## Abstract

The standard model of particle physics describes the vast majority of experiments and observations involving elementary particles. Any deviation from its predictions would be a sign of new, fundamental physics. One long-standing discrepancy concerns the anomalous magnetic moment of the muon, a measure of the magnetic field surrounding that particle. Standard-model predictions^{1} exhibit disagreement with measurements^{2} that is tightly scattered around 3.7 standard deviations. Today, theoretical and measurement errors are comparable; however, ongoing and planned experiments aim to reduce the measurement error by a factor of four. Theoretically, the dominant source of error is the leading-order hadronic vacuum polarization (LO-HVP) contribution. For the upcoming measurements, it is essential to evaluate the prediction for this contribution with independent methods and to reduce its uncertainties. The most precise, model-independent determinations so far rely on dispersive techniques, combined with measurements of the cross-section of electron–positron annihilation into hadrons^{3,4,5,6}. To eliminate our reliance on these experiments, here we use ab initio quantum chromodynamics (QCD) and quantum electrodynamics simulations to compute the LO-HVP contribution. We reach sufficient precision to discriminate between the measurement of the anomalous magnetic moment of the muon and the predictions of dispersive methods. Our result favours the experimentally measured value over those obtained using the dispersion relation. Moreover, the methods used and developed in this work will enable further increased precision as more powerful computers become available.

## Access options

Subscribe to Journal

Get full journal access for 1 year

$199.00

only $3.90 per issue

All prices are NET prices.

VAT will be added later in the checkout.

Tax calculation will be finalised during checkout.

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

## Data availability

The datasets used for the figures and tables are available from the corresponding author on request.

## Code availability

A CPU code for configuration production and measurements can be obtained from the corresponding author upon request. The Wilson flow evolution code, which was used to determine *w*_{0}, can be downloaded from https://arxiv.org/abs/1203.4469.

## References

- 1.
Tanabashi, M. et al. Review of particle physics.

*Phys. Rev. D***98**, 030001 (2018). - 2.
Bennett, G. W. et al. Final report of the muon E821 anomalous magnetic moment measurement at BNL.

*Phys. Rev. D***73**, 072003 (2006). - 3.
Davier, M., Hoecker, A., Malaescu, B. & Zhang, Z. A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to \(\alpha ({m}_{Z}^{2})\).

*Eur. Phys. J. C***80**, 241 (2020); erratum**80**, 410 (2020). - 4.
Keshavarzi, A., Nomura, D. & Teubner, T.

*g*− 2 of charged leptons, \(\alpha ({M}_{Z}^{2})\), and the hyperfine splitting of muonium.*Phys. Rev. D***101**, 014029 (2020). - 5.
Colangelo, G., Hoferichter, M. & Stoffer, P. Two-pion contribution to hadronic vacuum polarization.

*J. High Energy Phys*.**2019**, 006 (2019). - 6.
Hoferichter, M., Hoid, B. L. & Kubis, B. Three-pion contribution to hadronic vacuum polarization.

*J. High Energy Phys*.**2019**, 137 (2019). - 7.
Aoyama, T. et al. The anomalous magnetic moment of the muon in the Standard Model.

*Phys. Rep*.**887**, 1–166 (2020). - 8.
Bernecker, D. & Meyer, H. B. Vector correlators in lattice QCD: methods and applications.

*Eur. Phys. J. A***47**, 148 (2011). - 9.
Lautrup, B. E., Peterman, A. & de Rafael, E. Recent developments in the comparison between theory and experiments in quantum electrodynamics.

*Phys. Rep*.**3**, 193–259 (1972). - 10.
de Rafael, E. Hadronic contributions to the muon

*g*−2 and low-energy QCD.*Phys. Lett. B***322**, 239–246 (1994). - 11.
Blum, T. Lattice calculation of the lowest order hadronic contribution to the muon anomalous magnetic moment.

*Phys. Rev. Lett*.**91**, 052001 (2003). - 12.
Borsanyi, S. et al. High-precision scale setting in lattice QCD.

*J. High Energy Phys*.**2012**, 010 (2012). - 13.
Dowdall, R. J., Davies, C. T. H., Lepage, G. P. & McNeile, C.

*V*_{us}from*π*and*K*decay constants in full lattice QCD with physical*u*,*d*,*s*and*c*quarks.*Phys. Rev. D***88**, 074504 (2013). - 14.
Borsanyi, S. et al. Hadronic vacuum polarization contribution to the anomalous magnetic moments of leptons from first principles.

*Phys. Rev. Lett*.**121**, 022002 (2018). - 15.
Neff, H., Eicker, N., Lippert, T., Negele, J. W. & Schilling, K. On the low fermionic eigenmode dominance in QCD on the lattice.

*Phys. Rev. D***64**, 114509 (2001). - 16.
Giusti, L., Hernandez, P., Laine, M., Weisz, P. & Wittig, H. Low-energy couplings of QCD from current correlators near the chiral limit.

*J. High Energy Phys*.**2004**, 013 (2004). - 17.
DeGrand, T. A. & Schaefer, S. Improving meson two point functions in lattice QCD.

*Comput. Phys. Commun*.**159**, 185–191 (2004). - 18.
Shintani, E. et al. Covariant approximation averaging.

*Phys. Rev. D***91**, 114511 (2015). - 19.
Blum, T. et al. Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment.

*Phys. Rev. Lett*.**121**, 022003 (2018). - 20.
Aubin, C. et al. Light quark vacuum polarization at the physical point and contribution to the muon

*g*− 2.*Phys. Rev. D***101**, 014503 (2020). - 21.
de Divitiis, G. M. et al. Isospin breaking effects due to the up-down mass difference in Lattice QCD.

*J. High Energy Phys*.**2012**, 124 (2012). - 22.
de Divitiis, G. M. et al. Leading isospin breaking effects on the lattice.

*Phys. Rev. D***87**, 114505 (2013). - 23.
Colangelo, G., Durr, S. & Haefeli, C. Finite volume effects for meson masses and decay constants.

*Nucl. Phys. B***721**, 136–174 (2005). - 24.
Davoudi, Z. & Savage, M. J. Finite-volume electromagnetic corrections to the masses of mesons, baryons and nuclei.

*Phys. Rev. D***90**, 054503 (2014). - 25.
Borsanyi, S. et al. Ab initio calculation of the neutron-proton mass difference.

*Science***347**, 1452–1455 (2015). - 26.
Fodor, Z. et al. Quantum electrodynamics in finite volume and nonrelativistic effective field theories.

*Phys. Lett. B***755**, 245–248 (2016). - 27.
Aubin, C. et al. Finite-volume effects in the muon anomalous magnetic moment on the lattice.

*Phys. Rev. D***93**, 054508 (2016). - 28.
Bijnens, J. & Relefors, J. Vector two-point functions in finite volume using partially quenched chiral perturbation theory at two loops.

*J. High Energy Phys*.**2017**, 114 (2017). - 29.
Hansen, M. T. & Patella, A. Finite-volume effects in \({(g-2)}_{\mu }^{{\rm{HVP}},{\rm{LO}}}\).

*Phys. Rev. Lett*.**123**, 172001 (2019). - 30.
Jegerlehner, F. & Szafron, R. \({\rho }^{0}-\gamma \) mixing in the neutral channel pion form factor \({F}_{\pi }^{e}\) and its role in comparing

*e*^{+}e− with*τ*spectral functions.*Eur. Phys. J. C***71**, 1632 (2011). - 31.
Chakraborty, B. et al. The hadronic vacuum polarization contribution to

*a*_{μ}from full lattice QCD.*Phys. Rev. D***96**, 034516 (2017). - 32.
Gérardin, A. et al. The leading hadronic contribution to (

*g*− 2)_{μ}from lattice QCD with*N*_{f}= 2 + 1 flavours of O(*a*) improved Wilson quarks.*Phys. Rev. D***100**, 014510 (2019). - 33.
Davies, C. T. H. et al. Hadronic-vacuum-polarization contribution to the muon’s anomalous magnetic moment from four-flavor lattice QCD.

*Phys. Rev. D***101**, 034512 (2020). - 34.
Giusti, D., Lubicz, V., Martinelli, G., Sanfilippo, F. & Simula, S. Electromagnetic and strong isospin-breaking corrections to the muon

*g*− 2 from Lattice QCD+QED.*Phys. Rev. D***99**, 114502 (2019). - 35.
Giusti, D., Sanfilippo, F. & Simula, S. Light-quark contribution to the leading hadronic vacuum polarization term of the muon

*g*− 2 from twisted-mass fermions.*Phys. Rev. D***98**, 114504 (2018). - 36.
Shintani, E. et al. Hadronic vacuum polarization contribution to the muon

*g*− 2 with 2+1 flavor lattice QCD on a larger than (10 fm)^{4}lattice at the physical point.*Phys. Rev. D***100**, 034517 (2019). - 37.
Bazavov, A. et al. Gradient flow and scale setting on MILC HISQ ensembles.

*Phys. Rev. D***93**, 094510 (2016).

## Acknowledgements

We thank J. Charles, A. El-Khadra, M. Hoferichter, F. Jegerlehner, C. Lehner, M. Knecht, A. Kronfeld, E. de Rafael and participants of the online workshop ‘The hadronic vacuum polarization from lattice QCD at high precision’ (16–20 November 2020) for discussions. We thank J. Bailey, W. Lee and S. Sharpe for correspondence on staggered XPT. Special thanks to A. Keshavarzi for cross-section data and discussions, and to G. Colangelo and H. Meyer for constructive criticism. The computations were performed on JUQUEEN, JURECA, JUWELS and QPACE at Forschungszentrum Jülich, on SuperMUC and SuperMUC-NG at Leibniz Supercomputing Centre in Munich, on Hazel Hen and HAWK at the High Performance Computing Center in Stuttgart, on Turing and Jean Zay at CNRS IDRIS, on Joliot-Curie at CEA TGCC, on Marconi in Rome and on GPU clusters in Wuppertal and Budapest. We thank the Gauss Centre for Supercomputing, PRACE and GENCI (grant 52275) for awarding us computer time on these machines. This project was partially funded by DFG grant SFB/TR55, by BMBF grant 05P18PXFCA, by the Hungarian National Research, Development and Innovation Office grant KKP126769 and by the Excellence Initiative of Aix-Marseille University - A*MIDEX, a French “Investissements d’Avenir” programme, through grants AMX-18-ACE-005, AMX-19-IET-008 - IPhU and ANR-11-LABX-0060.

## Author information

### Affiliations

### Contributions

S.B., K.K.S. and B.C.T. wrote the codes and carried out the runs for configuration generation and measurements. S.B., Z.F., K.K.S., B.C.T. and L.V. were the main developers of the scale setting; L.L., K.K.S. and B.C.T. of the isospin breaking; F.S., K.K.S. and B.C.T. of the XPT; L.L., F.S. and C.T. of the MLLGS model; L.L., K.K.S. and B.C.T. of the RHO model; S.B., Z.F. and K.K.S. of the lattice finite-size study; K.K.S. and C.T. of the finite-size effects of isospin breaking; C.H., K.K.S. and B.C.T. of the overlap simulations. The global analysis strategy was developed by S.B., Z.F., S.D.K., L.L., F.S., K.K.S. and B.C.T. The global fits were carried out by S.B., J.N.G., S.D.K. and B.C.T. R-ratio and perturbative computations were done by Z.F., L.L., K.K.S. and C.T. Various crosschecks were performed by K.M., L.P., B.C.T. and C.T. S.B., Z.F., L.L., T.L. and K.K.S. were involved in acquisition of computer resources. Z.F., L.L. and K.K.S. wrote the main paper. Z.F. and K.K.S. coordinated the project.

### Corresponding author

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Additional information

**Peer review information** *Nature* thanks Gilberto Colangelo, Harvey Meyer and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

**Publisher’s note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Upper and lower bounds on the light isospin-symmetric component of *a*_{μ}, \({[{{\boldsymbol{a}}}_{{\boldsymbol{\mu }}}^{{\rm{l}}{\bf{ight}}}]}_{{\bf{0}}}\).

The bounds are computed using the lattice correlator below a time separation of *t*_{c} and an analytical formula describing the large-time behaviour above *t*_{c}.The results shown are obtained with the 4HEX action on two different lattice sizes, 56 × 84 and 96 × 96, both at *a* = 0.112 fm lattice spacing and *M*_{π} = 121 MeV Goldstone pion mass. We also carried out another simulation with *M*_{π} = 104 MeV mass. From these two, we interpolate to *M*_{π} = 110 MeV. This value ensures that a particular average of pion tastes is fixed to the physical value of the pion mass (see text). Error bars are statistical errors (s.e.m.).

### Extended Data Fig. 2 Isospin-symmetric component of \({{\boldsymbol{a}}}_{{\boldsymbol{\mu }}}^{{\rm{l}}{\bf{ight}}}\), computed with a sliding window.

The window starts at *t*_{1} and ends 0.5 fm later. The plot shows the difference between a fine and a coarse lattice with spacing *a* = 0.064 fm and *a* = 0.119 fm. The black squares with error bars are obtained from the simulation, and errors are statistical (s.e.m.). The coloured curves are the predictions of NLO,NNLO SXPT, and the SRHO and SMLLGS models. They are computed at the parameters (pion mass, taste violation, volume) of the simulations.

### Extended Data Fig. 3 Example continuum limits of \({{\boldsymbol{a}}}_{{\boldsymbol{\mu }}}^{{\rm{l}}{\bf{ight}}}\).

The light-green triangles labelled ‘none’ correspond to our lattice results with no taste improvement. The blue squares repesent data that have undergone no taste improvement for *t* < 1.3 fm and SRHO improvement above. The blue curves correspond to example continuum extrapolations of improved data to polynomials in *a*^{2}, up to and including *a*^{4}. We note that extrapolations in *a*^{2}*α*_{s}(1/*a*)^{3}, with *α*_{s}(1/*a*) the strong coupling at the lattice scale, are also considered in our final result. The red circles and curves are the same as the blue points, but correspond to SRHO taste improvement for *t* ≥ 0.4 fm and no improvement for smaller *t*. The purple histogram results from fits using the SRHO improvement, and the corresponding central value and error is the purple band. The darker grey circles correspond to results corrected with SRHO in the range 0.4–1.3 fm and with NNLO SXPT for larger *t*. These latter fits serve to estimate the systematic uncertainty of the SRHO improvement. The grey band includes this uncertainty, and the corresponding histogram is shown with grey. Errors are s.e.m.

### Extended Data Fig. 4 Comparison of the continuum extrapolation of \({{\boldsymbol{a}}}_{{\boldsymbol{\mu }}}^{{\boldsymbol{I}}={\rm{0}},{\rm{l}}{\bf{ight}}}\) to those of \({{\boldsymbol{a}}}_{{\boldsymbol{\mu }}}^{{\rm{l}}{\bf{ight}}}\) and \({{\boldsymbol{a}}}_{{\boldsymbol{\mu }}}^{{\bf{disc}}}\).

Top, grey points correspond to our uncorrected results for \(\frac{1}{10}{a}_{\mu }^{{\rm{light}}}\). The red symbols show the same results with our standard SRHO taste improvement. They have a much milder continuum limit that exhibits none of the nonlinear behaviour of the grey points. The red curves show typical examples of illustrative continuum extrapolations of those points. Bottom, grey and red points and curves are the same quantities, but for \({a}_{\mu }^{{\rm{disc}}}\). Combining the results from the two individual continuum extrapolations of \(\frac{1}{10}{a}_{\mu }^{{\rm{light}}}\) and \({a}_{\mu }^{{\rm{disc}}}\), according to equation (6), gives the result with statistical errors illustrated by the red band, with combined statistical and systematic errors indicated by the broader pink band. The blue points correspond to our results for \({a}_{\mu }^{I=0,{\rm{light}}}\) for each of our simulations, and are obtained by combining the two sets of grey points, according to equation (6). As these blue points show, the resulting continuum-limit behaviour of \({a}_{\mu }^{{\rm{light}}}\) is much milder than that of either the uncorrected \({a}_{\mu }^{{\rm{light}}}\) or \({a}_{\mu }^{{\rm{disc}}}\), and shows none of their curvature. This behaviour resembles much more that of the taste-improved red points. Moreover, all of the blue points, including typical continuum extrapolations drawn as blue lines, lie within the bands. This suggests that our taste improvements neither bias the central values of our continuum-extrapolated \({a}_{\mu }^{{\rm{light}}}\) and \({a}_{\mu }^{{\rm{disc}}}\), nor do they lead to an underestimate of uncertainties. Errors are s.e.m.

### Extended Data Fig. 5 Continuum extrapolations of the contributions to *w*_{0}*M*_{Ω}.

From top to bottom: isospin-symmetric, electromagnetic valence–valence, sea–valence and sea–sea components. The results are multiplied by \({10}^{4}/{M}_{\varOmega }^{\ast }\) and the electric derivatives are multiplied by \({e}_{* }^{2}\), where the asterisk denotes physical value. Error bars show statistical errors (s.e.m.). Dashed lines are continuum extrapolations, showing illustrative examples from our several thousand fits. Only the lattice spacing dependence is shown: the data points are moved to the physical light- and strange-quark mass point. This adjustment varies from fit to fit, and the red data points are obtained in an *a*^{2}-linear fit to all ensembles. If in a fit the adjusted points differ considerably from the red points, we show them with grey colour. The final result is obtained from a weighted histogram of the several thousand fits.

## Supplementary information

### Supplementary Information

This file contains Supplementary Information, including Supplementary Figures 1–30 and Supplementary Tables 1–20, and Supplementary References.

## Rights and permissions

## About this article

### Cite this article

Borsanyi, S., Fodor, Z., Guenther, J.N. *et al.* Leading hadronic contribution to the muon magnetic moment from lattice QCD.
*Nature* (2021). https://doi.org/10.1038/s41586-021-03418-1

Received:

Accepted:

Published:

## Further reading

## Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.