Editing Chiral anomaly (section)

==Informal introduction==
[[Image:Anomalous-pion-decay.png |right|thumb| Anomaly-induced neutral pion decay <math>\pi^0 \to\gamma\gamma~.</math> This is a one-loop [[Feynman diagram]]. The <math>\pi^0</math> coupling is a [[pseudoscalar]] coupling; the two photons couple as vectors. The triangle sums over all lepton generations.]]
The chiral anomaly originally referred to the anomalous [[decay rate]] of the [[neutral particle|neutral]] [[pion]], as computed in the [[current algebra]] of the [[chiral model]]. These calculations suggested that the decay of the pion was suppressed, clearly contradicting experimental results. The nature of the anomalous calculations was first explained in 1969 by [[Stephen L. Adler]]<ref name=Adler>
{{cite journal
 |last1=Adler |first1=S. L.
 |year=1969
 |title=Axial-Vector Vertex in Spinor Electrodynamics
 |journal=[[Physical Review]]
 |volume=177 |issue=5 |pages=2426–2438
 |bibcode=1969PhRv..177.2426A
 |doi=10.1103/PhysRev.177.2426
}}
</ref> and [[John Stewart Bell]] &amp; [[Roman Jackiw]].<ref name=bj>
{{cite journal
 |last1=Bell |first1=J. S.
 |last2=Jackiw |first2=R.
 |year=1969
 |title=A PCAC puzzle: &pi;<sup>0</sup>&rarr;&gamma;&gamma; in the &sigma;-model
 |journal=[[Il Nuovo Cimento A]]
 |volume=60 |issue=1
 |pages=47–61
 |doi=10.1007/BF02823296
 |bibcode=1969NCimA..60...47B
 |s2cid=125028356
 |url=https://cds.cern.ch/record/348417
}}
</ref> This is now termed the '''Adler–Bell–Jackiw anomaly''' of [[quantum electrodynamics]].<ref>
Roman W. Jackiw (2008) "[http://www.scholarpedia.org/article/Axial_anomaly Axial anomaly]", ''Shcolarpedia'' '''3'''(10):7302.
</ref><ref>
Claude Itzykson and Jean-Bernard Zuber, (1980) "Quantum Field Theory", McGraw-Hill. ''(See Chapter 11-5 pp 549–560)''
</ref> This is a symmetry of classical [[electrodynamics]] that is violated by quantum corrections.

The Adler–Bell–Jackiw anomaly arises in the following way. If one considers the classical (non-quantized) theory of [[electromagnetism]] coupled to massless [[fermion]]s (electrically charged [[Dirac spinor]]s solving the [[Dirac equation]]), one expects to have not just one but two [[conserved current]]s: the ordinary electrical current (the [[vector current]]), described by the Dirac field <math>j^\mu = \overline\psi\gamma^\mu\psi</math> as well as an [[axial current]] <math>j_5^\mu = \overline\psi\gamma^5\gamma^\mu\psi~.</math>  When moving from the classical theory to the quantum theory, one may compute the quantum corrections to these currents; to first order, these are the [[One-loop Feynman diagram|one-loop]] [[Feynman diagram]]s. These are famously divergent, and require a [[regularization (physics)|regularization]] to be applied, to obtain the [[renormalization|renormalized]] amplitudes. In order for the [[renormalization]] to be meaningful, coherent and consistent, the regularized diagrams must obey the same symmetries as the zero-loop (classical) amplitudes. This is the case for the vector current, but not the axial current: it cannot be regularized in such a way as to preserve the axial symmetry. The axial symmetry of classical electrodynamics is broken by quantum corrections. Formally, the [[Ward–Takahashi identity|Ward–Takahashi identities]] of the quantum theory follow from the [[gauge symmetry]] of the electromagnetic field; the corresponding identities for the axial current are broken.

At the time that the Adler–Bell–Jackiw anomaly was being explored in physics, there were related developments in [[differential geometry]] that appeared to involve the same kinds of expressions. These were not in any way related to quantum corrections of any sort, but rather were the exploration of the global structure of [[fiber bundle]]s, and specifically, of the [[Dirac operator]]s on [[spin structure]]s having [[curvature form]]s resembling that of the [[electromagnetic tensor]], both in four and three dimensions (the [[Chern–Simons theory]]). After considerable back and forth, it became clear that the structure of the anomaly could be described with bundles with a non-trivial [[homotopy group]], or, in physics lingo, in terms of [[instanton]]s. 

Instantons are a form of [[topological soliton]]; they are a solution to the ''classical'' field theory, having  the property that they are stable and cannot decay (into [[plane wave]]s, for example). Put differently: conventional field theory is built on the idea of a [[vacuum]] – roughly speaking, a flat empty space. Classically, this is the "trivial" solution; all fields vanish.  However, one can also arrange the (classical) fields in such a way that they have a non-trivial global configuration. These non-trivial configurations are also candidates for the vacuum, for empty space; yet they are no longer flat or trivial; they contain a twist, the instanton. The quantum theory is able to interact with these configurations; when it does so, it manifests as the chiral anomaly.  

In mathematics, non-trivial configurations are found during the study of [[Dirac operator]]s in their fully generalized setting, namely, on [[Riemannian manifold]]s in arbitrary dimensions. Mathematical tasks include finding and classifying structures and configurations. Famous results include the [[Atiyah–Singer index theorem]] for Dirac operators. Roughly speaking, the symmetries of [[Minkowski spacetime]], [[Lorentz invariance]], [[Laplacian]]s, Dirac operators and the U(1)xSU(2)xSU(3) [[fiber bundle]]s can be taken to be a special case of a far more general setting in [[differential geometry]]; the exploration of the various possibilities accounts for much of the excitement in theories such as [[string theory]]; the richness of possibilities accounts for a certain perception of lack of progress. 

The Adler–Bell–Jackiw anomaly is seen experimentally, in the sense that it describes the decay of the [[neutral pion]], and specifically, the [[decay width|width of the decay]] of the neutral pion into two [[photon]]s. The neutral pion itself was discovered in the 1940s; its decay rate (width) was correctly estimated by J. Steinberger in 1949.<ref>{{cite journal | last=Steinberger | first=J. | title=On the Use of Subtraction Fields and the Lifetimes of Some Types of Meson Decay | journal=Physical Review | publisher=American Physical Society (APS) | volume=76 | issue=8 | date=1949-10-15 | issn=0031-899X | doi=10.1103/physrev.76.1180 | pages=1180–1186| bibcode=1949PhRv...76.1180S }}</ref> The correct form of the anomalous divergence of the axial current is obtained by Schwinger in 1951 in a 2D model of electromagnetism and massless fermions.<ref>{{cite journal | last=Schwinger | first=Julian | title=On Gauge Invariance and Vacuum Polarization | journal=Physical Review | publisher=American Physical Society (APS) | volume=82 | issue=5 | date=1951-06-01 | issn=0031-899X | doi=10.1103/physrev.82.664 | pages=664–679| bibcode=1951PhRv...82..664S }}</ref> That the decay of the neutral pion is suppressed in the [[current algebra]] analysis of the [[chiral model]] is obtained by Sutherland and Veltman in 1967.<ref>{{cite journal | last=Sutherland | first=D.G. | title=Current algebra and some non-strong mesonic decays | journal=Nuclear Physics B | publisher=Elsevier BV | volume=2 | issue=4 | year=1967 | issn=0550-3213 | doi=10.1016/0550-3213(67)90180-0 | pages=433–440| bibcode=1967NuPhB...2..433S | url=https://cds.cern.ch/record/347030 | url-access=subscription }}</ref><ref>{{cite journal |last=Veltman|first=M.| title=I. Theoretical aspects of high energy neutrino interactions | journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences | publisher=The Royal Society | volume=301 | issue=1465 | date=1967-10-17 | issn=0080-4630 | doi=10.1098/rspa.1967.0193 | pages=107–112|bibcode=1967RSPSA.301..107V|s2cid=122755742}}</ref> An analysis and resolution of this anomalous result is provided by Adler<ref name=Adler/> and Bell &amp; Jackiw<ref name=bj/> in 1969. A general structure of the anomalies is discussed by Bardeen in 1969.<ref>{{cite journal | last=Bardeen | first=William A. | title=Anomalous Ward Identities in Spinor Field Theories | journal=Physical Review | publisher=American Physical Society (APS) | volume=184 | issue=5 | date=1969-08-25 | issn=0031-899X | doi=10.1103/physrev.184.1848 | pages=1848–1859| bibcode=1969PhRv..184.1848B }}</ref>

The [[quark model]] of the pion indicates it is a bound state of a quark and an anti-quark. However, the [[quantum number]]s, including parity and angular momentum, taken to be conserved, prohibit the decay of the pion, at least in the zero-loop calculations (quite simply, the amplitudes vanish.) If the quarks are assumed to be massive, not massless, then a [[chirality]]-violating decay is allowed; however, it is not of the correct size. (Chirality is not a [[constant of motion]] of massive spinors; they will change handedness as they propagate, and so mass is itself a chiral symmetry-breaking term. The contribution of the mass is given by the Sutherland and Veltman result; it is termed "PCAC", the [[partially conserved axial current]].) The Adler–Bell–Jackiw analysis provided in 1969 (as well as the earlier forms by Steinberger and Schwinger), do provide the correct decay width for the neutral pion. 

Besides explaining the decay of the pion, it has a second very important role. The one loop amplitude includes a factor that counts the grand total number of leptons that can circulate in the loop. In order to get the correct decay width, one must have exactly [[Generation (particle physics)|three generations]] of quarks, and not four or more. In this way, it plays an important role in constraining the [[Standard model]]. It provides a direct physical prediction of the number of quarks that can exist in nature.

Current day research is focused on similar phenomena in different settings, including non-trivial topological configurations of the [[electroweak theory]], that is, the [[sphaleron]]s. Other applications include the hypothetical non-conservation of [[baryon number]] in [[Grand_Unified_Theory|GUTs]] and other theories.