The muon is a short-lived elementary particle with spin 1/2 and a mass 207 times larger than that of the electron. Both particles create a magnetic field around them, characterized by a magnetic dipole moment. This moment is proportional to the spin and charge of the particle and inversely proportional to twice its mass. Dirac’s relativistic quantum mechanics predicts that the constant of proportionality, gμ, known as the Landé factor, is precisely 2. Relativistic quantum field theory introduces further small corrections induced not only by all particles and interactions of the standard model but also potentially by yet undiscovered ones. Because muons are more massive than electrons, quantum corrections associated with heavy particles are generically much larger for the former than for the latter5. This increased sensitivity to the effects of possible unknown particles is the reason for the present focus on the muon. The corrections to gμ are commonly called the anomalous magnetic moment and are quantified as aμ = (gμ − 2)/2.
When calculating aμ, the uncertainty comes almost exclusively from the strong interaction, described in the standard model by QCD. In particular, the dominant source of uncertainty comes from hadronic vacuum polarization (HVP) at leading order in the fine-structure constant (LO-HVP). More generally, HVP induces a modification in the propagation of a virtual photon in the vacuum, caused by the strong interaction.
Here we present a calculation of this LO-HVP contribution to aμ (\({a}_{\mu }^{\text{LO-HVP}}\)) with unprecedented accuracy. To that end, we apply numerical lattice quantum field theory techniques that allow QCD predictions to be made in the highly nonlinear regime that is relevant here. Mathematically, QCD is a generalized version of quantum electrodynamics (QED). However, QCD predicts physical phenomena that are very different from those described by QED. The Euclidean Lagrangian for a quark of mass m and charge q (in units of the positron charge, e), subject to strong and electromagnetic interactions, can be written as \({\mathcal{L}}=1/(4{e}^{2}){F}_{\mu \nu }{F}_{\mu \nu }+1/(2{g}^{2}){\rm{Tr}}{G}_{\mu \nu }{G}_{\mu \nu }+\bar{\psi }[{\gamma }_{\mu }({\partial }_{\mu }+{\rm{i}}q{A}_{\mu }+{\rm{i}}{G}_{\mu })+m]\psi \), in which Fμν = ∂μAν − ∂νAμ, Gμν = ∂μGν − ∂νGμ + i[Gμ, Gν] and g is the QCD coupling constant. The fermionic quark fields ψ have an extra ‘colour’ index in QCD, which runs from 1 to 3. Different ‘flavours’ of quarks are represented by independent fermionic fields, with different masses and charges. In QED, the gauge potential Aμ is a real-valued field, whereas in QCD, Gμ is a 3 × 3 traceless Hermitian matrix field acting in ‘colour’ space. In the present work, we include both QCD and QED as well as four nondegenerate quark flavours (up, down, strange and charm) in a fully dynamical, staggered-fermion formulation. We also consider the tiny contribution of the bottom quark. Its error is subdominant and we repeat the treatment of our earlier analysis1.
To calculate the LO-HVP contribution to aμ, we start with the zero-three-momentum, two-point function of the quark electromagnetic current in Euclidean time t (ref. 6). In this so-called time-momentum representation, it is given by
$$G(t)=-\frac{1}{3{e}^{2}}\sum _{\mu =1,2,3}\int {{\rm{d}}}^{3}x\langle \,{J}_{\mu }(\overrightarrow{x},t){J}_{\mu }(0)\rangle ,$$
(1)
in which Jμ is the quark electromagnetic current with \({J}_{\mu }/e\,=\) \(\frac{2}{3}\bar{{\rm{u}}}{\gamma }_{\mu }{\rm{u}}-\frac{1}{3}\bar{{\rm{d}}}{\gamma }_{\mu }{\rm{d}}-\frac{1}{3}\bar{{\rm{s}}}{\gamma }_{\mu }{\rm{s}}+\frac{2}{3}\bar{{\rm{c}}}{\gamma }_{\mu }{\rm{c}}\). u, d, s and c are the up, down, strange and charm quark fields, respectively. The angle brackets stand for the QCD + QED expectation value to order e2. It is convenient to decompose G(t) into light (u and d), strange, charm and disconnected components, which have very different statistical and systematic uncertainties. Performing a weighted integral of the one-photon-irreducible part, G1γI(t), of G(t) from t = 0 to infinity yields the LO-HVP contribution to aμ (ref. 6). The weight is a known kinematic function, K(tmμ) (refs. 6,7,8,9). Thus:
$${a}_{\mu }^{\text{LO-HVP}}={\alpha }^{2}{\int }_{0}^{\infty }{\rm{d}}tK(t{m}_{\mu }){G}_{1\gamma {\rm{I}}}(t),$$
(2)
in which α is the fine-structure constant at vanishing recoil and mμ is the mass of the muon.
Reducing the uncertainty in the calculation of \({a}_{\mu }^{\text{LO-HVP}}\) to below half a percent is a notable challenge. In particular, several contributions to this uncertainty must be controlled. They are: (1) statistical uncertainties; (2) those associated with the finite spatial size L and time T of the lattice; (3) with the extrapolation to the continuum limit; (4) with fixing the five parameters of four-flavour QCD; (5) with isospin symmetry breaking. The progress made in our successive lattice calculations of \({a}_{\mu }^{\text{LO-HVP}}\) is illustrated in Fig. 1, in which those contributions to the uncertainty are shown. In the present work, we focus on reducing the two largest ones in our 2020 calculation, which are (3) and (2). We discuss all of these contributions ((1)–(5)) in detail now.
Fig. 1: Main uncertainties and their reduction in our successive lattice calculations of \({{\boldsymbol{a}}}_{{\boldsymbol{\mu }}}^{{\bf{LO-HVP}}}\).
The alternative text for this image may have been generated using AI.
Their sources are labelled (1)–(5) in the text and are given a short descriptive title below the bars in the plot. Their approximate size relative to the total LO-HVP contribution obtained in the present work is also shown. The blue bars on the left of each group correspond to our 2017 result24, the pink bars to our 2020 findings1 and the orange bars to the work presented here. The isospin-breaking uncertainty (5) in this work is slightly larger than in 2020 owing to changes in the way we set the physical scale. We moved from using the Ω− baryon to the pion decay rate, which reduced other uncertainties but increased the isospin-breaking uncertainty. Note: the statistical error (1) refers to that of the isospin-symmetric contribution in finite volume. The finite-size (2) and isospin-breaking (5) errors also contain statistical components of 0.08% and 0.16%, respectively.
(1) Statistical uncertainties in the light-quark-connected and disconnected contributions to the correlation function of equation (1), associated with the stochastic evaluation of the QCD and QED path integrals, increase exponentially at large Euclidean times t. As well as the many improvements made in ref. 1, to reduce those uncertainties further, we use mock analyses to determine which ensembles required more statistics. In particular, we increase the statistics on the lattices that have the smallest lattice spacings and are critical for controlling the necessary continuum extrapolations. Moreover, to control the statistical uncertainties at large t, we replace the lattice calculation of the contribution to \({a}_{\mu }^{\text{LO-HVP}}\) from G(t) above t ≥ 2.8 fm by a state-of-the-art, data-driven determination, by means of the HVPTools set-up10,11,12,13. (Such a combination was originally proposed in ref. 14. There, however, lattice results were replaced by e+e− → hadrons data above a much earlier Euclidean time, t ≥ 1 fm.) Here and in the rest of the paper, the expression ‘data-driven’ refers to predictions based on measurements of the hadron spectrum in e+e− annihilation and τ-decay experiments. Before combining the two results, we verify that the lattice and the data-driven determinations of part of this long-distance ‘tail’ contribution agree within errors. We compute this tail contribution using the most precise measurements of the two-pion spectrum by BaBar15,16, KLOE17,18,19,20 and CMD-3 (ref. 2), as well as the one obtained from hadronic τ decays21,22. These experiments almost fully cover the relevant energy range. For estimating the uncertainty of the tail observable, we also use other experiments with partial coverage. The two-pion spectra of these experiments are supplemented by the contributions from all of the other hadronic final states, as described in ref. 23. The tail contribution is dominated by centre-of-mass energies below the ρ-meson peak, a region in which all of the measurements agree very well. The tail only accounts for less than 5% of our final, lattice-dominated result for \({a}_{\mu }^{\text{LO-HVP}}\). The Supplementary Information describes our determination of this contribution and further justifies its use in our calculation.
(2) Finite L and T corrections gave the largest contribution to the error in 2017 (ref. 24). Even in our 2020 calculation1, it was still a substantial source of uncertainty. Here our determination of the tail contribution using a data-driven approach reduces those corrections by a factor of about two and the associated uncertainties by roughly three. We compute those corrections using the dedicated simulations of ref. 1, supplemented by next-to-next-to-leading order chiral perturbation theory for distances beyond 11 fm (refs. 1,25). Those results are checked against nonperturbative analytical approaches to finite-volume corrections26,27,28,29,30 that we complement with experimental π+π− cross-section data below 1.3 GeV. Details are given in the Supplementary Information.
(3) The continuum extrapolation of the isovector contribution to \({a}_{\mu }^{\text{LO-HVP}}\) was the largest source of uncertainty in our 2020 computation1 and we have dedicated substantial resources to further control it. The uncertainties were mainly because of long-distance, taste-breaking effects that are present in staggered-fermion computations. Here we add a new, finer lattice spacing. The corresponding simulations have a numerical cost close to that required for the full 2020 computation. In ref. 1 the smallest lattice spacing was 0.064 fm. The new lattice spacing is 0.048 fm. Because the leading discretization effects are proportional to the square of the lattice spacing, results at this new lattice spacing have cut-off effects reduced by a factor of nearly two. We further account for the fact that different t regions in G(t) have different cut-off effects by dividing the integral of equation (2) into four t intervals delimited by sigmoid functions. Such intervals or ‘windows’ were first proposed in ref. 14. The first window corresponds to the Euclidean-time interval 0.0 to 0.4 fm, known as the short-distance window14,31 and denoted \({a}_{\mu ,00-04}^{\text{LO-HVP}}\) here. We use three more intervals between 0.4 and 2.8 fm (separated at 2.0 and 2.4 fm) because this choice yields a reduced uncertainty on the final result for \({a}_{\mu }^{\text{LO-HVP}}\). We carry out the continuum extrapolation in those windows separately. We then add the individual extrapolated results to obtain the contribution to \({a}_{\mu }^{\text{LO-HVP}}\) from the Euclidean-time interval from 0 to 2.8 fm, taking correlations into account. The uncertainty on the light-connected contribution is decreased by the new ensembles by 37% and by using the data-driven approach to compute the tail by an extra 22%. The whole procedure is detailed in the Supplementary Information.
(4) We improve the determination of the physical point, which is now based on a very precise computation of the muonic decay rate of the charged pion. As a cross-check, we also perform the determination using the mass of the Ω− baryon as input and find good agreement between the two approaches. The uncertainty associated with the physical point determination was already small in ref. 1 and is even smaller here. For details, see the Supplementary Information.
(5) The uncertainties on the isospin-symmetry-breaking contributions obtained in ref. 1 were already sufficiently small to reach the precision sought here. Our error on this contribution is now slightly increased: the isospin-breaking error on the pion decay rate is larger than it was on the Ω− baryon. Also we perform a variety of cross-checks that confirm our earlier results on the isospin-breaking contributions. Our present uncertainty details are given in the Supplementary Information.
By far the largest contributions to the various windows considered in this work come from connected light-quark diagrams. We focus on these here and discuss the other contributions in the Supplementary Information.
For the connected contribution of the light u and d quarks to the intermediate-distance (ID) window, we find \({a}_{\mu ,04-10}^{\text{LO-HVP,light}}\,=\)\(206.92(37)(34)[50]\times 1{0}^{-10}\), in which the first and second numbers in parentheses refer to the statistical and systematic uncertainties, respectively, and the number in square brackets is their quadrature sum, the total uncertainty. As shown in Fig. 2a, our result agrees with eight other lattice calculations of this quantity1,32,33,34,35,36,37,38, including our previous determination, within less than one standard deviation.
Fig. 2: Comparison of our intermediate-window results with others in the literature.
The alternative text for this image may have been generated using AI.
a, Light contribution to the ID window, \({a}_{\mu ,04-10}^{\text{LO-HVP,light}}\). Our result is the orange square and the pink squares correspond to other lattice computations: Fermilab Lattice/HPQCD/MILC ’24 (ref. 45), RBC/UKQCD ’23 (ref. 38), ETM ’22 (ref. 36), Mainz/CLS ’22 (ref. 35), Aubin et al. ’22 (ref. 34), χQCD ’22 (ref. 33), Lehner and Meyer ’20 (ref. 32) and our previous result BMW ’20 (ref. 1). The blue circles denote data-driven determinations of Benton et al. ’23 (ref. 39) and BMW ’20 (ref. 1). These two results are based on the KNT19 data compilation40,41. b, Full ID window, \({a}_{\mu ,04-10}^{\text{LO-HVP}}\). Here, in the data-driven case, we show results23 that use the measurements of the two-pion spectrum obtained in individual electron–positron annihilation experiments and in τ decays, as explained in ref. 23. The error bars correspond to the standard error of the mean.
On the other hand, our new result for \({a}_{\mu ,04-10}^{\text{LO-HVP,light}}\) differs from the data-driven one presented in ref. 1 by 4.3σ. This number was obtained by using the total result \({a}_{\mu ,04-10}^{\text{LO-HVP}}\) from the data-driven approach and subtracting all but the light-connected contributions measured in our 2020 lattice simulations. There is another published result using the data-driven approach by Benton et al.39. These two results for the light-connected ID window are the only data-driven ones published. They are both based on the KNT data compilation40,41 that does not include the more recent CMD-3 measurement nor the ones from τ decays. Their difference with our new result, as shown in Fig. 2a, reinforces the disagreement between the lattice and data-driven determinations found in ref. 1, which was a first strong indication that the lattice1 and reference predictions for \({a}_{\mu }^{\text{LO-HVP}}\) (ref. 31) could not both be correct.
Note that the exact value of \({a}_{\mu ,04-10}^{\text{LO-HVP,light}}\) depends on the scheme used to define the isospin-symmetric limit of QCD. Our scheme, originally defined in ref. 1, is specified in the Supplementary Information. In ref. 38, it is shown that the difference between the value of \({a}_{\mu ,04-10}^{\text{LO-HVP,light}}\) obtained in the RBC/UKQCD scheme and in our scheme is approximately 0.10(24) × 10−10, smaller than even our present uncertainties. The differences with other schemes used by the other collaborations are probably on the same level. However, we emphasize that this scheme dependence in no way affects our final result for \({a}_{\mu }^{\text{LO-HVP}}\) nor for the full value of \({a}_{\mu ,04-10}^{\text{LO-HVP}}\) that includes all flavour, isospin-breaking contributions. Both are unambiguous physical quantities.
In Fig. 2b, we show a comparison of our result for the full ID window contribution, \({a}_{\mu ,04-10}^{\text{LO-HVP}}=236.29(41)(39)[57]\), with the five other lattice determinations of that quantity. Here the results do not depend on any scheme choice and agreement is still excellent. Also plotted are the individual data-driven results23 obtained using the same datasets as for computing the central value of the tail. Those results show notable tensions that forbid an overall comparison between the lattice and data-driven approaches. However, important progress is being made on understanding the sources of those differences and we expect that the situation on the data-driven side will be clarified soon. The differences may be partly because of the treatment of radiative corrections, as explained in refs. 23,42. Although the difference of our lattice result with that obtained using KLOE’s measurement17,18,19,20 is 6.2σ, it reduces to 3.5σ for the BaBar measurement15,16 and even to 1.3σ for the one by CMD-3 (ref. 2). Compared with the determination obtained through τ decays21,22, the difference is 2.7σ. With an alternative evaluation of the τ data43, the difference is even smaller. These numbers illustrate the known discrepancies between measurements at energies around the ρ-meson peak. Note that these contributions are highly suppressed in the tail observable. Nevertheless, we take into account these discrepancies by performing the analysis of the tail with and without the most extreme experiments. The associated uncertainty is an order of magnitude below our final error on \({a}_{\mu }^{\text{LO-HVP}}\). Details can be found in the Supplementary Information.
Our result for the light-connected contribution to the short-distance window, \({a}_{\mu ,00-04}^{\text{LO-HVP,light}}=47.85(5)(13)[14]\times 1{0}^{-10}\), is in excellent agreement with five other lattice computations of this quantity36,38,44,45,46. We also consider the window observable proposed in ref. 34, from 1.5 to 1.9 fm, and we obtain \({a}_{\mu ,15-19}^{\text{LO-HVP,light}}=97.57(1.76)(1.17)[2.11]\times 1{0}^{-10}\). Again we find a good agreement with the other two computations of this quantity34,37. A more detailed comparison of our results for the above windows is provided in the Supplementary information.
Now, summing the connected-light and disconnected contributions obtained in our four chosen Euclidean-time intervals and combining them with all of the other required contributions, including the data-driven tail, we obtain \({a}_{\mu }^{\text{LO-HVP}}=715.1(2.5)(2.3)[3.4]\times 1{0}^{-10}\), as detailed in the Supplementary Information. This result agrees with our earlier 2017 and 2020 determinations but reduces uncertainties by a factor of 5.5 compared with the former and of 1.6 to the latter. The difference between our result and the 2020 result is 7.6 × 10−10, with an uncertainty of 5.2 × 10−10, indicating that the new result is 1.5σ higher. To obtain that result, we assume zero correlation among some of the systematics. When assuming full correlation, the uncertainty becomes 4.5 × 10−10 and, in this case, the new result is 1.7σ higher.
Adding our determination of \({a}_{\mu }^{\text{LO-HVP}}\) to the other standard-model contributions compiled in ref. 3 yields aμ = 11,659,205.2(3.6) × 10−10. In Fig. 3, we compare this result with the world average of the direct measurements of the magnetic moment of the muon4. Our prediction differs from that measurement by −0.5σ. Also given are the Muon g − 2 Theory Initiative combinations from the years 2020 (ref. 31) and 2025 (ref. 3), in which the \({a}_{\mu }^{\text{LO-HVP}}\) contribution was obtained only from the data-driven and only from the lattice approach, respectively. As well as these combinations, we also provide individual results in both approaches. As the figure shows, some of the data-driven results are in serious tension both with our and the lattice-only estimates. Our \({a}_{\mu }^{\text{LO-HVP}}\) is in good agreement with the latest Theory Initiative combination and our uncertainty is a factor of 1.8 smaller.
Fig. 3: Comparison of standard-model predictions for the muon anomalous magnetic moment with its measured value.
The alternative text for this image may have been generated using AI.
Top, world-average measurement of aμ (ref. 4) and the standard-model prediction of this work. The latter is denoted by the orange band and is obtained by adding the value of \({a}_{\mu }^{\text{LO-HVP}}\) computed here to the results for all of the other contributions summarized in ref. 3. Middle, predictions using recent lattice computations for \({a}_{\mu }^{\text{LO-HVP}}\), RBC/UKQCD (refs. 14,38,51), Mainz/CLS52 and our previous computation1. The Muon g − 2 Theory Initiative combination from 2025 (ref. 3), which is obtained using lattice results for \({a}_{\mu }^{\text{LO-HVP}}\), is labelled ‘White paper ’25’. Bottom, predictions using the data-driven approach for \({a}_{\mu }^{\text{LO-HVP}}\) including the most precise measurements of the two-pion spectrum in electron–positron annihilation and τ-decay experiments23. These correspond to BaBar15,16, KLOE17,18,19,20 and CMD-3 (ref. 2) for e+e− annihilation and Tau for τ decays21,22. The earlier Theory Initiative combination from 2020 (ref. 31), which is obtained using the data-driven results, is labelled ‘White paper ’20’. Note, all standard-model predictions include non-HVP contributions from ‘White paper ’25’, except for ‘White paper ’20’. The error bars are the standard error of the mean.
In the near future, we expect more data for the e+e− → π+π− cross-section47. Beyond consolidating our present understanding23,42 of the tensions in the measurements of that cross-section, these new data should improve the data-driven determination of \({a}_{\mu }^{\text{LO-HVP}}\). Also, the possibility of directly measuring HVP in the space-like region is being investigated by the MUonE collaboration48. Finally, combinations of lattice and data-driven results, beyond the simple one presented here, ought to be pursued, following, for example, the methods put forward in ref. 49. Investigations along all of those lines are underway.
The precise measurement and standard-model prediction for the muon anomalous magnetic moment reflect substantial scientific progress. Experimentally, Fermilab’s ‘Muon g − 2’ collaboration has measured aμ to 0.127 ppm (ref. 4). Furthermore, there is the ‘Muon g − 2/EDM’ experiment under development at KEK’s J-PARC50 to measure this quantity using a completely new and independent experimental approach. On the theoretical side, physicists from around the world have performed complex calculations (see, for example, ref. 3), some based on further precise measurements, incorporating all aspects of the standard model and many quantum field theory refinements. It is notable that the electromagnetic, electroweak and strong interactions, which require very different computational tools, can be combined into a single calculation with such precision. The result for \({a}_{\mu }^{\text{LO-HVP}}\) presented here, combined with other contributions to aμ summarized in ref. 3, provides a standard-model prediction with a precision of 0.31 ppm. At such a level of precision, the agreement found between experiment and theory, to within less than one standard deviation, is a great success for the standard model and, from a broader perspective, for renormalized quantum field theory.