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A hybrid lattice-plus-data calculation brings the Standard Model prediction for the muon's magnetic moment to within 0.5σ of experiment.

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2026-08-02 18:59 UTC pith:7CGLELBN

load-bearing objection This is a proceedings summary of BMW/DMZ's hybrid HVP result; the genuinely new 0.5σ claim lives in arXiv:2407.10913, not here, and the data-driven tail carries a common-normalization caveat that the paper treats conservatively but cannot by itself dispel. the 3 major comments →

arxiv 2603.03835 v2 pith:7CGLELBN submitted 2026-03-04 hep-lat hep-phhep-th

BMW/DMZ calculation of the hadronic vacuum polarisation for the muon magnetic moment

Finn M. Stokes , Alessandro Cotellucci , Michel Davier , Zoltan Fodor , Fabian Frech , Davide Giusti , Andrey Yu. Kotov , Laurent Lellouch
show 6 more authors
Bogdan Malaescu Sophie Mutzel Kalman K. Szabo Balint C. Toth Gen Wang Zhiqing Zhang
This is my paper
classification hep-lat hep-phhep-th
keywords muon g-2hadronic vacuum polarisationlattice QCDe+e- data-drivenhybrid calculationEuclidean-time windowsanomalous magnetic momentStandard Model validation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that the hadronic vacuum polarisation contribution to the muon's magnetic moment can be computed at 0.45% precision by combining lattice QCD at short and intermediate distances with electron-positron data for the very long-distance tail. With that hybrid value, the full Standard Model prediction lands only 0.5σ away from the measured value, validating the Standard Model to 0.31 parts per million. If correct, the twenty-year-old 'muon g-2 anomaly' would disappear: the discrepancy would be explained by shortcomings of earlier data-driven evaluations, not by new physics. This matters because it would close a longstanding open question about whether the Standard Model is complete for this observable.

Core claim

On the paper's own terms, the discovery is that the conflict between lattice and data-driven determinations of the hadronic vacuum polarisation is confined to the intermediate Euclidean-time window that emphasizes the rho resonance, while the long-distance tail beyond 2.8 fm — dominated by low-energy states well below the rho — gives mutually consistent results from both methods. Exploiting this, the authors build a hybrid a_mu^HVP that uses the lattice for everything except the tail, replaces that tail with the data-driven value, and obtains a result whose uncertainty is smaller than either method alone. Combined with the other Standard Model contributions, this yields a prediction that dis

What carries the argument

The central object is the Euclidean-time window decomposition of the HVP integral. The integral over the vector-vector correlator C(t) with kinematic kernel K(t) is split into short (<0.4 fm), intermediate (0.4–1.0 fm), and long-distance (1.0–2.8 fm lattice; >2.8 fm tail) windows. The paper's key move is the hybrid construction: in the far tail, where lattice errors grow but the kernel suppresses the rho-peak region where e+e- experiments disagree, it substitutes the data-driven value, adding a doubled systematic to cover variations between experimental treatments. This lets each method contribute where it is strongest and removes the finite-volume and statistical errors that dominate the la

Load-bearing premise

The result stands or falls on the premise that the e+e- data in the long-distance tail (Euclidean times beyond 2.8 fm, energies well below the rho resonance) have no hidden common systematic, so substituting that data for the lattice tail cannot bias the final value.

What would settle it

A re-analysis of the low-energy e+e- data that introduces a common normalisation parameter finds a shift in the tail integral larger than the doubled uncertainty the paper assigns; or a new independent measurement of the two-pion cross section below about 0.6 GeV moves the tail average by more than roughly 10% of the HVP total, which would move the final prediction away from the measured value.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the hybrid value is correct, the Standard Model and the measured muon magnetic moment agree within 0.5σ, removing the need for new physics to explain this observable.
  • The remaining methodological dispute about the HVP is now localised to the rho-peak region; resolving the experimental disagreements there would reconcile the lattice and data-driven approaches entirely.
  • With the tail essentially fixed by data, further improvement of the lattice calculation can focus on the short and intermediate windows, where the continuum limit controls the uncertainty.
  • The 0.31 ppm validation means the full Standard Model stack — QED, electroweak, hadronic light-by-light, and this HVP — is consistent with experiment at the level tested.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same hybrid logic may transfer to other low-energy observables, such as the running of the electromagnetic coupling, wherever lattice and data-driven determinations overlap in a region free of known tensions.
  • A dedicated, independent low-energy measurement of the two-pion cross section below roughly 0.6 GeV would directly test the region the hybrid leans on; a significant shift there would propagate into the final prediction.
  • If the agreement survives, any future hint of new physics in muon g-2 would have to enter through quantities other than the HVP contribution, since that part now matches experiment.
  • Recomputing the tail with a qualitatively different lattice discretisation at higher statistics would test whether the data-driven tail is unbiased or whether the 0.5σ agreement reflects the specific choice of window.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This proceedings contribution from the BMW/DMZ collaboration describes an updated lattice-QCD calculation of the hadronic vacuum polarisation (HVP) contribution to the muon anomalous magnetic moment, combined with a data-driven evaluation of the very-long-distance tail (t > 2.8 fm). The central claim, stated in the abstract, Section 1 and Section 10, is that the hybrid determination reaches 0.45% precision, differs from the experimental measurement by only 0.5σ, and validates the Standard Model to 0.31 ppm. The paper describes the lattice ensembles, the analysis strategy based on AIC-weighted global fits, the Euclidean-time window decomposition, the hybrid tail replacement, the new f_π scale-setting procedure, and the changes relative to the 2020 BMW result. It also discloses several conservative systematics and the blinded nature of the 2024 analysis.

Significance. If the hybrid result is correct, it would be a major step toward closing the long-standing muon g−2 tension: a first-principles lattice HVP, supplemented by data only in a small low-energy tail, would agree with the Fermilab measurement, implying no new physics is needed in this observable. The paper has clear methodological strengths: the analysis was performed blind; scale setting was cross-checked with the independent f_π input after the Ω-based scale setting was identified as potentially contaminated by KΞ states; the hybrid tail systematics are deliberately conservative and overestimated; and the window results agree with independent lattice and data-driven determinations. The main weaknesses are that the actual numerical result and uncertainty budget are not reported in this text, and the hybrid tail replacement relies on an assumption about common systematic errors in e+e− data that is not quantitatively tested. These issues prevent the central claim from being fully evaluated from this manuscript alone.

major comments (3)
  1. [Abstract, §1, §10] The central quantitative claims — a 0.45%-precision hybrid a_mu^HVP, a 0.5σ difference from experiment, and 0.31 ppm Standard Model validation — are stated in the abstract, §1 and §10, but no numerical value, uncertainty budget, or comparison plot is given in this proceedings. Table 1 is the white-paper consensus, not the hybrid result. The reader is referred to arXiv:2407.10913 [1], but a paper whose title announces a 'calculation' should at least give the central value and its main components in a table. As written, the central claim cannot be checked from this text.
  2. [§7 (Hybrid Approach)] The replacement of the t>2.8 fm tail by data-driven results is justified by the statement that this region is dominated by low-energy states and that the rho-region tensions are suppressed. However, the variations listed to estimate the associated uncertainty (pre- vs post-integration averaging, inclusion of partial-coverage experiments, CMD-3 in/out, BaBar vs KLOE) are all internal choices among data sets; they do not constrain a common normalization or radiative-correction offset shared by e+e− experiments. The tau-driven point in Fig. 9 is a useful external check but relies on isospin-breaking corrections. Since the tail contributes roughly 5% of a_mu^HVP (≈35×10^-10), a common 3% offset would shift the hybrid by ≈1×10^-10, which is non-negligible against a claimed total uncertainty of ≈3×10^-10 and could move the 0.5σ statement by about 0.3σ. Doubling the tail uncertainty from the li
  3. [§8 (Scale Setting)] The disclosure that the Ω-based scale setting has a possible uncontrolled contamination from KΞ states, and that the 2024 analysis therefore moved to f_π, is welcome. But the proceedings do not state the resulting f_π scale-setting uncertainty or its contribution to the final 0.45% error budget. Given that scale setting was previously a significant systematic in BMW analyses, this omission is part of the missing numerical support for the headline claim. A summary table with the scale-setting error and its treatment in the hybrid combination should be included, or the relevant numbers from [1] reproduced.
minor comments (5)
  1. [References] References [31] and [38] appear to be the same BMW 2018 paper (Phys. Rev. Lett. 121, 022002) listed twice. One should be removed or the citation differentiated.
  2. [Figures 5-10] Several window labels in the figures and captions use corrupted glyphs (e.g., 'a_{μ,04}^{□10}', 'a_{μ,00}^{□04}'). These should be typeset using the same window notation as in the text.
  3. [§9] The quoted correlated change between the 2020 and 2024 results, 7.6(5.2)×10^-10, is presented without specifying how the correlation between the two analyses was estimated. A sentence describing the correlation model would make the 1.5σ statement reproducible.
  4. [§5] The uncertainty estimate takes the larger of half the 68% CDF width and a quarter of the 95% CDF width. Some PDFs, e.g., Fig. 8, are visibly bimodal. It would help to state explicitly which estimator was used for the final quoted uncertainty and how bimodality is handled.
  5. [Abstract/§1] 'an 0.45% precision' should read 'a 0.45% precision'.

Circularity Check

0 steps flagged

No significant circularity: the hybrid result combines an independently confirmed lattice calculation with external e+e−/tau data, and the 0.5σ agreement is a genuine comparison rather than a fitted output.

full rationale

I walked the derivation chain from Sections 3, 6, 7, and 10. The central quantity, the hybrid a_mu^HVP, is obtained by combining (i) the BMW lattice calculation of the short- and intermediate-distance windows and the 1.0–2.8 fm region, and (ii) a data-driven determination of the t>2.8 fm tail. The lattice input is largely cited to the collaboration's own [1] and [3], but this is not load-bearing circularity: the text notes independent confirmation by Mainz [7] and RBC/UKQCD [6], and describes a blinded analysis ('Our 2024 calculation was performed completely blind, with all HVP data multiplied by unknown random numbers before analysis commenced'). The data-driven tail is based on external experimental inputs (BaBar, KLOE, CMD-3, tau) through dispersive relations; it is not derived from the experimental value of a_mu. The selection of the t>2.8 fm window is justified physically by kernel suppression of the rho-region tensions and by the observed agreement in Fig. 9, but the final comparison 'differs from the experimental measurement by only 0.5σ' is not obtained by fitting to the experimental a_mu. The manuscript also explicitly accounts for procedural variants and doubles the tail uncertainty, stating this 'gives an overestimate of the final uncertainties.' The common-normalization concern raised by the skeptic is a legitimate systematic-risk caveat, not a circularity: it does not make any equation reduce to an input by construction. No step in this paper exhibits the specific reduction required for self-definitional, fitted-input, or self-citation circularity.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The central result rests on the standard lattice QCD machinery of the collaboration's own prior work ([1],[3]) and the data-driven dispersive inputs of [10,17-19]; the new hybrid step adds the assumption that the far-tail region is discrepancy-free. Free parameters are the continuum-fit coefficients/exponent per window and the hand-chosen window boundaries; no new physical entities are postulated.

free parameters (3)
  • Continuum extrapolation coefficients A0, A2, A4, A6 and exponent γ = A_i and γ ∈ {0, 0.5, ..., 2.5} fitted per window; combined via AIC weights
    Eq (3): the central a→0 extrapolation uses these fitted coefficients. The AIC weighting (Eq 5) and the exclusion of coarsest spacings define the systematic; the 1.5σ shift between 2020 and 2024 (Section 9) shows sensitivity to these choices.
  • Euclidean-time window boundaries (0.4 fm, 1.0 fm, 2.8 fm) = not fitted; chosen by hand
    Sections 6-7: boundaries are chosen so short/intermediate/long windows separate the dominant systematic sources. The 2.8 fm hybrid cutoff is the point where the data-driven tail is judged reliable; a different split would change the hybrid composition.
  • Data-driven averaging procedure variants (pre/post integration average, CMD-3 in/out, BaBar vs KLOE) = n/a (treated as systematics)
    Section 7: these procedure choices are varied and folded into an inflated tail uncertainty (factor 2). The central value of the hybrid shifts by an unstated amount between variants; the paper reports only the inflated error.
axioms (6)
  • standard math Time-momentum representation a_mu^HVP = α² ∫ K(t) C(t) dt (Eq 2)
    Bernecker-Meyer [30]; the integral identity is exact up to the numerical truncation of the integral, a standard result the paper builds on.
  • domain assumption Staggered fermion discretization with stout smearing reproduces the continuum HVP at the β values used
    Section 4: tree-level Symanzik + one-link staggered action; taste breaking is assumed controlled by smearing, and the continuum limit (Eq 3) is assumed to remove remaining discretization effects, including the logarithmic short-window terms handled via perturbation theory (Fig 7).
  • domain assumption The continuum extrapolation ansatz O(a² α_s^γ) with AIC model averaging covers the true systematic error
    Section 5, Eq (3): polynomial in a²α_s^γ with γ range and spacing-exclusion choices. If the true scaling is outside the ansatz family, the central value shifts; Section 9's correlated 1.5σ shift between 2020 and 2024 indicates the modelling is still evolving.
  • domain assumption Perturbative QCD matches the lattice short-distance window below 0.3 fm
    Section 6, Fig 7: perturbation theory is used to fix the logarithmic discretization terms in the short-window fit. The displayed agreement is good, but the assumption that residual lattice terms follow the modeled form is load-bearing for the 00-04 window.
  • domain assumption The data-driven dispersive evaluation of the t > 2.8 fm tail is reliable and free of the rho-region tensions
    Section 7: the hybrid replaces the lattice tail with data-driven input because 'the experimental tensions that plague the data-driven approach in the region of the rho peak are significantly suppressed by this kernel'. If the e+e- data in this window carry a common hidden systematic, the hybrid inherits it; the <5% weight limits but does not eliminate the impact.
  • domain assumption Finite-volume effects in the 11 fm box are exponentially suppressed, with residual effects covered by dedicated simulations
    Section 3(b): large-volume DBW2 ensembles at a single lattice spacing are used to correct finite-size effects; the assumption that remaining FVE are negligible in the windowed analysis is standard but not directly proven for each window.

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read the original abstract

For twenty years, a persistent discrepancy between experimental measurements and theoretical calculations of the muon anomalous magnetic moment have provided tantalising hints of new physics. In recent years, improvements to the experimental precision have appeared to make the tension stronger and stronger. However, at the same time, our lattice calculation overturned the theoretical consensus, completely eliminating the tension. I will present the latest results from the Budapest-Marseille-Wuppertal (BMW) and DMZ collaborations, with a hybrid determination of the hadronic vacuum polarisation contribution to a precision of 0.45%

Figures

Figures reproduced from arXiv: 2603.03835 by Alessandro Cotellucci, Andrey Yu. Kotov, Balint C. Toth, Bogdan Malaescu, Davide Giusti, Fabian Frech, Finn M. Stokes, Gen Wang, Kalman K. Szabo, Laurent Lellouch, Michel Davier, Sophie Mutzel, Zhiqing Zhang, Zoltan Fodor.

Figure 1
Figure 1. Figure 1: Comparison of Standard Model predictions for the muon anomalous magnetic moment with its measured value, taken from Ref. [1]. The top panel shows a comparison of the world-average experimental measurement of 𝑎𝜇 [2] with the Standard Model prediction obtained by the BMW collaboration [1], denoted by the red band. The middle panel shows a predictions based on the earlier BMW result from Ref. [3] as well as r… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of lattice values for the leading HVP contribution to the muon magnetic moment. The blue circle is the Standard Model consensus from the 2020 White Paper [9], and the squares are lattice results. The older lattice results [4, 31–37] are consistent with both the Standard Model consensus and the experimental measurement. The four most precise results (from BMW [1, 3], Mainz [7], and RBC/UKQCD [6])… view at source ↗
Figure 3
Figure 3. Figure 3: Main uncertainties and their reduction in the BMW collaboration’s successive lattice calculations of 𝑎 LO-HVP 𝜇 . Their sources are labelled (a-e) 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 orange bars on the left of each group correspond to the 2017 BMW resu… view at source ↗
Figure 4
Figure 4. Figure 4: Spread of our ensembles around the physical point, as defined by the masses of the pseudo￾scalar mesons 𝑀𝑙𝑙 and 𝑀𝑠𝑠. Different colours denote different lattice spacings. The black point denotes the (isospin-symmetric) physical point, with error bars corresponding to the uncertainties from our determination of 𝑀𝑙𝑙 and 𝑀𝑠𝑠 in physical units. Our 2024 calculation was performed completely blind, with all HVP d… view at source ↗
Figure 5
Figure 5. Figure 5: Light-connected window observable 𝑎 light 𝜇,04−10. Top: continuum extrapolations, coloured according to the weight of the fit in our analysis. Lower left: the probability distribution function, which shows a single well-defined peak. The shaded bands correspond to the one- and two-sigma confidence bands. Lower right: we compare our result with others from the literature, both lattice [3, 5, 36, 37, 50–53] … view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of our full intermediate-window results, 𝑎 LO-HVP 𝜇,04−10, with others in the literature. In the top panel, we show lattice results [1, 3, 5, 51–53]. In the lower panel, we show data-driven results [10] that use the measurements of the two-pion spectrum obtained in individual electron-positron annihilation experiments and in 𝜏-decays, as explained in Ref. [10]. on specific regions in 𝑡. It is st… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the short-distance window observables 𝑎 light 𝜇,00−𝑡1 between the lattice results and those obtained from perturbation theory. For ease of comparison, the central value of the lattice results has been subtracted off, and a slight horizontal offset has been applied. approach in the region of the rho peak are significantly suppressed by this kernel. This makes this window one of the most reliab… view at source ↗
Figure 8
Figure 8. Figure 8: Light-connected window observable 𝑎 light 𝜇,00−04. Top: continuum extrapolations, coloured according to the weight of the fit in our analysis. We plot fits with two different kernel functions, denoted by 𝑞 and 𝑞ˆ, which correspond to two different discretisations for the momentum, and also show fits to data either corrected or not corrected by tree-level perturbation theory. Lower left: the probability dis… view at source ↗
Figure 9
Figure 9. Figure 9: A comparison of data-driven results for the long-distance tail of the HVP with two-pion production data restricted to specific experimental data sets, including results from tau decays [1]. The shaded bands represent an average of the data-driven points, and all points are relative to the central value of that average. Left: the entire tail from 2.8 fm to infinity. Right: a comparison with lattice for the … view at source ↗
Figure 10
Figure 10. Figure 10: Light-connected window observable 𝑎 light 𝜇,10−28. Top: continuum extrapolations, coloured accord￾ing to the weight of the fit in our analysis, with no, NNLO XPT and SRHO taste improvements. Bottom: the probability distribution function. With all of these factors taken together, we estimate the correlated difference between the old and new results to be 7.6(5.2) × 10−10. This means the new result is 1.5𝜎 … view at source ↗
Figure 11
Figure 11. Figure 11: Ground state Ω baryon effective mass obtained through a GEVP, along with three excited states, at 𝛽 = 4.1479. The band through the ground state is from a single-exponential fit to the projected ground state propagator. The line through the first excitation shows the mass of the Ω excitation observed at Belle [57]. 0.169 0.170 0.171 0.172 0.173 0.174 0.175 w0 / fm This work (fπ) This work (MΩ) FLAG ’24 ETM… view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the gradient-flow scale 𝑤0 using either 𝑀Ω or 𝑓𝜋 for scale setting. The lower panel shows earlier determinations [3, 61–65] and the grey band shows the latest FLAG average [66]. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗

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Reference graph

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