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{{Short description|Value in  quantum electrodynamics}}
{{Use American English|date = February 2019}}
{{Use dmy dates|date=February 2023}}

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.

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
<math display="block"> a = \frac{g-2}{2} </math>

== Electron ==
[[File:Vertex correction.svg|thumb|right|One-loop correction to a [[fermion]]'s magnetic dipole moment.]]

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 adjacent diagram. The calculation is relatively straightforward{{hsp}}<ref>
{{cite book
 |last1=Peskin |first1=M. E.
 |last2=Schroeder |first2=D. V.
 |year=1995
 |title=An Introduction to Quantum Field Theory
 |url=https://archive.org/details/introductiontoqu0000pesk |url-access=registration |publisher=[[Addison-Wesley]]
 |isbn=978-0-201-50397-5
 |section=Section 6.3
}}</ref> and the one-loop result is:
<math display="block"> a_\text{e} = \frac{\alpha}{2 \pi} \approx 0.001\,161\,4 ,</math>
where <math>\alpha</math> is the [[fine-structure constant]]. This result was first found by [[Julian Schwinger]] in 1948{{hsp}}<ref>
{{cite journal
 |last1=Schwinger |first1=J.
 |year=1948
 |title=On Quantum-Electrodynamics and the Magnetic Moment of the Electron
 |url=http://web.ihep.su/dbserv/compas/src/schwinger48b/eng.pdf
 |journal=[[Physical Review]]
 |volume=73 |issue=4 |pages=416–417
 |bibcode=1948PhRv...73..416S
 |doi=10.1103/PhysRev.73.416
 |doi-access=free
}}</ref> and is engraved on [[Julian Schwinger#Death|his tombstone]]. As of 2016, the coefficients of the QED formula for the anomalous magnetic moment of the electron are known analytically up to <math>\alpha^3</math>{{hsp}}<ref>
{{cite journal
 |last1=Laporta |first1=S.
 |last2=Remiddi |first2=E.
 |year=1996
 |title=The analytical value of the electron ({{nowrap|''g'' − 2}}) at order ''α''<sup>3</sup> in QED
 |journal=[[Physics Letters B]]
 |volume=379 |issue=1–4
 |pages=283–291
 |arxiv=hep-ph/9602417
 |bibcode=1996PhLB..379..283L
 |doi=10.1016/0370-2693(96)00439-X
}}</ref> and have been calculated up to order <math>\alpha^5</math>:<ref>
{{cite journal 
 |last1=Aoyama |first1=T.
 |last2=Hayakawa |first2=M.
 |last3=Kinoshita |first3=T.
 |last4=Nio |first4=M.
 |year=2012
 |title=Tenth-Order QED Contribution to the Electron ''g''−2 and an Improved Value of the Fine Structure Constant
 |journal=[[Physical Review Letters]]
 |volume=109 |issue=11 |pages=111807
 |arxiv=1205.5368
 |bibcode=2012PhRvL.109k1807A
 |doi=10.1103/PhysRevLett.109.111807 |pmid=23005618
 |s2cid=14712017
}}</ref><ref>
{{cite journal
 |first1=Tatsumi |last1=Aoyama
 |first2=Masashi |last2=Hayakawa
 |first3=Toichiro |last3=Kinoshita
 |first4=Makiko |last4=Nio
 |title=Tenth-Order Electron Anomalous Magnetic Moment – Contribution of Diagrams without Closed Lepton Loops
 |journal=[[Physical Review D]]
 |volume=91 |issue=3 |date=1 February 2015
 |page=033006
 |arxiv=1412.8284
 |doi=10.1103/PhysRevD.91.033006
 |bibcode=2015PhRvD..91c3006A
 |s2cid=119024825
}}</ref><ref>
{{cite conference 
 |first=Makiko |last=Nio
 |title=QED tenth-order contribution to the electron anomalous magnetic moment and a new value of the fine-structure constant
 |conference=Fundamental Constants Meeting 2015 |date=3 February 2015
 |location=Eltville, Germany
 |url=http://www.bipm.org/cc/CODATA-TGFC/Allowed/2015-02/Nio.pdf
 |conference-url=http://www.bipm.org/cc/AllowedDocuments.jsp?cc=CODATA-TGFC
}}</ref>
<math display="block">a_\text{e} = 0.001\,159\,652\,181\,643(764)</math><!-- Paper give it as ...181 643(25)(23)(16)(763). sqrt(25**2+23**2+26**2+763**2) = 763.92-->

The QED prediction agrees with the experimentally measured value to more than 10 significant figures, making the magnetic moment of the electron one of the most accurately verified predictions in the history of [[physics]]. (See ''[[Precision tests of QED]]'' for details.)

The current experimental value and uncertainty is:<ref>{{Cite journal |last1=Fan |first1=X. |last2=Myers |first2=T. G. |last3=Sukra |first3=B. A. D. |last4=Gabrielse |first4=G. |date=2023-02-13 |title=Measurement of the Electron Magnetic Moment |url=https://link.aps.org/doi/10.1103/PhysRevLett.130.071801 |journal=Physical Review Letters |volume=130 |issue=7 |pages=071801 |doi=10.1103/PhysRevLett.130.071801|pmid=36867820 |arxiv=2209.13084 |bibcode=2023PhRvL.130g1801F |s2cid=123962197 }}</ref>
<math display="block">a_\text{e} = 0.001\,159\,652\,180\,59(13)</math>
According to this value, <math>a_\text{e}</math> is known to an accuracy of around 1 part in 10 billion (10<sup>10</sup>). This required measuring [[g-factor (physics)|<math>g</math>]] to an accuracy of around 1 part in 10 trillion (10<sup>13</sup>).

== Muon ==
[[File:The muon g-2.svg|thumb|300px|Proposed [[Minimal Supersymmetric Standard Model]] one-loop corrections to the muon ''g''−2 involving particles beyond the standard model: a [[neutralino]] and a [[smuon]], and a [[chargino]] and a muon [[sneutrino]] respectively.]]

The anomalous magnetic moment of the [[muon]] is calculated in a similar way to the electron. The prediction for the value of the muon anomalous magnetic moment includes three parts:<ref name=PDG />
<math display="block">\begin{align}a_\mu^\mathrm{SM} &= a_\mu^\mathrm{QED} + a_\mu^\mathrm{EW} + a_\mu^\mathrm{hadron}\\ &= 0.001\,165\,918\,04(51)\end{align}</math>

Of the first two components, <math>a_\mu^\mathrm{QED}</math> represents the photon and lepton loops, and <math>a_\mu^\mathrm{EW}</math> the W&nbsp;boson, Higgs boson and Z&nbsp;boson loops; both can be calculated precisely from first principles. The third term, <math>a_\mu^\mathrm{hadron}</math>, represents hadron loops; it 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 [[electron]]–[[antielectron]] ({{subatomic particle|electron-}}–{{subatomic particle|electron+}}) collisions. As of July&nbsp;2017, the measurement disagrees with the [[Standard Model]] by 3.5&nbsp;[[standard deviation]]s,<ref>{{cite journal |last1=Giusti |first1=D. |last2=Lubicz |first2=V. |last3=Martinelli |first3=G. |last4=Sanflippo |first4=F. |last5=Simula |first5=S. |title=Strange and charm HVP contributions to the muon (''g''&nbsp;−&nbsp;2) including QED corrections with twisted-mass fermions |arxiv=1707.03019 |year=2017 |journal=Journal of High Energy Physics |volume=2017 |issue=10 |page=157 |doi-access=free |doi=10.1007/JHEP10(2017)157|bibcode=2017JHEP...10..157G }}</ref> suggesting [[physics beyond the Standard Model]] may be having an effect (or that the theoretical/experimental errors are not completely under control). This is one of the long-standing discrepancies between the Standard Model and experiment.

The [[Muon g-2#Muon g − 2 at Brookhaven National Laboratory|E821 Experiment]] at [[Brookhaven National Laboratory]] (BNL) studied the precession of [[muon]] and [[antimuon]] in a constant external magnetic field as they circulated in a confining storage ring.<ref>{{cite web |title=The E821 Muon (g−2) Home Page |url=http://www.g-2.bnl.gov/ |publisher=[[Brookhaven National Laboratory]] |access-date=2014-07-01}}</ref> The E821 Experiment reported the following average value<ref name=PDG>{{cite journal |last1=Patrignani |first1=C. |last2=Agashe |first2=K. |title=Review of Particle Physics |journal=Chinese Physics C |publisher=IOP Publishing |volume=40 |issue=10 |year=2016 |issn=1674-1137 |doi=10.1088/1674-1137/40/10/100001 |page=100001 |bibcode=2016ChPhC..40j0001P |s2cid=125766528 |url=http://inspirehep.net/record/1489868/files/openaccess_cpc_40_10_100001.pdf}}</ref>
<math display="block">a_\mu = 0.001\; 165\; 920\; 9 (6).</math>

In 2024, the [[Fermilab#Muon g−2|Fermilab]] collaboration "[[Muon g−2|Muon ''g''−2]]" doubled the accuracy of this value over the group’s previous measurements from the 2018 data set. The data for the experiment were collected during the 2019–2020 runs. The independent value came in at <math>a_\mu = 0.001\,165\,920\,57(25)</math> (0.21 ppm), which, combined with measurements from Brookhaven National Laboratory, yields a world average of <math>a_\mu = 0.001\,165\,920\,59(22)</math> (0.19 ppm).<ref>{{cite journal|url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.110.032009 |journal= Physical Review D|title= Detailed report on the measurement of the positive muon anomalous magnetic moment to 0.20 ppm |date=8 August 2024 |publisher=Fermilab |access-date=16 March 2015}}</ref>

In April 2021, an international group of fourteen physicists reported that by using ab-initio [[quantum chromodynamics]] and [[quantum electrodynamics]] simulations they were able to obtain a theory-based approximation agreeing more with the experimental value than with the previous theory-based value that relied on the electron–positron annihilation experiments.<ref>{{cite journal |last1=Borsany |first1=Szabolcs |title=Leading hadronic contribution to the muon magnetic moment from lattice QCD |journal=Nature |date=7 April 2021 |volume=593 |issue=7857 |pages=51–55 |doi=10.1038/s41586-021-03418-1 |pmid=33828303|display-authors=etal|arxiv=2002.12347 |bibcode=2021Natur.593...51B |s2cid=221151004 }}</ref>

== Tau ==
The Standard Model prediction for the [[tauon|tau]]'s anomalous magnetic dipole moment is<ref>{{Cite journal|last1=Eidelman|first1=S.|last2=Passera|first2=M.|date=2007-01-30|title=THEORY OF THE τ LEPTON ANOMALOUS MAGNETIC MOMENT|journal=Modern Physics Letters A|language=en|volume=22|issue=3|pages=159–179|doi=10.1142/S0217732307022694|arxiv=hep-ph/0701260|bibcode=2007MPLA...22..159E|s2cid=18000680|issn=0217-7323}}</ref>
<math display="block">a_\tau=0.001\,177\,21(5) ,</math>
while the best measured bound for <math>a_\tau</math> is<ref>{{Cite journal |last=CMS Collaboration |first=The |date=2024-10-01 |title=Observation of γ γ → τ τ in proton–proton collisions and limits on the anomalous electromagnetic moments of the τ lepton |url=https://iopscience.iop.org/article/10.1088/1361-6633/ad6fcb |journal=Reports on Progress in Physics |volume=87 |issue=10 |pages=107801 |doi=10.1088/1361-6633/ad6fcb |issn=0034-4885|arxiv=2406.03975 }}</ref>
<math display="block">a_\tau = {0.0009}_{-0.0031}^{+0.0032},</math>reported by the CMS experiment at the CERN LHC.

== Composite particles ==
[[Composite particle]]s often have a huge anomalous magnetic moment. The [[nucleons]], [[proton]]s and [[neutron]]s, both composed of [[quark]]s, are examples. The [[nucleon magnetic moment]]s are both large and were unexpected; the proton's magnetic moment is much too large for an elementary particle, while the [[neutron]]'s magnetic moment was expected to be zero due to its charge being zero.

== See also ==
* [[Anomalous electric dipole moment]]
* [[Electron magnetic moment]]
* [[G-factor (physics)|''g''-factor (physics)]]
* [[Gordon decomposition]]

== References ==
{{reflist}}

== Bibliography ==
* {{cite book | author=Sergei Vonsovsky |title=Magnetism of Elementary Particles | url=https://archive.org/details/MagnetismOfElementaryParticles | publisher=Mir Publishers | year=1975 |author-link=Sergei Vonsovsky }}

== External links ==
{{Commons category}}
* [http://www.g-2.bnl.gov/physics/index.html Overview of the ''g−2'' experiment]
* {{cite journal
 |last1=Kusch |first1=P.
 |last2=Foley |first2=H. M.
 |year=1948
 |title=The Magnetic Moment of the Electron
 |journal=[[Physical Review]]
 |volume=74 |issue=3 |pages=250–263
 |bibcode=1948PhRv...74..250K
 |doi=10.1103/PhysRev.74.250
}}
* {{cite journal
 |last1=Aoyama|first1=T.
 |last2=Asmussen|first2=N.
 |display-authors=1
 |year=2020
 |title=The anomalous magnetic moment of the muon in the Standard Model
 |journal=[[Physics Reports]]
 |volume=887 |pages=1–166
 |doi=10.1016/j.physrep.2020.07.006
 |arxiv=2006.04822
 |bibcode=2020PhR...887....1A
 |doi-access=free
}}

{{QED}}

[[Category:Magnetic moment]]
[[Category:Quantum electrodynamics]]
[[Category:Quantum field theory]]
[[Category:Standard Model]]