View source for Anomaly (physics)

{{Short description|Asymmetry of classical and quantum action}}
{{Quantum field theory|cTopic=Tools}}
{{distinguish|Anomaly (natural sciences)}}
In [[quantum physics]] an '''anomaly''' or '''quantum anomaly''' is the failure of a [[symmetry]] of a theory's classical [[action (physics)|action]] to be a symmetry of any [[regularization (physics)|regularization]] of the full quantum theory.<ref>
{{cite journal
 |last=Bardeen |first=William |year=1969
 |title=Anomalous Ward identities in spinor field theories
 |journal=[[Physical Review]]
 |volume=184
 |issue=5 |pages=1848–1859
 |doi=10.1103/physrev.184.1848|bibcode = 1969PhRv..184.1848B }}</ref><ref>{{cite book
 |last1=Cheng |first1=T.P.|last2=Li |first2=L.F. |date=1984 |title=Gauge Theory of Elementary Particle Physics|publisher=Oxford Science Publications }}</ref>  
In [[classical physics]], a '''classical anomaly''' is the failure of a symmetry to be restored in the limit in which the symmetry-breaking parameter goes to zero. Perhaps the first known anomaly was the dissipative anomaly<ref>{{cite web|title=Dissipative Anomalies in Singular Euler Flows|url=https://www-n.oca.eu/etc7/EE250/presentations/Eyink.pdf}}</ref> in [[turbulence]]: time-reversibility remains broken (and energy dissipation rate finite) at the limit of vanishing [[viscosity]]. 

In quantum theory, the first anomaly discovered was the [[Adler–Bell–Jackiw anomaly]], wherein the [[Chiral_anomaly|axial vector current]] is conserved as a classical symmetry of [[electrodynamics]], but is broken by the quantized theory. The relationship of this anomaly to the [[Atiyah–Singer index theorem]] was one of the celebrated achievements of the theory. Technically, an anomalous symmetry in a quantum theory is a symmetry of the [[action (physics)|action]], but not of the [[measure (physics)|measure]], and so not of the [[partition function (quantum field theory)|partition function]] as a whole.

==Global anomalies==

A global anomaly is the quantum violation of a global symmetry current conservation. 
A global anomaly can also mean that a non-perturbative global anomaly cannot be captured by one loop or any loop perturbative Feynman diagram calculations—examples include the [[#Witten anomaly and Wang–Wen–Witten anomaly|Witten anomaly and Wang–Wen–Witten anomaly]].

===Scaling and renormalization===

The most prevalent global anomaly in physics is associated with the violation of [[scale invariance]] by quantum corrections, quantified in [[renormalization]].
Since regulators generally introduce a distance scale, the classically scale-invariant theories are subject to [[renormalization group]] flow, i.e., changing behavior with energy scale.  For example, the large strength of the [[strong nuclear force]] results from a theory that is weakly coupled at short distances flowing to a strongly coupled theory at long distances, due to this scale anomaly.

===Rigid symmetries===

Anomalies in [[commutative|abelian]] global symmetries pose no problems in a [[quantum field theory]], and are often encountered (see the example of the [[chiral anomaly]]).  In particular the corresponding anomalous symmetries  can be fixed by fixing the [[boundary condition]]s of the [[path integral formulation|path integral]].

===Large gauge transformations===

Global anomalies in [[symmetries]] that approach the identity sufficiently quickly at [[infinity]] do, however, pose problems.  In known examples such symmetries correspond to disconnected components of gauge symmetries.  Such symmetries and possible anomalies occur, for example, in theories with chiral fermions or self-dual [[differential form]]s coupled to [[gravity]] in 4''k''&nbsp;+&nbsp;2 dimensions, and also in the [[#Witten anomaly and Wang–Wen–Witten anomaly|Witten anomaly]] in an ordinary 4-dimensional SU(2) gauge theory.

As these symmetries vanish at infinity, they cannot be constrained by boundary conditions and so must be summed over in the path integral.  The sum of the gauge orbit of a state is a sum of phases which form a subgroup of U(1).  As there is an anomaly, not all of these phases are the same, therefore it is not the identity subgroup.  The sum of the phases in every other subgroup of U(1) is equal to zero, and so all path integrals are equal to zero when there is such an anomaly and a theory does not exist.

An exception may occur when the space of configurations is itself disconnected, in which case one may have the freedom to choose to integrate over any
subset of the components.  If the disconnected gauge symmetries map the system between disconnected configurations, then there is in general a consistent truncation of a theory in which one integrates only over those connected components that are not related by large gauge transformations.  In this case the large gauge transformations do not act on the system and do not cause the path integral to vanish.

====Witten anomaly and Wang–Wen–Witten anomaly====

In SU(2) [[gauge theory]] in 4 dimensional [[Minkowski space]], a gauge transformation corresponds to a choice of an element of the [[special unitary group]] SU(2) at each point in spacetime.  The group of such gauge transformations is connected.

However, if we are only interested in the subgroup of gauge transformations that vanish at infinity, we may consider the 3-sphere at infinity to be a single point, as the gauge transformations vanish there anyway.  If the 3-sphere at infinity is identified with a point, our Minkowski space is identified with the 4-sphere.  Thus we see that the group of gauge transformations vanishing at infinity in Minkowski 4-space is [[isomorphic]] to the group of all gauge transformations on the 4-sphere.

This is the group which consists of a continuous choice of a gauge transformation in SU(2) for each point on the 4-sphere.  In other words, the gauge symmetries are in one-to-one correspondence with maps from the 4-sphere to the 3-sphere, which is the group manifold of SU(2).  The space of such maps is ''not'' connected, instead the connected components are classified by the fourth [[homotopy group]] of the 3-sphere which is the [[cyclic group]] of order two.  In particular, there are two connected components.  One contains the identity and is called the ''identity component'', the other is called the ''disconnected component''.

When a theory contains an odd number of flavors of chiral fermions, the actions of gauge symmetries in the identity component and the disconnected component of the gauge group on a physical state differ by a sign.  Thus when one sums over all physical configurations in the [[functional integration|path integral]], one finds that contributions come in pairs with opposite signs.  As a result, all path integrals vanish and a theory does not exist.

The above description of a global anomaly is for the SU(2) gauge theory coupled to an odd number of (iso-)spin-1/2 Weyl fermion in 4 spacetime dimensions. This is known as the Witten SU(2) anomaly.<ref name="An SU(2) Anomaly">{{cite journal | last=Witten | first=Edward | title=An SU(2) Anomaly | journal=Phys. Lett. B | volume=117 | issue=5 | date= November 1982 | doi=10.1016/0370-2693(82)90728-6 | page=324 | bibcode=1982PhLB..117..324W }}</ref> In 2018, it is found by Wang, Wen and Witten that the SU(2) gauge theory coupled to an odd number of (iso-)spin-3/2 Weyl fermion in 4 spacetime dimensions has a further subtler non-perturbative global anomaly detectable on certain non-spin manifolds without [[spin structure]].<ref name="1810.00844">{{cite journal | last1=Wang | first1=Juven | last2=Wen | first2=Xiao-Gang | last3=Witten | first3=Edward | title=A New SU(2) Anomaly | journal=Journal of Mathematical Physics | volume=60 | issue=5 | date= May 2019 | issn= 1089-7658 | doi=10.1063/1.5082852 | page=052301 |arxiv=1810.00844| bibcode=2019JMP....60e2301W | s2cid=85543591 }}</ref> This new anomaly is called the new SU(2) anomaly. Both types of anomalies<ref name="An SU(2) Anomaly"/> <ref name=1810.00844/> have analogs of (1) dynamical gauge anomalies for dynamical gauge theories and (2) the 't Hooft anomalies of global symmetries. In addition, both types of anomalies are mod 2 classes (in terms of classification, they are both finite groups '''Z'''<sub>''2''</sub>  of order 2 classes), and have analogs in 4 and 5 spacetime dimensions.<ref name=1810.00844/> More generally, for any natural integer N, it can be shown that an odd number of fermion multiplets in representations of (iso)-spin 2N+1/2 can have the SU(2) anomaly; an odd number of fermion multiplets in representations of (iso)-spin 4N+3/2 can have the new SU(2) anomaly.<ref name=1810.00844/> For fermions in the half-integer spin representation, it is shown that there are only these two types of SU(2) anomalies and the linear combinations of these two anomalies; these classify all global SU(2) anomalies.<ref name=1810.00844/> This new SU(2) anomaly also plays an important rule for confirming the consistency of [[SO(10)]] grand unified theory, with a Spin(10) gauge group and chiral fermions in the 16-dimensional spinor representations, defined on non-spin manifolds.<ref name=1810.00844/><ref name="1809.11171">{{cite journal | last1=Wang | first1=Juven | last2=Wen | first2=Xiao-Gang | title=Nonperturbative definition of the standard models | journal=Physical Review Research  | volume=2 | issue=2 | date=1 June 2020 | issn=2469-9896 | doi=10.1103/PhysRevResearch.2.023356 | page=023356 |arxiv=1809.11171| bibcode= 2018arXiv180911171W| s2cid=53346597 }}</ref>

===Higher anomalies involving higher global symmetries: Pure Yang–Mills gauge theory as an example===

The concept of global symmetries can be generalized to higher global symmetries,<ref name="1412.5148">{{cite journal | last1=Gaiotto | first1=Davide | last2=Kapustin | first2=Anton | last3=Seiberg | first3=Nathan  | last4=Willett | first4=Brian | title=Generalized Global Symmetries  | journal=JHEP | volume=2015 | issue=2 | date=February 2015 | page=172 | issn=1029-8479 | doi=10.1007/JHEP02(2015)172 |arxiv=1412.5148| bibcode=2015JHEP...02..172G | s2cid=37178277 }}</ref> such that the charged object for the ordinary 0-form symmetry is a particle, while the charged object for the n-form symmetry is an n-dimensional extended operator. It is found that the 4 dimensional pure Yang–Mills theory with only SU(2) gauge fields with a topological theta term <math>\theta=\pi,</math> can have a mixed higher 't Hooft anomaly between the 0-form time-reversal symmetry and 1-form '''Z'''<sub>''2''</sub> center symmetry.<ref name="1703.00501">{{cite journal | last1=Gaiotto | first1=Davide | last2=Kapustin | first2=Anton   | last3=Komargodski | first3=Zohar | last4=Seiberg | first4=Nathan | title=Theta, Time Reversal, and Temperature
  | journal=JHEP | volume=2017 | issue=5 | date=May 2017 | page=91 | issn=1029-8479 | doi=10.1007/JHEP05(2017)091 |arxiv=1412.5148| bibcode=2017JHEP...05..091G | s2cid=119528151 }}</ref> The 't Hooft anomaly of 4 dimensional pure Yang–Mills theory can be precisely written as a 5 dimensional invertible topological field theory or mathematically a 5 dimensional bordism invariant, generalizing the anomaly inflow picture to this '''Z'''<sub>''2''</sub> class of global anomaly involving higher symmetries.<ref name="1904.00994">{{cite journal | last1=Wan | first1=Zheyan | last2=Wang | first2=Juven | last3=Zheng | first3=Yunqin | title=Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory  | journal=Physical Review D | volume=100 | issue=8 | date=October 2019 | issn=2470-0029 | doi=10.1103/PhysRevD.100.085012 | page= 085012 |arxiv=1904.00994| bibcode=2019PhRvD.100h5012W | s2cid=201305547 }}</ref> In other words, we can regard the 4 dimensional pure Yang–Mills theory with a topological theta term <math>\theta=\pi</math> live as a boundary condition of a certain '''Z'''<sub>''2''</sub> class  invertible topological field theory, in order to match their higher anomalies on the 4 dimensional boundary.<ref name="1904.00994">{{cite journal | last1=Wan | first1=Zheyan | last2=Wang | first2=Juven | last3=Zheng | first3=Yunqin | title=Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory  | journal=Physical Review D | volume=100 | issue=8 | date=October 2019 | issn=2470-0029 | doi=10.1103/PhysRevD.100.085012 | page= 085012 |arxiv=1904.00994| bibcode=2019PhRvD.100h5012W | s2cid=201305547 }}</ref>

==Gauge anomalies==

{{Main article|Gauge anomaly}}

Anomalies in gauge symmetries lead to an inconsistency, since a gauge symmetry is required in order to cancel unphysical degrees of freedom with a negative norm (such as a [[photon]] polarized in the time direction). An attempt to cancel them—i.e., to build theories [[consistent]] with the gauge symmetries—often leads to extra constraints on the theories (such is the case of the [[gauge anomaly]] in the [[Standard Model]] of particle physics). Anomalies in [[gauge theory|gauge theories]] have important connections to the [[topology]] and [[geometry]] of the [[gauge group]].

Anomalies in gauge symmetries can be calculated exactly at the one-loop level.  At tree level (zero loops), one reproduces the classical theory.  [[Feynman diagrams]] with more than one loop always contain internal [[boson]] propagators.  As bosons may always be given a mass without breaking gauge invariance, a [[Pauli–Villars regularization]] of such diagrams is possible while preserving the symmetry.  Whenever the regularization of a diagram is consistent with a given symmetry, that diagram does not generate an anomaly with respect to the symmetry.

Vector gauge anomalies are always [[chiral anomaly|chiral anomalies]]. Another type of gauge anomaly is the [[gravitational anomaly]].

==At different energy scales==

{{Main article|Anomaly matching condition}}

Quantum anomalies were discovered via the process of [[renormalization]], when some [[ultraviolet divergence|divergent integrals]] cannot be [[regularization (physics)|regularized]] in such a way that all the symmetries are preserved simultaneously. This is related to the high energy physics. However, due to [[Gerard 't Hooft]]'s [[anomaly matching condition]], any [[chiral anomaly]] can be described either by the UV degrees of freedom (those relevant at high energies) or by the IR degrees of freedom (those relevant at low energies). Thus one cannot cancel an anomaly by a [[UV completion]] of a theory—an anomalous symmetry is simply not a symmetry of a theory, even though classically it appears to be.

==Anomaly cancellation==
[[File:Triangle diagram.svg|left]]
Since cancelling anomalies is necessary for the consistency of gauge theories, such cancellations are  of central importance in constraining the fermion content of the [[standard model]], which is a chiral gauge theory.

For example, the vanishing of the [[mixed anomaly]] involving two SU(2) generators and one U(1) hypercharge constrains all charges in a fermion generation to add up to zero,<ref>Bouchiat, Cl,  Iliopoulos, J, and  Meyer, Ph (1972) . "An anomaly-free version of Weinberg's model." ''Physics Letters'' '''B38''', 519-523.</ref><ref>{{cite journal | last1 = Minahan | first1 = J. A. | last2 = Ramond | first2 = P. | last3 = Warner | first3 = R. C. | year = 1990 | title = Comment on anomaly cancellation in the standard model | journal = Phys. Rev. D | volume = 41 | issue = 2| pages = 715–716 | doi = 10.1103/PhysRevD.41.715 | pmid = 10012386 |bibcode = 1990PhRvD..41..715M }}</ref> and thereby dictates that the sum of the proton plus the sum of the electron vanish: the ''charges of quarks and leptons must be commensurate''.
Specifically, for two external gauge fields {{math|''W<sup>a</sup>''}}, {{math|''W<sup>b</sup>''}} and one hypercharge {{mvar|B}} at the vertices of the triangle diagram, cancellation of the triangle requires
:<math>\sum_{all ~doublets}\!\!\!\! \mathrm{Tr} ~T^a T^b Y \propto \delta^{ab}  \sum_{all ~doublets} Y=\sum_{all ~doublets} Q =0 ~,  </math>  
so, for each generation, the charges of the leptons and quarks are balanced, <math>-1+3\times\frac{2-1}{3}=0 </math>, whence {{math|1=''Q''<sub>p</sub> + ''Q''<sub>e</sub> = 0}}{{Citation needed|date=April 2020|reason=Why should that follow?}}.

The anomaly cancelation in SM was also used to predict a quark from 3rd generation, the [[top quark]].<ref>{{Cite book|last=Conlon|first=Joseph|url=https://www.taylorfrancis.com/books/9781482242492|title=Why String Theory?|date=2016-08-19|publisher=CRC Press|isbn=978-1-315-27236-8|edition=1|language=en|doi=10.1201/9781315272368|page=81}}</ref>

Further such mechanisms include:
* [[Axion]]
* [[Chern–Simons]]
* [[Green–Schwarz mechanism]]
* Liouville action

==Anomalies and cobordism==

In the modern description of anomalies classified by [[cobordism]] theory,<ref name="1604.06527">{{cite journal | last1= Freed | first1=Daniel S. | last2=Hopkins | first2=Michael J. | title=Reflection positivity and invertible topological phases
 | journal=Geometry & Topology | year=2021 | volume=25 | issue=3 | pages=1165–1330 | doi=10.2140/gt.2021.25.1165 |issn=1465-3060 |arxiv=1604.06527| bibcode= 2016arXiv160406527F| s2cid=119139835 }}</ref> the [[Feynman diagram|Feynman-Dyson graphs]] only captures the perturbative local anomalies classified by integer '''Z''' classes  also known as the free part. There exists nonperturbative global anomalies classified by [[cyclic group|cyclic groups]] '''Z'''/''n'''''Z''' classes also known as the torsion part.

It is widely known and checked in the late 20th century that the [[standard model]] and chiral gauge theories are free from perturbative local anomalies (captured by [[Feynman diagram|Feynman diagrams]]). However, it is not entirely clear whether there are any nonperturbative global anomalies for the [[standard model]] and chiral gauge theories. 
Recent developments <ref name="1808.00009">{{cite journal | last1=García-Etxebarria | first1=Iñaki | last2=Montero | first2=Miguel | title=Dai-Freed anomalies in particle physics  | journal=JHEP | volume=2019 | issue=8 | date=August 2019 | page=3 | issn=1029-8479 | doi=10.1007/JHEP08(2019)003 |arxiv=1808.00009| bibcode=2019JHEP...08..003G | s2cid=73719463 }}</ref>
<ref name="1910.11277">{{cite journal | last1=Davighi | first1=Joe | last2=Gripaios | first2=Ben | last3=Lohitsiri | first3=Nakarin | title=Global anomalies in the Standard Model(s) and Beyond  | journal=JHEP | volume=2020 | issue=7 | date=July 2020 | page=232 | issn=1029-8479 | doi=10.1007/JHEP07(2020)232 |arxiv=1910.11277| bibcode=2020JHEP...07..232D | s2cid=204852053 }}</ref>
<ref name="1910.14668">{{cite journal | last1=Wan | first1=Zheyan | last2=Wang | first2=Juven | title=Beyond Standard Models and Grand Unifications: Anomalies, Topological Terms, and Dynamical Constraints via Cobordisms  | journal=JHEP | volume=2020 | issue=7 |  date=July 2020 | page=62 | issn=1029-8479 | doi=10.1007/JHEP07(2020)062 |arxiv=1910.14668| bibcode=2020JHEP...07..062W | s2cid=207800450 }}</ref>
based on the [[cobordism theory]] examine this problem, and several additional nontrivial global anomalies found can further constrain these gauge theories. There is also a formulation of both perturbative local and nonperturbative global description of anomaly inflow in terms of [[Michael Atiyah|Atiyah]], [[Vijay Kumar Patodi|Patodi]], and [[Isadore Singer|Singer]]
<ref name="APS">{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Patodi | first2=V. K. | last3=Singer | first3=I. M. | title=Spectral asymmetry and Riemannian geometry | doi=10.1112/blms/5.2.229  | mr=0331443  | year=1973 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=5 | issue=2 | pages=229–234| citeseerx=10.1.1.597.6432 }}</ref>
<ref name="APS1">{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Patodi | first2=V. K. | last3=Singer | first3=I. M. | title=Spectral asymmetry and Riemannian geometry. I | doi=10.1017/S0305004100049410 | mr=0397797  | year=1975 | journal=Mathematical Proceedings of the Cambridge Philosophical Society | issn=0305-0041 | volume=77 | issue=1 | pages=43–69| bibcode=1975MPCPS..77...43A | s2cid=17638224 }}</ref> [[eta invariant]] in one higher dimension. This [[eta invariant]] is a cobordism invariant whenever the perturbative local anomalies vanish. <ref name="1909.08775">{{cite arXiv| last1=Witten | first1=Edward | last2=Yonekura | first2=Kazuya | title=Anomaly Inflow and the eta-Invariant  | year=2019 | class=hep-th |eprint=1909.08775}}</ref>

== Examples ==
* [[Chiral anomaly]]
* [[Conformal anomaly]] (anomaly of [[scale invariance]])
* [[Gauge anomaly]]
* [[Global anomaly]]
* [[Gravitational anomaly]] (also known as ''diffeomorphism anomaly'')
* [[Konishi anomaly]]
* [[Mixed anomaly]]
* [[Parity anomaly]]
* [[Anomaly matching condition|'t Hooft anomaly]]

==See also==
* [[Anomalon]]s, a topic of some debate in the 1980s, anomalons were found in the results of some [[high-energy physics]] experiments that seemed to point to the existence of anomalously highly interactive states of matter. The topic was controversial throughout its history.

==References==
;Citations
{{reflist}}
;General
* Gravitational Anomalies by [[Luis Alvarez-Gaumé]]: This classic paper, which introduces pure [[gravitational anomaly|gravitational anomalies]], contains a good general introduction to anomalies and their relation to [[regularization (physics)|regularization]] and to [[conserved current]]s.  All occurrences of the number 388 should be read "384". Originally at: ccdb4fs.kek.jp/cgi-bin/img_index?8402145. Springer https://link.springer.com/chapter/10.1007%2F978-1-4757-0280-4_1
{{String theory topics |state=collapsed}}

[[Category:Anomalies (physics)| ]]