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| In [[quantum electrodynamics]], the '''anomalous magnetic moment''' of a particle is a contribution of effects of [[quantum mechanics]], expressed by [[Feynman diagram]]s with loops, to the [[magnetic moment]] of that particle. (The ''magnetic moment'', also called ''magnetic dipole moment'', is a measure of the strength of a magnetic source.)
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| The "Dirac" [[magnetic moment]], corresponding to tree-level Feynman diagrams (which can be thought of as the classical result), can be calculated from the [[Dirac equation]]. It is usually expressed in terms of the [[g-factor (physics)|g-factor]]; the Dirac equation predicts <math>g = 2</math>. For particles such as the [[electron]], this classical result differs from the observed value by a small fraction of a percent. The difference is the anomalous magnetic moment, denoted <math> a </math> and defined as
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| :<math> a = \frac{g-2}{2} </math>
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| ==Anomalous magnetic moment of the electron==
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| [[Image:Vertex correction.svg|thumb|right|One-loop correction to the fermion's magnetic dipole moment.]]
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| The one-loop contribution to the anomalous magnetic moment—corresponding to the first and largest quantum mechanical correction—of the electron is found by calculating the [[vertex function]] shown in the diagram on the right. The calculation is relatively straightforward<ref>See section 6.3 of Michael E. Peskin and Daniel V. Schroeder, ''An Introduction to Quantum Field Theory'', Addison-Wesley, Reading, 1995.</ref> and the one-loop result is:
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| :<math> a = \frac{\alpha}{2 \pi} \approx 0.0011614 </math>
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| where <math> \alpha </math> is the [[fine structure constant]].
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| This result was first found by [[Julian Schwinger|Schwinger]] in 1948.<ref>J. Schwinger, ''Phys. Rev.'' '''73''', 416L (1948)</ref> As of 2009, the coefficients of the QED formula for the anomalous magnetic moment of the electron have been calculated through order <math>\alpha^4</math>,<ref>{{Cite journal| journal = Physical Review D | volume = 77 | pages = 053012 | year = 2008 | doi = 10.1103/PhysRevD.77.053012 | author = Aoyama ''et al.'' | title = Revised value of the eighth-order QED contribution to the anomalous magnetic moment of the electron | last2 = Hayakawa | first2 = M. | last3 = Kinoshita | first3 = T. | last4 = Nio | first4 = M.|bibcode = 2008PhRvD..77e3012A| issue = 5 }}</ref> and are known ''analytically'' up to <math>\alpha^3</math>.<ref>{{Cite journal| journal = Physics Letters B | volume = 379 | pages = 283–291 | author = Laporta and Remiddi | doi = 10.1016/0370-2693(96)00439-X | title = The analytical value of the electron (g − 2) at order α3 in QED | year = 1996 | last2 = Remiddi | first2 = E|arxiv = hep-ph/9602417 |bibcode = 1996PhLB..379..283L }}</ref>
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| The QED prediction agrees with the experimentally measured value to more than 10 significant figures, making the magnetic moment of the electron the most accurately verified prediction in the history of [[physics]]. (See [[precision tests of QED]] for details.)
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| The current experimental value and uncertainty is:<ref>
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| {{Cite journal
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| | last = Hanneke
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| | first = D.
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| | coauthors = S. Fogwell Hoogerheide, G. Gabrielse
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| | title = Cavity Control of a Single-Electron Quantum Cyclotron: Measuring the Electron Magnetic Moment
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| | date = 24 September 2010
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| | arxiv = arXiv:1009.4831v1 [physics.atom-ph]
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| }}
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| </ref>
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| :<math>a = 0.00115965218073 (28)</math>
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| According to this value, ''a'' is known to an accuracy of around 1 part in 1 billion (10<sup>9</sup>). This required measuring [[g-factor (physics)|''g'']] to an accuracy of around 1 part in 1 trillion (10<sup>12</sup>).
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| ==Anomalous magnetic moment of the muon==
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| [[Image:The muon g-2.svg|thumb|300px|One-loop [[Minimal Supersymmetric Standard Model|MSSM]] corrections to the muon g-2 involving a [[neutralino]] and a [[smuon]], and a [[chargino]] and a muon [[sneutrino]] respectively.]]
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| The anomalous magnetic moment of the [[muon]] is calculated in a similar way; its measurement provides a [[precision tests of electroweak theory|precision test]] of the [[Standard Model]]. The prediction for the value of the muon anomalous magnetic moment includes three parts: α<sub>μ</sub><sup>SM</sup> = α<sub>μ</sub><sup>QED</sup> + α<sub>μ</sub><sup>EW</sup> + α<sub>μ</sub><sup>had</sup>. The first two components represent the photon and lepton loops, and the W boson and Z boson loops, respectively, and can be calculated precisely from first principles. The third term represents hadron loops, and cannot be calculated accurately from theory alone. It is estimated from experimental measurements of the ratio of hadronic to muonic cross sections ([[R (cross section ratio)|R]]) in e<sup>+</sup>e<sup>-</sup> collisions. As of November 2006, the measurement disagrees with the Standard Model by 3.4 [[standard deviation]]s,<ref>
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| {{Cite journal
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| | last = Hagiwara
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| | first = K.
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| | coauthors = A. D. Martin; Daisuke Nomura; T. Teubner
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| | title = Improved predictions for g−2 of the muon and <math>\alpha_\mathrm{QED}(M_Z^2)</math>
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| | date = 31 May 2007
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| | arxiv = hep-ph/0611102
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| | journal = [[Physics Letters B]]
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| | volume = 649
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| | issue = 2–3
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| | pages = 173–179
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| | doi = 10.1016/j.physletb.2007.04.012|bibcode = 2007PhLB..649..173H }}
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| </ref> suggesting [[beyond the Standard Model]] physics may be having an effect (or theoretical/experimental errors not completely under control).
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| The [http://www.g-2.bnl.gov/ E821 experiment] at [[Brookhaven National Laboratory]] (BNL) studied the precession of muon and anti-muon in a constant external magnetic field as they circulated in a confining storage ring. The E821 Experiment reported the following average value [http://pdg.lbl.gov/2011/reviews/rpp2011-rev-g-2-muon-anom-mag-moment.pdf (from the July 2009 review by Particle Data Group)]
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| :<math>a = \frac{g-2}{2} = 0.00116592089(54)(33)</math>
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| where the first errors are statistical and the second systematic.
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| ==Anomalous magnetic moment of composite particles==
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| [[Composite particle]]s often have a huge anomalous magnetic moment. This is true for the [[proton]], which is made up of charged [[quark]]s, and the [[neutron]], which has a magnetic moment even though it is electrically neutral.
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| ==Notes==
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| <references />
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| ==See also==
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| * [[Anomalous electric dipole moment]]
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| * [[G-factor (physics)|G-factor]]
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| ==External links==
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| *[http://www.g-2.bnl.gov/physics/index.html Overview of the ''g-2'' experiment]
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| *{{Cite journal
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| | last=Kusch | first=P. | coauthors=Foley, H. M.
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| | title=The Magnetic Moment of the Electron
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| | journal=Physical Review | year=1948 | volume=74
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| | pages=250–263 | doi=10.1103/PhysRev.74.250 |bibcode = 1948PhRv...74..250K
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| | issue=3 }}
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| {{QED}}
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| {{DEFAULTSORT:Anomalous Magnetic Dipole Moment}}
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| [[Category:Quantum field theory]]
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| [[Category:Quantum electrodynamics]]
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