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| |bibcode = 1969PhRv..177.2426A
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| | issue = 5 }}
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| *{{cite journal
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| | author = J.S. Bell and R. Jackiw
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| | year = 1969
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| | title=A PCAC puzzle: π<sup>0</sup>→γγ in the σ-model
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| | journal = [[Il Nuovo Cimento A]]
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| | volume = 60 | pages = 47
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| | doi=10.1007/BF02823296
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| |bibcode = 1969NCimA..60...47B }}
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| *{{cite journal
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| | author = P.H. Frampton and T.W. Kephart
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| | year = 1983
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| | title = Explicit Evaluation of Anomalies in Higher Dimensions
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| | journal =[[Physical Review Letters]]
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| | volume = 50 |pages=1343
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| | doi =10.1103/PhysRevLett.50.1343
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| | bibcode=1983PhRvL..50.1343F
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| | issue = 18
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| }}
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| *{{cite journal
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| | author = P.H. Frampton and T.W. Kephart
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| | year = 1983
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| | title = Analysis of anomalies in higher space-time dimensions
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| | journal = Physical Review
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| | volume = D28 |pages=1010
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| | doi =10.1103/PhysRevD.28.1010
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| |bibcode = 1983PhRvD..28.1010F
| |
| | issue = 4 }}
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| *{{cite journal
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| | author = Alan R. White
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| | year = 2004
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| | title = Electroweak High-Energy Scattering and the Chiral Anomaly
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| | journal = Physical Review
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| | volume = D69 | pages = 096002
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| | doi = 10.1103/PhysRevD.69.096002
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| | arxiv = hep-ph/0308287
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| |bibcode = 2004PhRvD..69i6002W
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| | issue = 9 }}
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| *{{cite journal
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| | author = T. Csörgő, R. Vértesi and J. Sziklai
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| | year = 2010
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| | title = Indirect Observation of an In-Medium η′ Mass Reduction in sqrt(s_{NN})=200 GeV Au+Au Collisions
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| | journal = [[Physical Review Letters]]
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| | volume = 105 | pages = 182301
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| | doi = 10.1103/PhysRevLett.105.182301
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| | arxiv = 0912.5526
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| |bibcode = 2010PhRvL.105r2301C }}
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| ===Textbooks===
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| * {{cite book
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| | author=K. Fujikawa and H. Suzuki
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| | title=Path Integrals and Quantum Anomalies
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| | publisher=Clarendon Press
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| |date=May 2004
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| | isbn = 0-19-852913-9
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| }}
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| * {{cite book
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| | author=S. Weinberg
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| | title=The Quantum Theory of Fields. Volume II: Modern Applications
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| | publisher=Cambridge University Press
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| | year=2001
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| | isbn = 0-521-55002-5
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| }}
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| ===Preprints===
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| *{{cite arXiv
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| |eprint=hep-ph/0309311
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| |last1=Yang | first1=J.-F.
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| |title=Trace and chiral anomalies in QED and their underlying theory interpretation
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| |class=hep-ph
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| |year=2003
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| }}
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| *{{cite journal
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| |last1=Yang | first1=J.-F.
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| |title=Trace anomalies and chiral Ward identities
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| |year=2004
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| |doi=10.1088/0256-307X/21/5/008
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| |journal=Chinese Physics Letters
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| |volume=21
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| |issue=5
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| |pages=792–794
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| |arxiv=hep-ph/0403173
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| |bibcode = 2004ChPhL..21..792Y }}
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| *{{cite journal
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| |last1=Gozzi | first1=E.
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| |last2=Mauro | first2=D.
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| |last3=Silvestri | first3=A.
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| |title=Chiral Anomalies via Classical and Quantum Functional Methods
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| |year=2004
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| |doi=10.1142/S0217751X05025085
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| |journal=International Journal of Modern Physics A [Particles and Fields; Gravitation; Cosmology; Nuclear Physics]
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| |volume=20
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| |issue=20 & 21
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| |pages=5009
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| |arxiv=hep-th/0410129
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| |bibcode = 2005IJMPA..20.5009G }}
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| *{{cite arXiv
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| |eprint=hep-ph/9707419
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| |last1=Dolgov | first1=A.D.
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| |title=Baryogenesis, 30 Years after
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| |class=hep-ph
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| |year=1997
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| }}
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| [[Category:Anomalies in physics]]
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| [[Category:Quantum chromodynamics]]
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| [[Category:Standard Model]]
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| [[Category:Conservation laws]]
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Switching from band blueprint that will help your besprinkle blueprint shows some sort of added authentic picture. The main accumbent time arbor is considered to be scaled evenly. But it's adamantine to constitute able to acquaint what's activity now within specific bottom-left bend now. The ethics are as bunched up you will certainly not acquaint them very far nowadays.
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