Muon Anomalous Magnetic Moment

The final Fermilab measurement makes the experimental muon anomaly exceptionally precise, but it does not by itself establish a discrepancy with the Standard Model. At this cutoff, lattice-based Standard Model evaluations agree with experiment, while dispersive evaluations based on incompatible hadronic cross-section data do not support one stable comparison. The live issue is therefore the reconciliation of hadronic vacuum polarization (HVP), not a method-independent signal of new physics.

Evidence cutoff. 11 August 2026. Reassess by 11 November 2026 or when a new HVP combination, independent lattice average, or official muon measurement appears.

Required background. Lepton magnetic moments defines $a_\mu$ and its perturbative decomposition; vacuum polarization and the running charge supplies the correlator and dispersion structure behind HVP.

Helpful background. Euclidean correlators and spectral information explains the lattice-to-spectral interface; correlated uncertainty propagation is needed to combine multi-stage calculations; subtracted dispersion relations fixes what experimental cross sections determine.

The comparison being tested

For a declared Standard Model evaluation $S$, the relevant residual is

$$\Delta a_\mu(S)=a_\mu^{\mathrm{exp}}-a_\mu^{\mathrm{SM}}(S).$$

The scope is the inclusive magnetic anomaly of a positive muon in the on-shell convention, with QED, electroweak, HVP, and hadronic light-by-light contributions included. The assessment compares published experiment and Standard Model evaluations; it does not fit new-physics models or produce a new HVP average.

HVP is the decisive branch point. A dispersive evaluation integrates measured $e^+e^-\to\text{hadrons}$ cross sections with a known kernel. A lattice evaluation computes the corresponding Euclidean current correlator and controls continuum, volume, quark-mass, scale-setting, and long-distance errors. These are meaningfully different routes, but lattice calculations may share ensembles, scale inputs, radiative corrections, and analysis ideas, while dispersive evaluations share experiments and radiative-correction pipelines.

Evidence matrix

Source and method	Relation to the bounded claim	Independence	Result and stated uncertainty	Main limitation
Fermilab Muon $g-2$, 2025, storage-ring precession, Runs 1–6	fixes the experimental side	final E989 dataset; shares the storage-ring method and some systematics across run periods	$a_\mu=116592070.5(14.8)\times10^{-11}$, 127 ppb	not a Standard Model test until paired with a declared theory evaluation
E821 plus final E989 world average in Fermilab Muon $g-2$, 2025	supports experimental precision and cross-facility consistency	two facilities, but similar magnetic-storage-ring observable and external inputs	$116592071.5(14.5)\times10^{-11}$, 124 ppb	the E989 result dominates the weight; this is not two equally precise tests
Muon $g-2$ Theory Initiative, 2025, lattice-HVP Standard Model synthesis	supplies the declared 2025 evaluation for comparison with experiment	synthesis of several calculations, with correlations and common inputs estimated rather than absent	$a_\mu^{\mathrm{SM}}=116592033(62)\times10^{-11}$; paired here with the final world average, $\Delta a_\mu=39(64)\times10^{-11}$, about $0.6\sigma$	the larger HVP uncertainty reflects unresolved consistency and correlation questions
BMW Collaboration, 2026, lattice plus data-assisted HVP windows	independently qualifies the comparison and localizes disagreement	distinct lattice analysis, but overlaps conceptually with lattice inputs considered by the Initiative	leading-order HVP $715.1(3.4)\times10^{-10}$; derived Standard Model value agrees with experiment within about $0.5\sigma$	a hybrid window construction is not wholly independent of experimental spectral data
CMD-3, 2024, $e^+e^-\to\pi^+\pi^-$ cross section	evidence against treating the older dispersive input set as internally settled	new detector and analysis, but measures the same dominant channel as KLOE and BaBar	a higher two-pion contribution than the older generally lower-valued datasets, which already had internal tensions	the source does not alone determine the full HVP integral; discrepancies may reside in normalization or radiative treatment
Muon $g-2$ Theory Initiative, 2020, predominantly data-driven HVP	historical context for the former headline discrepancy	predates final E989 and the present lattice/data conflict	supplied the lower-uncertainty Standard Model value used for the widely quoted 2021 discrepancy	superseded as a current consensus comparison by new experimental and HVP information

Assessment and dependence structure

The final experimental result is internally consistent across E989 run periods and compatible with BNL E821. The 2025 Theory Initiative’s declared lattice-HVP evaluation and the 2026 BMW calculation both place the Standard Model near the experimental value. Their agreement is important, but it is not fully independent: both belong to the same broad Euclidean-HVP program, and the Initiative synthesis incorporates lattice results with overlapping ensembles and scale-setting knowledge.

The dispersive route has a different dominant evidence base and therefore supplies the most valuable methodological cross-check. It is also the route with the clearest unresolved internal conflict. CMD-3’s two-pion cross section substantially increases the incompatibility among inputs, especially relative to the older generally lower-valued KLOE and BaBar data, which were themselves not tension-free. A small-error dispersive average that simply combines those inputs would obscure that conflict. The 2025 Initiative accordingly did not promote such an average to its reference Standard Model value.

This changes the scientific question. The evidence no longer supports the unqualified statement “the muon anomaly disagrees with the Standard Model.” It supports two narrower statements: the experiment is precise, and the HVP determinations need a cross-method explanation that survives correlations and radiative-systematics checks.

What this evidence does not establish

• Agreement of a lattice-based evaluation with experiment does not prove that every lattice systematic is below its stated error.
• Disagreement among hadronic cross-section datasets is not evidence for a muon-specific new interaction.
• The older data-driven discrepancy remains a legitimate historical comparison, but not a cutoff-independent significance.
• The calculation constrains combinations of possible new interactions; it does not identify a model without other observables.

What would change the assessment

The conclusion would strengthen toward a robust Standard Model discrepancy if independently reproduced lattice calculations and a reconciled dispersive program converged on the same lower HVP value while experimental central values remained stable. It would strengthen toward consistency if the cross-section discrepancy were resolved in favor of the larger HVP contribution and independent lattice calculations retained their present range.

The most discriminating near-term evidence is therefore: corrected and cross-checked $e^+e^-$ data with released covariance and radiative treatment; lattice results with transparent ensemble and scale correlations; window-by-window comparisons using common kernels; and the experimentally different J-PARC ultracold-muon program.

Source selection

The finite set includes the final E989 result, the current Theory Initiative synthesis, the latest high-precision lattice/window determination, the cross-section result that drives the dispersive conflict, and the 2020 synthesis needed to interpret the historical headline. Sources were screened for a published numerical result and enough method information to identify dominant dependencies. New-physics scans, conference-only projections, and calculations that merely reuse one of these central values were excluded from the assessment.

Related research pages

• EFT inference, power counting, and truncation for translating a residual into controlled operator constraints.
• Euclidean lattice fields and continuum extrapolation for the HVP error decomposition.
• Amplitudes and precision scattering for the broader precision-prediction culture.

References

• Aoyama, Tatsumi, et al. “The Anomalous Magnetic Moment of the Muon in the Standard Model.” Physics Reports 887 (2020): 1–166. DOI.
• Boccaletti, A., et al. “Hybrid Calculation of Hadronic Vacuum Polarization in Muon $g-2$ to 0.48%.” Nature 653 (2026): 373–377. DOI.
• CMD-3 Collaboration. “Measurement of the Pion Form Factor with CMD-3 Detector and Its Implication to the Hadronic Contribution to Muon $(g-2)$.” Physical Review Letters 132 (2024): 231903. DOI.
• Muon $g-2$ Collaboration. “Measurement of the Positive Muon Anomalous Magnetic Moment to 127 ppb.” Physical Review Letters 135 (2025): 101802. DOI; arXiv.
• Muon $g-2$ Theory Initiative. “The Anomalous Magnetic Moment of the Muon in the Standard Model: An Update.” Physics Reports 1143 (2025): 1–158. DOI; arXiv.