Editing Chiral anomaly

{{Use American English|date=January 2019}}{{Short description|Non-conservation of chiral current in physics}}
In [[theoretical physics]], a '''chiral anomaly''' is the [[anomaly (physics)|anomalous]] [[nonconservation]] of a [[chirality (physics)|chiral]] current. In everyday terms, it is analogous to a sealed box that contained equal numbers of left and right-handed [[Bolt (fastener)|bolts]], but when opened was found to have more left than right, or vice versa.

Such events are expected to be prohibited according to classical [[conservation law]]s, but it is known there must be ways they can be broken, because we have evidence of [[CP violation|charge–parity non-conservation ]] ("CP violation"). It is possible that other imbalances have been caused by breaking of a ''chiral law'' of this kind. Many physicists suspect that the fact that the observable universe contains [[baryon asymmetry|more matter than antimatter]] is caused by a chiral anomaly.<ref>
{{cite journal
 |last1=Dolgov |first1=A. D.
 |year=1997
 |title=Baryogenesis, 30 years after
 |journal=[[Surveys in High Energy Physics]]
 |volume=13 |issue=1–3 |pages=83–117
 |arxiv=hep-ph/9707419
 |bibcode=1998SHEP...13...83D
 |doi=10.1080/01422419808240874
|s2cid=119499400
 }}
</ref> Research into [[chiral symmetry breaking]] laws is a major endeavor in particle physics research at this time.{{Citation needed|date=March 2024}}{{When|date=March 2024}}

==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.

==General discussion==
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. The non-conservation happens in a process of [[Quantum tunneling|tunneling]] from one [[Vacuum#Quantum mechanics|vacuum]] to another. Such a process is called an [[instanton]].

In the case of a symmetry related to the conservation of a [[particle number operator|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, in the [[path integral formulation]], an '''anomalous symmetry''' is a symmetry of the [[action (physics)|action]] <math>\mathcal A</math>, but not of the [[measure (physics)|measure]] {{mvar| μ}} and  therefore ''not'' of the [[partition function (quantum field theory)|generating functional]] 
:<math>\mathcal Z=\int\! {\exp (i \mathcal A/\hbar) ~ \mathrm{d} \mu}</math>

of the quantized theory ({{mvar|ℏ}} is Planck's action-quantum divided by 2{{mvar|π}}). The measure <math>d\mu</math> consists of a part depending on the fermion field <math>[\mathrm{d}\psi]</math> and a part depending on its complex conjugate <math>[\mathrm{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>\gamma^5</math> is the chiral [[gamma matrix]] acting on <math>\psi</math>. From the formula for <math>\mathcal Z</math> one also sees explicitly that in the [[classical limit]], {{nowrap|{{mvar|ℏ}} → 0,}} anomalies don't come into play, since in this limit only the extrema of <math>\mathcal A</math> remain relevant.

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 for the theory to be consistent.)

==Calculation==
The chiral anomaly can be calculated exactly by [[one-loop Feynman diagram]]s, e.g.  Steinberger's "triangle diagram", contributing to the [[pion]] decays,   and <math>\pi^0\to e^+e^-\gamma</math>. The amplitude 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 condition'''.

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: baryon number non-conservation==
The Standard Model of [[electroweak]] interactions has all the necessary ingredients for successful [[baryogenesis]], although these interactions have never been observed<ref>{{cite journal | last1=Eidelman | first1=S. | last2=Hayes | first2=K.G. | last3=Olive | first3=K.A. | last4=Aguilar-Benitez | first4=M. | last5=Amsler | first5=C. | last6=Asner | first6=D. | last7=Babu | first7=K.S. | last8=Barnett | first8=R.M. | last9=Beringer | first9=J. | last10=Burchat | first10=P.R. | last11=Carone | first11=C.D. | last12=Caso | first12=S. | last13=Conforto | first13=G. | last14=Dahl | first14=O. | last15=D'Ambrosio | first15=G. | last16=Doser | first16=M. | last17=Feng | first17=J.L. | last18=Gherghetta | first18=T. | last19=Gibbons | first19=L. | last20=Goodman | first20=M. | display-authors=5|collaboration=Particle Data Group| title=Review of Particle Physics | journal=Physics Letters B | publisher=Elsevier BV | volume=592 | issue=1–4 | year=2004 | issn=0370-2693 | doi=10.1016/j.physletb.2004.06.001|arxiv=astro-ph/0406663 | pages=1–5| bibcode=2004PhLB..592....1P }}</ref> and may be insufficient to explain the total [[baryon number]] of the observed universe if the initial baryon number of the universe at the time of the Big Bang is zero. 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 (field theory)|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 known as the [[sphaleron]] destroy this [[conservation law]]:  instead of zero in the right hand side of this equation, there is a non-vanishing quantum term,
:<math>\partial^\mu J_\mu^B = \frac{g^2 C}{16\pi^2} G^{\mu\nu a} \tilde{G}_{\mu\nu}^a,</math>
where {{mvar|C}} is a numerical constant vanishing for ℏ =0,
:<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>

Electroweak sphalerons can only change the baryon and/or lepton number by 3 or multiples of 3 (collision of three baryons into three leptons/antileptons and vice versa).

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.

===Geometric form===

In the language of [[differential form]]s, to any self-dual curvature form <math>F_A</math> we may assign the abelian 4-form <math>\langle F_A\wedge F_A\rangle:=\operatorname{tr}\left(F_A\wedge F_A\right)</math>.  [[Chern–Weil theory]] shows that this 4-form is locally ''but not globally'' exact, with potential given by the [[Chern–Simons form|Chern–Simons 3-form]] locally:

:<math>d\mathrm{CS}(A)=\langle F_A\wedge F_A\rangle</math>.

Again, this is true only on a single chart, and is false for the global form <math>\langle F_\nabla\wedge F_\nabla\rangle</math> unless the instanton number vanishes.

To proceed further, we attach a "point at infinity" {{math|''k''}} onto <math>\mathbb{R}^4</math> to yield <math>S^4</math>, and use the [[clutching construction]] to chart principal A-bundles, with one chart on the neighborhood of {{math|''k''}} and a second on <math>S^4-k</math>.  The thickening around {{math|''k''}}, where these charts intersect, is trivial, so their intersection is essentially <math>S^3</math>.  Thus instantons are classified by the third [[homotopy group]] <math>\pi_3(A)</math>, which for <math>A = \mathrm{SU(2)}\cong S^3</math> is simply [[Homotopy groups of spheres|the third 3-sphere group]] <math>\pi_3(S^3)=\mathbb{Z}</math>.

The divergence of the baryon number current is (ignoring numerical constants)

:<math>\mathbf{d}\star j_b = \langle F_\nabla\wedge F_\nabla\rangle</math>,

and the instanton number is

:<math>\int_{S^4} \langle F_\nabla\wedge F_\nabla\rangle\in\mathbb{N}</math>.

==See also==
* [[Anomaly (physics)]]
*[[Chiral magnetic effect]]
* [[Global anomaly]]
* [[Gravitational anomaly]]
* [[strong CP problem#Strong CP problem|Strong CP problem]]

==References==
<references />

==Further reading==

===Published articles===
*{{cite journal
 |last1=Frampton |first1=P. H.
 |last2=Kephart |first2=T. W.
 |year=1983
 |title=Explicit Evaluation of Anomalies in Higher Dimensions
 |journal=[[Physical Review Letters]]
 |volume=50 |issue=18 |pages=1343–1346
 |bibcode=1983PhRvL..50.1343F
 |doi=10.1103/PhysRevLett.50.1343
}}
*{{cite journal
 |last1=Frampton |first1=P. H.
 |last2=Kephart |first2=T. W.
 |year=1983
 |title=Analysis of anomalies in higher space-time dimensions
 |journal=[[Physical Review]]
 |volume=D28 |issue=4 |pages=1010–1023
 |bibcode=1983PhRvD..28.1010F
 |doi=10.1103/PhysRevD.28.1010
}}
*{{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
 |journal=[[International Journal of Modern Physics A]]
 |volume=20 |issue=20–21 |pages=5009
 |arxiv=hep-th/0410129
 |bibcode=2005IJMPA..20.5009G
 |doi=10.1142/S0217751X05025085
|s2cid=15616630
 }}
*{{cite journal
 |last1=White |first1=A. R.
 |year=2004
 |title=Electroweak High-Energy Scattering and the Chiral Anomaly
 |journal=[[Physical Review D]]
 |volume=69 |issue=9 |pages=096002
 |arxiv=hep-ph/0308287
 |bibcode=2004PhRvD..69i6002W
 |doi=10.1103/PhysRevD.69.096002
|s2cid=10863873
 }}
*{{cite journal
 |last1=Yang |first1=J.-F.
 |year=2004
 |title=Trace anomalies and chiral Ward identities
 |journal=[[Chinese Physics Letters]]
 |volume=21 |issue=5 |pages=792–794
 |arxiv=hep-ph/0403173
 |bibcode=2004ChPhL..21..792Y
 |doi=10.1088/0256-307X/21/5/008
|s2cid=119021732
 }}
*{{cite journal
 |last1=Csörgő |first1=T.
 |last2=Vértesi |first2=R.
 |last3=Sziklai |first3=J.
 |year=2010
 |title=Indirect Observation of an In-Medium η′ Mass Reduction in {{sqrt|s<sub>NN</sub>}}=200 GeV Au+Au Collisions
 |journal=[[Physical Review Letters]]
 |volume=105 |issue=18 |pages=182301
 |doi=10.1103/PhysRevLett.105.182301
 |pmid=21231099
 |arxiv=0912.5526
 |bibcode=2010PhRvL.105r2301C
|s2cid=9011271
 }}
*{{cite journal
 |last1=AlMasri|first1= M. W.
 |year=2019
 |title=Axial-anomaly in noncommutative QED and Pauli–Villars regularization
 |journal=[[International Journal of Modern Physics A]]
 |volume=34 |issue=26 
|doi=10.1142/S0217751X19501501
 |arxiv=1909.10280
|bibcode= 2019IJMPA..3450150A
 |s2cid= 202719219
 }}

===Textbooks===
* {{cite book
 |last1=Fujikawa |first1=K.
 |last2=Suzuki |first2=H.
 |date=2004
 |title=Path Integrals and Quantum Anomalies
 |publisher=[[Clarendon Press]]
 |isbn=978-0-19-852913-2
 }}
* {{cite book
 |last=Weinberg |first=S.
 |year=2001
 |title=The Quantum Theory of Fields. Volume II: Modern Applications
 |publisher=[[Cambridge University Press]]
 |isbn=978-0-521-55002-4
}}

===Preprints===
*{{cite arXiv
 |last1=Yang | first1=J.-F.
 |year=2003
 |title=Trace and chiral anomalies in QED and their underlying theory interpretation
 |eprint=hep-ph/0309311
}}

[[Category:Anomalies (physics)]]
[[Category:Quantum chromodynamics]]
[[Category:Standard Model]]
[[Category:Conservation laws]]