# Anomalous magnetic dipole moment

In quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams 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; the Dirac equation predicts ${\displaystyle g=2}$. 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 ${\displaystyle a}$ and defined as

${\displaystyle a={\frac {g-2}{2}}}$

## Electron

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 [1] and the one-loop result is:

${\displaystyle a_{e}={\frac {\alpha }{2\pi }}\approx 0.001\;161\;4}$

where ${\displaystyle \alpha }$ is the fine structure constant. This result was first found by Julian Schwinger in 1948 [2] and is engraved on his tombstone. As of 2016, the coefficients of the QED formula for the anomalous magnetic moment of the electron are known analytically up to ${\displaystyle \alpha ^{3}}$[3] and have been calculated up to order ${\displaystyle \alpha ^{5}}$:[4][5][6]

${\displaystyle a_{e}=0.001\;159\;652\;181\;643(764)}$

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.)

The current experimental value and uncertainty is:[7]

${\displaystyle a_{e}=0.001\;159\;652\;180\;73(28)}$

According to this value, ${\displaystyle a_{e}}$ is known to an accuracy of around 1 part in 1 billion (109). This required measuring ${\displaystyle g}$ to an accuracy of around 1 part in 1 trillion (1012).

An empirical formula has been reported for the anomalous magnetic moment:[8]

${\displaystyle \delta \mu _{e}={\frac {e\hbar }{2c}}\left[{\frac {3}{{\sqrt {2}}\left(m_{p}-\exp \left(1\right)\delta m\right)}}\right]}$,

where ${\displaystyle m_{p}}$ is the proton mass and ${\displaystyle \delta m}$ is difference between the neutron and proton masses ${\displaystyle \left(m_{n}-m_{p}\right)}$. Therefore, the value ${\displaystyle a_{e}}$ is equal to

${\displaystyle a_{e}={\frac {3m_{e}}{{\sqrt {2}}\left(m_{p}-\exp \left(1\right)\delta m\right)}}=0.001\;159\;652\;3..}$,

where ${\displaystyle m_{e}}$ is the electron mass.

## Muon

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:[9]

{\displaystyle {\begin{aligned}a_{\mu }^{\mathrm {SM} }&=a_{\mu }^{\mathrm {QED} }+a_{\mu }^{\mathrm {EW} }+a_{\mu }^{\mathrm {Hadron} }\\&=0.001\;165\;918\;04(51)\end{aligned}}}

Of the first two components, ${\displaystyle a_{\mu }^{\mathrm {QED} }}$ represents the photon and lepton loops, and ${\displaystyle a_{\mu }^{\mathrm {EW} }}$ the W boson, Higgs boson and Z boson loops; both can be calculated precisely from first principles. The third term, ${\displaystyle a_{\mu }^{\mathrm {Hadron} }}$, 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) in electronantielectron (${\displaystyle e^{-}e^{+}}$) collisions. As of July 2017, the measurement disagrees with the Standard Model by 3.5 standard deviations,[10] 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 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.[11] The E821 Experiment reported the following average value[9]

${\displaystyle a_{\mu }=0.001\;165\;920\;9(6).}$

A new experiment at Fermilab called "Muon g−2" using the E821 magnet will improve the accuracy of this value.[12] Data taking began in March 2018 and is expected to end in September 2022.[13] An interim result released on April 7, 2021[14] yields ${\displaystyle a_{\mu }=0.001\;165\;920\;40(54)}$ which, in combination with existing measurements, gives a more precise estimate ${\displaystyle a_{\mu }=0.001\;165\;920\;61(41)}$, exceeding the Standard Model prediction by 4.2 standard deviations. Also, experiment E34 at J-PARC plans to start its first run in 2024.[15]

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.[16]

## Tau

The Standard Model prediction for tau's anomalous magnetic dipole moment is[17]

${\displaystyle a_{\tau }=0.001\;177\;21(5)}$,

while the best measured bound for ${\displaystyle a_{\tau }}$ is[18]

${\displaystyle -0.052.

## Composite particles

Composite particles often have a huge anomalous magnetic moment. This is true for the proton, which is made up of charged quarks, and the neutron, which has a magnetic moment even though it is electrically neutral.

## Notes

1. ^ Peskin, M. E.; Schroeder, D. V. (1995). "Section 6.3". An Introduction to Quantum Field Theory. Addison-Wesley. ISBN 978-0-201-50397-5.
2. ^ Schwinger, J. (1948). "On Quantum-Electrodynamics and the Magnetic Moment of the Electron" (PDF). Physical Review. 73 (4): 416–417. Bibcode:1948PhRv...73..416S. doi:10.1103/PhysRev.73.416.
3. ^ Laporta, S.; Remiddi, E. (1996). "The analytical value of the electron (g − 2) at order α3 in QED". Physics Letters B. 379 (1–4): 283–291. arXiv:hep-ph/9602417. Bibcode:1996PhLB..379..283L. doi:10.1016/0370-2693(96)00439-X.
4. ^ Aoyama, T.; Hayakawa, M.; Kinoshita, T.; Nio, M. (2012). "Tenth-Order QED Contribution to the Electron g−2 and an Improved Value of the Fine Structure Constant". Physical Review Letters. 109 (11): 111807. arXiv:1205.5368. Bibcode:2012PhRvL.109k1807A. doi:10.1103/PhysRevLett.109.111807. PMID 23005618. S2CID 14712017.
5. ^ Aoyama, Tatsumi; Hayakawa, Masashi; Kinoshita, Toichiro; Nio, Makiko (1 February 2015). "Tenth-Order Electron Anomalous Magnetic Moment — Contribution of Diagrams without Closed Lepton Loops". Physical Review D. 91 (3): 033006. arXiv:1412.8284. Bibcode:2015PhRvD..91c3006A. doi:10.1103/PhysRevD.91.033006. S2CID 119024825.
6. ^ Nio, Makiko (3 February 2015). QED tenth-order contribution to the electron anomalous magnetic moment and a new value of the fine-structure constant (PDF). Fundamental Constants Meeting 2015. Eltville, Germany.
7. ^ Hanneke, D.; Fogwell Hoogerheide, S.; Gabrielse, G. (2011). "Cavity Control of a Single-Electron Quantum Cyclotron: Measuring the Electron Magnetic Moment" (PDF). Physical Review A. 83 (5): 052122. arXiv:1009.4831. Bibcode:2011PhRvA..83e2122H. doi:10.1103/PhysRevA.83.052122. S2CID 16902741.
8. ^ Efimov, S.P. (2021). "Relation of the Anomalous Magnetic Moment of the Electron with Proton and Neutron Masses". Russ. Phys. J. 64 (6): 978–983. Bibcode:2021RuPhJ..64..978E. doi:10.1007/s11182-021-02417-z. S2CID 240269120.
9. ^ a b Patrignani, C.; Agashe, K. (2016). "Review of Particle Physics" (PDF). Chinese Physics C. IOP Publishing. 40 (10): 100001. Bibcode:2016ChPhC..40j0001P. doi:10.1088/1674-1137/40/10/100001. ISSN 1674-1137.
10. ^ Giusti, D.; Lubicz, V.; Martinelli, G.; Sanflippo, F.; Simula, S. (2017). "Strange and charm HVP contributions to the muon (g − 2) including QED corrections with twisted-mass fermions". Journal of High Energy Physics. 2017 (10): 157. arXiv:1707.03019. Bibcode:2017JHEP...10..157G. doi:10.1007/JHEP10(2017)157.
11. ^ "The E821 Muon (g−2) Home Page". Brookhaven National Laboratory. Retrieved 1 July 2014.
12. ^ "Revolutionary muon experiment to begin with 3,200 mile move of 50 foot-wide particle storage ring" (Press release). Fermilab. 8 May 2013. Retrieved 16 March 2015.
13. ^ "Current Status of Muon g-2 Experiment at Fermilab" (PDF). indico.cern.ch. Retrieved 28 September 2020.
14. ^ Abi, B.; et al. (April 2021). "Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm". Physical Review Letters. 126 (14): 141801. arXiv:2104.03281. Bibcode:2021PhRvL.126n1801A. doi:10.1103/PhysRevLett.126.141801. PMID 33891447.
15. ^ G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, JHEP 04, 161 (2017), arXiv:1702.07347 [hep-ph].
16. ^ Borsany, Szabolcs; et al. (7 April 2021). "Leading hadronic contribution to the muon magnetic moment from lattice QCD". Nature. 593 (7857): 51–55. arXiv:2002.12347. Bibcode:2021Natur.593...51B. doi:10.1038/s41586-021-03418-1. PMID 33828303. S2CID 221151004.
17. ^ Eidelman, S.; Passera, M. (30 January 2007). "THEORY OF THE τ LEPTON ANOMALOUS MAGNETIC MOMENT". Modern Physics Letters A. 22 (3): 159–179. arXiv:hep-ph/0701260. Bibcode:2007MPLA...22..159E. doi:10.1142/S0217732307022694. ISSN 0217-7323. S2CID 18000680.
18. ^ The DELPHI Collaboration (June 2004). "Study of tau-pair production in photon-photon collisions at LEP and limits on the anomalous electromagnetic moments of the tau lepton". The European Physical Journal C. 35 (2): 159–170. arXiv:hep-ex/0406010. Bibcode:2004EPJC...35..159D. doi:10.1140/epjc/s2004-01852-y. ISSN 1434-6044. S2CID 195317696.