56. Muon Anomalous Magnetic Moment1

56. Muon Anomalous Magnetic Moment
Revised August 2023 by A. Höcker (CERN) and W.J. Marciano (BNL).
~ with gyromagnetic ratio~ = gµ e S,
The Dirac equation predicts a muon magnetic moment, M
2mµgµ = 2. Quantum loop effects lead to a small calculable deviation from gµ = 2, parameterized by
the magnetic anomaly1gµ − 2aµ ≡.
(56.1)
2That quantity can be accurately measured and, within the Standard Model (SM) framework,
precisely predicted. Hence, comparison of experiment and theory tests the SM at its quantum
loop level. A deviation in aexp
from the SM expectation would signal effects of new physics, withµcurrent sensitivity reaching up to mass scales of O(TeV) [1]. For recent thorough muon g−2 reviews,
see e.g. Refs. [2–4].
The E821 experiment at Brookhaven National Lab (BNL) studied during the years 1997−2001
the precession of µ+ and µ− in a constant external magnetic field as they circulated in a confining
storage ring. It found2 [6]aexp,BNL= 116 592 040(60)(50) × 10−11 ,µ+aexp,BNL= 116 592 150(80)(30) × 10−11 ,µ−(56.2)

where the first errors are statistical and the second systematic. Assuming CPT invariance andtaking into account correlations between systematic uncertainties, one finds for their average [5, 6]
aexp,BNL= 116 592 089(54)(33) × 10−11 .
µ(56.3)

These results represent about a factor of 14 improvement over the classic CERN experiments of
the 1970s [7].
Further improvement of the measurement by a factor of four by setting up the E821 storage
ring at the Fermilab National Accelerator Laboratory (FNAL), and utilizing a cleaner and more
intense muon beam and improved detectors [8] is in progress. After six runs, the experiment hascompleted data taking in 2023. A first analysis with positive muons based on a fraction of thedata collected in 2018 was released in 2021 confirming the BNL result with similar precision [9].
A second analysis released in 2023 was based on a four times larger positive muon data sample
collected in 2019 and 2020, and benefitted from instrumental improvements and reduced systematic
uncertainties [10]. The result is dominated by statistical uncertainty and consistent with but more
precise than the previous one. Their combination yields [10]= 116 592 055(24) × 10−11 .
aexp,FNALµ+(56.4)The ongoing analysis of the remaining 2021–2023 dataset is expected to further reduce the uncertainty by a factor of two.
The FNAL result agrees with the BNL measurement and has improved precision. Their averageassuming CPT invariance reads [10]
−11
aexp
,
µ = 116 592 059(22) × 10
1(56.5)

Also referred to as anomalous magnetic moment despite being dimensionless.
The results reported by the experiment have been updated in Eqs. (56.2) and (56.3) to the newest value for theabsolute muon-to-proton magnetic ratio λ = 3.183 345 142(71) [5]. The change induced in aexp
with respect to theµvalue of λ = 3.183 345 39(10) used in Ref. [6] amounts to +10 × 10−11 .
2

S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:14am

56. Muon Anomalous Magnetic Moment

2

γ

γ

γ
W

γ
µ

γ
W

ν

Z
µ µ

µ µ

γ
µ µ

had

γ
µ

Figure 56.1: Representative diagrams contributing to aSMµ . From left to right: first order QED(Schwinger term), lowest-order weak, lowest-order hadronic.
providing a relative precision of 0.19 parts per million.
Another muon g−2 experiment with similar sensitivity but using an alternative zero-electric-fieldtechnique with a low-emittance and low-momentum muon beam is currently under construction at
J-PARC in Japan [11].
is generally divided into three parts (see Fig. 56.1 for representative
The SM prediction aSMµ
Feynman diagrams)
QED
aSM
+ aEW
+ aHad.
(56.6)µ = aµµµIn the following discussion we use the numerical estimates provided by the Muon g−2 Theory
Initiative White Paper [4].
The QED part includes all photonic and leptonic (e, µ, τ ) loops starting with the classic α/2π
Schwinger contribution [12]. It has been computed through five loops [4, 13, 14]
 2

α
α 3
+ 0.765 857 420(13)+ 24.050 509 85(23)
2π
π
π
 4
 5
α
α
+ 130.8782(60)+ 751.0(9)
+ ···
π
π

α
aQED
=
µ

 

(56.7)

with little change in the coefficients since our last update of this review. Employing α−1 =137.035 999 046(27), obtained from the precise measurements of h/mCs [15], the Rydberg constant,and mCs /me leads to3 [13]aQED
= 116 584 718.93(0.10) × 10−11 ,µ(56.8)

where the small error results mainly from the uncertainties in the estimate of the six loop contribution and in α.
Loop contributions involving heavy W ± , Z or Higgs particles are collectively labeled as aEW
µ .
2
2
−9
They are suppressed by at least a factor of (α/π) · (mµ /mW ) ' 4 × 10 . At 1-loop order [19]Gµ m2µEWaµ [1-loop] = √ 2
8 2π

"

2
m2µ
5 1
2
+
1 − 4 sin θW + O
2
3 3
MW

= 194.8 × 10−11 ,

!

m2µ
+O
m2H

!#

(56.9)

3
A recent measurement using Rb-87 atoms [16] resulted in α−1 = 137.035 999 206(11), which exhibits a larger
than 5σ discrepancy with the Cs-133 based result. Using this value in Eq. (56.7) leads to a reduction of aQEDµ
by 0.14 × 10−11 , which is larger than the quoted uncertainty in Eq. (56.8), but still negligible compared to otheruncertainties affecting aSMµ . This discrepancy impacts, however, the SM prediction of the magnetic moment of
the electron [14, 17], which differs by respectively −3.9σ and +2.1σ from the most recent experimental value [18],depending on whether the Cs-133 or Rb-87 based α value is used.

31st May, 2024

3

56. Muon Anomalous Magnetic Moment

2 /M 2 ' 0.223, and where G ' 1.166 × 10−5 GeV−2 is the Fermi coupling
for sin2 θW ≡ 1 − MWµZ
constant. Two-loop corrections are relatively large and negative [20]. For a Higgs boson mass of−11 [20], where the uncertainty stems from
125 GeV it amounts to aEWµ [2-loop] = −41.2(1.0) × 10
quark triangle loops. The 3-loop leading logarithms are negligible, O(10−12 ) [20, 21]. Overall one
finds
aEW
= 153.6(1.0) × 10−11 .
(56.10)µ

A recent complete 2-loop numerical evaluation of the electroweak correction [22] when adjusted forappropriate light quark masses confirmed the result in Eq. (56.10).
Hadronic (quark and gluon) loop contributions to aSMµ give rise to its main uncertainty. One traditionally relies on a data-driven dispersion relation approach to evaluate the lowest-order O(α2 )hadronic vacuum polarization contribution aHad
µ [LO] from corresponding cross section measurements [23]
α
3 π

1
aHad
µ [LO] =

!2 Z∞

ds

K(s) (0)R (s) ,
s(56.11)m2π

where K(s) is a QED kernel function [24], and where R(0) (s) denotes the ratio of the bare4 crosssection for e+ e− annihilation into hadrons to the pointlike muon-pair cross section at center-of-mass√energy s. The function K(s) ∼ 1/s in Eq. (56.11) emphasizes the low-energy part of the integral+ −so that aHad
µ [LO] is dominated by the ρ(770) → π π resonance.
The analysis of Eq. (56.11) results in the representative value [4, 25–30]
−11
aHad
,
µ [LO] = 6 931(40) × 10

(56.12)

whose error is dominated by systematic uncertainties in the e+ e− → hadrons cross-section data,
and also includes a small uncertainty due to perturbative QCD, which is used at intermediateand large energies in the dispersion integral to predict the contribution from the quark-antiquark
continuum. The experimental precision of Eq. (56.12) is limited by a discrepancy between themost precise π + π − data from the BABAR and KLOE experiments [27, 29]. Recent preliminary
π + π − data from the CMD-3 experiment [31] give a significantly larger contribution to aHad
µ [LO]
but disagree with essentially all other datasets. New or updated measurements are expected fromseveral experiments during the next years.
Alternatively, one can use precise vector spectral functions from τ → ντ + hadrons decays [32]that can be related to isovector e+ e− → hadrons cross sections by isospin symmetry. Analyses replaced e+ e− data in the two-pion and four-pion channels by the corresponding isospin- transformed
τ data, and applied isospin-violating corrections [33]. Owing to the progress in the precision of thee+ e− data, the τ data are now less precise and less reliable due to additional theoretical uncertainties, so that recent aHad
µ [LO] evaluations ignored them.
Progress has been achieved on the evaluation of aHad
µ [LO] using lattice QCD, which allows to
directly compute the real part of the two-point correlation function without invoking the resonances
occurring on the imaginary axis [34–36]. The required sub-percent precision represents a challenge
requiring to master systematic and extrapolation uncertainties. A combination of recent lattice
QCD evaluations [37, 38] gives [4]
−11
aHad
,
µ [LO] = 7 116(184) × 10

(56.13)

4
The bare cross section is defined as the measured cross section corrected for initial-state radiation, electronvertex loop contributions and vacuum-polarization effects in the photon propagator. However, QED effects in the
hadron vertex and final state, as photon radiation, are included.

31st May, 2024

56. Muon Anomalous Magnetic Moment

4with an uncertainty that is not yet competitive with the data-driven approach. Another recent
lattice QCD calculation with much reduced uncertainty found [39]
−11
aHad
,
µ [LO] = 7 075(55) × 10

(56.14)whose error is dominated by systematic uncertainties. It exhibits a 2.1σ tension with the datadriven result, growing up to 3.7σ for the modified observable aµ,win that has favourable properties
for computation on the lattice5 [38, 39, 41]. Because of the complexity and challenges of sucha calculation, it is critical that it is confronted with independent lattice results of comparable
precision [38, 41, 42].
Higher order hadronic loop contributions are also obtained from dispersion relations using thesame e+ e− → hadrons data but with different integration kernels [43]. Recent evaluations foundaµHad,Disp [NLO] = (−98.3 ± 0.7) × 10−11 [4, 30] and aµHad,Disp [NNLO] = (12.4 ± 0.1) × 10−11 [44].
In the case of hadronic light-by-light scattering contributions, which enter at NLO, recent studies[3,4,45,46] based on short-distance QCD, dispersion relations and lattice QCD calculations lead tothe improved prediction aHad,LBL[NLO] = 92(18) × 10−11 [4].6 One thus finds for the sum of theµthree higher order hadronic terms
−11
aHad
,
µ [N(N)LO] = 6(18) × 10

(56.15)where the error is dominated by the hadronic light-by-light scattering uncertainty.
Adding Eqs. (56.8), (56.10), (56.12) and (56.15) gives the representative SM prediction
−11
aSM
,
µ = 116 591 810(1)(40)(18) × 10

(56.16)

whose errors are due to the electroweak, lowest-order hadronic, and higher-order hadronic contributions, respectively. The difference between experiment and theorySM−11∆aµ = aexp,
µ − aµ = 249(22)(43) × 10

(56.17)

where the errors are from experiment and theory prediction (with all errors combined in quadrature,taking into account a small anticorrelation [4] between the data-driven lowest and higher ordercontributions), exhibits a discrepancy of 5.2σ.7 The new FNAL measurement has increased thediscrepancy with the SM prediction from previously 3.7σ (BNL only) and 4.2σ (including thefirst FNAL result), respectively. Despite the large significance, the situation remains inconclusive
due to the discrepancies in the hadronic cross section data underlying the dispersive aHad
µ [LO]
determination.
Figure 56.2 represents the difference between experimental and predicted aµ values based on
the most recent data-driven and lattice QCD based evaluations of aHad
µ [LO].
A leading candidate explanation for the deviation observed in ∆aµ has been supersymmetry
particle loop contributions. Such a scenario is quite natural, since generically, supersymmetricmodels predict [1] an additional contribution to aSMµ
aSUSY' ± 130 × 10−11 ·µ5100 GeV 2tanβ ,
mSUSY
(56.18)A dispersive approach similar to that for aHad
[LO] is used to determine the hadronic contribution to the runningµof the electromagnetic fine structure constant at the Z-boson mass ∆αhad (mZ ), which is a critical ingredient of
the global electroweak fit. One may therfore ask if the goodness of that fit would break down were aHad
[LO] inµEq. (56.12) be brought in compatibility with the result (56.14). However, the difference in the relevant integration
kernels (whereas 73% of the contribution to aHad
[LO] stems from the π + π − channel, it is only 13% in the case ofµHad∆αhad (mZ )) allows to absorb large effects in aµ [LO] without creating a significant tension in the electroweak fit.
Detailed discussions are available in the literature [40].
6
A very recent lattice QCD based calculation of aHad,LBL[NLO] [47], not included in [4], gives a consistent resultµ
with [4, 46] using a different approach.
7
The probabilistic interpretation of the discrepancy requires caution due to the systematic nature of the aSMµ
uncertainty.
31st May, 2024

56. Muon Anomalous Magnetic Moment

5

FNAL '21−'23+ BNL '02−'06

WP 2020 (DD)−249 ± 43(5.2σ)

0 ± 22

DHMZ 2019 (DD)(5.0σ)

Experiment

SM predictions−240 ± 43

KNT 2019 (DD)−253 ± 30(6.8σ)WP 2020 (LQ)
−64 ± 185(0.3σ)

BMW 2021 (LQ)−105 ± 58−600(1.7σ)

−500

−400

−300

−200

−100

0

100

200
−11

aµ − aexp[ × 10µ]Figure 56.2: Compilation of recent results for aµ (in units of 10−11 ), subtracted by the central valueof the experimental average (56.3). The light blue vertical band indicates the total experimental
uncertainty. The SM predictions shown differ only in the value used for the aHad
µ [LO] contribution.
They are taken from: WP 2020 [4] (Eq. (56.16), also indicated by the light grey vertical band),
DHMZ 2019 [29], KNT 2019 [30], all three based on the data-driven (DD) aHad
µ [LO] evaluation, andthe lattice QCD (LQ) based evaluations WP 2020 [4, 37, 38] and BMW 2021 [39]. Note that thequoted errors in the figure do not include the uncertainty on the subtracted experimental value.
To obtain for each theory calculation a result equivalent to Eq. (56.17), the errors from theory andexperiment must be added in quadrature.
where mSUSY is a representative supersymmetric mass scale, tanβ ' 3–40 a potential enhancement
factor, and ±1 corresponds to the sign of the µ term in the supersymmetric Lagrangian. Supersymmetric particles in the mass range 100–500 GeV could be the source of the deviation ∆aµ . If
so, those particles should be directly observable at the Large Hadron Collider at CERN. So far,there is however no direct evidence in support of the supersymmetry interpretation.
New physics effects [1] other than supersymmetry could also explain a non-vanishing ∆aµ . A
popular scenario involves the “dark photon”, a relatively light hypothetical vector boson from thedark matter sector that couples to our world of particle physics through mixing with the ordinary
photon [48–50]. As a result, it couples to ordinary charged particles with strength ε · e and gives
rise to an additional muon magnetic anomaly contribution
photon
adark
=
µ

α 2
ε F (mV /mµ ) ,2π(56.19)where F (x) = 01 2z(1−z)2 /[(1−z)2 +x2 z] dz. For values of ε ∼ 1–2×10−3 and mV ∼ 10–100 MeV,
the dark photon, which was originally motivated by cosmology, can provide a viable solution to theR

31st May, 2024

6

56. Muon Anomalous Magnetic Momentmuon g−2 discrepancy. However, experimental constraints appear to rule out the simplest modelin which the dark photon has equal couplings to electrons and muons and decays primarily to e+ e−pairs [51] or invisible dark particles [52] that give rise to missing event energy. One can expand thedark photon scenario into a more complete theory in which several new particles contribute to themuon g−2. Direct searches for a dark photon, therefore, continue to be well motivated, but withprimary guidance coming from astrophysics [53].
Recent popular solutions to the muon anomaly discrepancy have also focused on loop contributions induced by relatively light scalar or pseudo scalar particle appendages from physics beyond
the SM [54, 55].
References[1] A. Czarnecki and W. J. Marciano, Phys. Rev. D 64, 013014 (2001), [hep-ph/0102122];
M. Davier and W. Marciano, Ann. Rev. Nucl. Part. Sci. 54, 115 (2004).
[2] J. P. Miller, E. de Rafael and B. Roberts, Rept. Prog. Phys. 70, 795 (2007), [hep-ph/0703049];
F. Jegerlehner and A. Nyffeler, Phys. Rept. 477, 1 (2009), [arXiv:0902.3360]; J. P. Miller et al.,Ann. Rev. Nucl. Part. Sci. 62, 237 (2012).
[3] F. Jegerlehner, Springer Tracts Mod. Phys. 274, 1 (2017).
[4] T. Aoyama et al., Phys. Rept. 887, 1 (2020), [arXiv:2006.04822].
[5] P. J. Mohr, D. B. Newell and B. N. Taylor, Rev. Mod. Phys. 88, 3, 035009 (2016),[arXiv:1507.07956].
[6] G. Bennett et al. (Muon g-2), Phys. Rev. Lett. 89, 101804 (2002), [Erratum: Phys.Rev.Lett.
89, 129903 (2002)], [hep-ex/0208001]; G. Bennett et al. (Muon g-2), Phys. Rev. Lett. 92,
161802 (2004), [hep-ex/0401008]; G. Bennett et al. (Muon g-2), Phys. Rev. D 73, 072003(2006), [hep-ex/0602035].
[7] J. Bailey et al. (CERN-Mainz-Daresbury), Nucl. Phys. B 150, 1 (1979).
[8] J. Grange et al. (Muon g-2) (2015), [arXiv:1501.06858].
[9] B. Abi et al. (Muon g-2), Phys. Rev. Lett. 126, 14, 141801 (2021), [arXiv:2104.03281].
[10] D. P. Aguillard et al. (Muon g-2) (2023), [arXiv:2308.06230].
[11] M. Abe et al., PTEP 2019, 5, 053C02 (2019), [arXiv:1901.03047].
[12] J. S. Schwinger, Phys. Rev. 73, 416 (1948).
[13] T. Aoyama et al., Phys. Rev. Lett. 109, 111808 (2012), [arXiv:1205.5370]; T. Aoyama
et al., Phys. Rev. Lett. 109, 111807 (2012), [arXiv:1205.5368]; T. Kinoshita and M. Nio,
Phys. Rev. D 73, 013003 (2006), [hep-ph/0507249]; T. Aoyama et al., Phys. Rev. Lett.
99, 110406 (2007), [arXiv:0706.3496]; T. Kinoshita and M. Nio, Phys. Rev. D 70, 113001(2004), [hep-ph/0402206]; T. Kinoshita, Nucl. Phys. B Proc. Suppl. 144, 206 (2005); T. Kinoshita and M. Nio, Phys. Rev. D 73, 053007 (2006), [hep-ph/0512330]; S. Laporta, Phys.
Lett. B 772, 232 (2017), [arXiv:1704.06996]; A. Kurz et al., Phys. Rev. D 93, 5, 053017(2016), [arXiv:1602.02785]; A. Kataev, in “12th Lomonosov Conference on Elementary Particle Physics,” 345–349 (2006), [hep-ph/0602098]; M. Passera, J. Phys. G 31, R75 (2005),[hep-ph/0411168].
[14] T. Aoyama, T. Kinoshita and M. Nio, Atoms 7, 1, 28 (2019).
[15] R. H. Parker et al., Science 360, 191 (2018), [arXiv:1812.04130].
[16] L. Morel et al., Nature 588, 7836, 61 (2020).
[17] T. Aoyama et al., Phys. Rev. D 91, 3, 033006 (2015), [Erratum: Phys.Rev.D 96, 019901(2017)], [arXiv:1412.8284].
31st May, 2024

7

56. Muon Anomalous Magnetic Moment[18] X. Fan et al., Phys. Rev. Lett. 130, 7, 071801 (2023), [arXiv:2209.13084].
[19] R. Jackiw and S. Weinberg, Phys. Rev. D 5, 2396 (1972); G. Altarelli, N. Cabibbo and
L. Maiani, Phys. Lett. B 40, 415 (1972); I. Bars and M. Yoshimura, Phys. Rev. D 6, 374(1972); K. Fujikawa, B. Lee and A. Sanda, Phys. Rev. D 6, 2923 (1972).
[20] C. Gnendiger, D. Stöckinger and H. Stöckinger-Kim, Phys. Rev. D 88, 053005 (2013),[arXiv:1306.5546]; A. Czarnecki, W. J. Marciano and A. Vainshtein, Phys. Rev. D 67,
073006 (2003), [Erratum: Phys.Rev.D 73, 119901 (2006)], [hep-ph/0212229]; S. Heinemeyer,
D. Stockinger and G. Weiglein, Nucl. Phys. B 699, 103 (2004), [hep-ph/0405255]; T. Gribouk and A. Czarnecki, Phys. Rev. D 72, 053016 (2005), [hep-ph/0509205]; A. Czarnecki,
B. Krause and W. J. Marciano, Phys. Rev. Lett. 76, 3267 (1996), [hep-ph/9512369]; A. Czarnecki, B. Krause and W. J. Marciano, Phys. Rev. D 52, 2619 (1995), [hep-ph/9506256]; S. Peris,
M. Perrottet and E. de Rafael, Phys. Lett. B 355, 523 (1995), [hep-ph/9505405]; T. Kukhto
et al., Nucl. Phys. B 371, 567 (1992).
[21] G. Degrassi and G. Giudice, Phys. Rev. D 58, 053007 (1998), [hep-ph/9803384].
[22] T. Ishikawa, N. Nakazawa and Y. Yasui, Phys. Rev. D 99, 7, 073004 (2019), [arXiv:1810.13445].
[23] C. Bouchiat and L. Michel, J. Phys. Radium 22, 2, 121 (1961); M. Gourdin and E. De Rafael,Nucl. Phys. B 10, 667 (1969).
[24] S. J. Brodsky and E. De Rafael, Phys. Rev. 168, 1620 (1968).
[25] M. Davier et al., Eur. Phys. J. C77, 12, 827 (2017), [arXiv:1706.09436].
[26] A. Keshavarzi, D. Nomura and T. Teubner, Phys. Rev. D 97, 11, 114025 (2018),[arXiv:1802.02995].
[27] G. Colangelo, M. Hoferichter and P. Stoffer, JHEP 02, 006 (2019), [arXiv:1810.00007].
[28] M. Hoferichter, B.-L. Hoid and B. Kubis, JHEP 08, 137 (2019), [arXiv:1907.01556].
[29] M. Davier et al., Eur. Phys. J. C 80, 3, 241 (2020), [Erratum: Eur.Phys.J.C 80, 410 (2020)],[arXiv:1908.00921].
[30] A. Keshavarzi, D. Nomura and T. Teubner, Phys. Rev. D 101, 1, 014029 (2020),[arXiv:1911.00367].
[31] F. V. Ignatov et al. (CMD-3) (2023), [arXiv:2302.08834].
[32] R. Alemany, M. Davier and A. Hocker, Eur. Phys. J. C 2, 123 (1998), [hep-ph/9703220].
[33] M. Davier et al., Eur. Phys. J. C 71, 1515 (2011), [Erratum: Eur.Phys.J.C 72, 1874 (2012)],[arXiv:1010.4180].
[34] T. Blum, Phys. Rev. Lett. 91, 052001 (2003), [hep-lat/0212018].
[35] D. Bernecker and H. B. Meyer, Eur. Phys. J. A 47, 148 (2011), [arXiv:1107.4388].
[36] W. A. Bardeen, R. Gastmans and B. E. Lautrup, Nucl. Phys. B 46, 319 (1972).
[37] B. Chakraborty et al. (Fermilab Lattice, LATTICE-HPQCD, MILC), Phys. Rev. Lett. 120, 15,152001 (2018), [arXiv:1710.11212]; S. Borsanyi et al. (Budapest-Marseille-Wuppertal), Phys.
Rev. Lett. 121, 2, 022002 (2018), [arXiv:1711.04980]; T. Blum et al. (RBC, UKQCD), Phys.
Rev. Lett. 121, 2, 022003 (2018), [arXiv:1801.07224]; D. Giusti et al. (ETM), Phys. Rev. D99,
11, 114502 (2019), [arXiv:1901.10462]; E. Shintani and Y. Kuramashi, Phys. Rev. D100,
3, 034517 (2019), [arXiv:1902.00885]; C. T. H. Davies et al. (Fermilab Lattice, LATTICEHPQCD, MILC), Phys. Rev. D101, 3, 034512 (2020), [arXiv:1902.04223]; A. Gérardin et al.,Phys. Rev. D100, 1, 014510 (2019), [arXiv:1904.03120].
[38] C. Aubin et al., Phys. Rev. D101, 1, 014503 (2020), [arXiv:1905.09307]; D. Giusti and S. Simula, PoS LATTICE2019, 104 (2019), [arXiv:1910.03874].
31st May, 2024

856. Muon Anomalous Magnetic Moment[39] S. Borsanyi et al., Nature 593, 7857, 51 (2021), [arXiv:2002.12347].
[40] A. Crivellin et al., Phys. Rev. Lett. 125, 9, 091801 (2020), [arXiv:2003.04886]; A. Keshavarzi
et al., Phys. Rev. D 102, 3, 033002 (2020), [arXiv:2006.12666]; E. de Rafael, Phys. Rev. D
102, 5, 056025 (2020), [arXiv:2006.13880]; B. Malaescu and M. Schott, Eur. Phys. J. C 81, 1,
46 (2021), [arXiv:2008.08107]; G. Colangelo, M. Hoferichter and P. Stoffer, Phys. Lett. B 814,
136073 (2021), [arXiv:2010.07943].
[41] T. Blum et al. (RBC, UKQCD), Phys. Rev. Lett. 121, 2, 022003 (2018), [arXiv:1801.07224].
[42] C. Lehner and A. S. Meyer, Phys. Rev. D 101, 074515 (2020), [arXiv:2003.04177].
[43] B. Krause, Phys. Lett. B 390, 392 (1997), [hep-ph/9607259].
[44] A. Kurz et al., Phys. Lett. B 734, 144 (2014), [arXiv:1403.6400].
[45] K. Melnikov and A. Vainshtein, Phys. Rev. D70, 113006 (2004), [hep-ph/0312226]; P. Masjuan
and P. Sánchez-Puertas, Phys. Rev. D95, 5, 054026 (2017), [arXiv:1701.05829]; G. Colangelo et al., JHEP 04, 161 (2017), [arXiv:1702.07347]; M. Hoferichter et al., JHEP 10, 141(2018), [arXiv:1808.04823]; A. Gérardin, H. B. Meyer and A. Nyffeler, Phys. Rev. D100, 3,034520 (2019), [arXiv:1903.09471]; J. Bijnens, N. Hermansson-Truedsson and A. RodríguezSánchez, Phys. Lett. B798, 134994 (2019), [arXiv:1908.03331]; G. Colangelo et al., JHEP
03, 101 (2020), [arXiv:1910.13432]; V. Pauk and M. Vanderhaeghen, Eur. Phys. J. C74,
8, 3008 (2014), [arXiv:1401.0832]; I. Danilkin and M. Vanderhaeghen, Phys. Rev. D95,
1, 014019 (2017), [arXiv:1611.04646]; M. Knecht et al., Phys. Lett. B787, 111 (2018),[arXiv:1808.03848]; G. Eichmann, C. S. Fischer and R. Williams, Phys. Rev. D101, 5, 054015(2020), [arXiv:1910.06795]; P. Roig and P. Sánchez-Puertas, Phys. Rev. D101, 7, 074019(2020), [arXiv:1910.02881]; G. Colangelo et al., Phys. Lett. B735, 90 (2014), [arXiv:1403.7512].
[46] T. Blum et al., Phys. Rev. Lett. 124, 13, 132002 (2020), [arXiv:1911.08123].
[47] E.-H. Chao et al., Eur. Phys. J. C 81, 7, 651 (2021), [arXiv:2104.02632].
[48] P. Fayet, Phys. Rev. D 75, 115017 (2007), [hep-ph/0702176].
[49] M. Pospelov, Phys. Rev. D 80, 095002 (2009), [arXiv:0811.1030].
[50] D. Tucker-Smith and I. Yavin, Phys. Rev. D 83, 101702 (2011), [arXiv:1011.4922].
[51] J. Batley et al. (NA48/2), Phys. Lett. B 746, 178 (2015), [arXiv:1504.00607].
[52] Y. M. Andreev et al. (NA64), Phys. Rev. Lett. 126, 21, 211802 (2021), [arXiv:2102.01885];
J. P. Lees et al. (BaBar), Phys. Rev. Lett. 119, 13, 131804 (2017), [arXiv:1702.03327].
[53] J. Alexander et al. (2016), [arXiv:1608.08632].
[54] K.-m. Cheung, C.-H. Chou and O. C. W. Kong, Phys. Rev. D 64, 111301 (2001), [hepph/0103183].
[55] D. Chang et al., Phys. Rev. D 63, 091301 (2001), [hep-ph/0009292].

31st May, 2024