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# Leading hadronic contribution to the muon magnetic moment from lattice QCD

## 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 predictions1 exhibit disagreement with measurements2 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 hadrons3,4,5,6. 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.

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## 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 w0, can be downloaded from https://arxiv.org/abs/1203.4469.

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## 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

Authors

### 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

Correspondence to Z. Fodor.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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 aμ, $${[{{\boldsymbol{a}}}_{{\boldsymbol{\mu }}}^{{\rm{l}}{\bf{ight}}}]}_{{\bf{0}}}$$.

The bounds are computed using the lattice correlator below a time separation of tc and an analytical formula describing the large-time behaviour above tc.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 Mπ = 121 MeV Goldstone pion mass. We also carried out another simulation with Mπ = 104 MeV mass. From these two, we interpolate to Mπ = 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 t1 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 a2, up to and including a4. We note that extrapolations in a2αs(1/a)3, with αs(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 w0MΩ.

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 a2-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.

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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

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