We present a new measurement of the positive muon magnetic anomaly, a µ ≡ (gµ − 2)/2, from the Fermilab Muon g −2 Experiment based on data collected in 2019 and 2020. We have analyzed more than four times the number of positrons from muon decay than in our previous result from 2018 data. The systematic error is reduced by more than a factor of two due to better running conditions, a more stable beam, and improved knowledge of the magnetic field weighted by the muon distribution, ω′ p , and of the anomalous precession frequency corrected for beam dynamics effects, ωa. From the ratio ωa/ω ′ p , together with precisely determined external parameters, we determine a µ = 116 592 057(25) × 10 −11 (0.21 ppm). Combining this result with our previous result from the 2018 data, we obtain a µ (FNAL) = 116 592 055(24) × 10 −11 (0.20 ppm). The new experimental world average is aµ(Exp) = 116 592 059(22) × 10 −11 (0.19 ppm), which represents a factor of two improvement in precision.

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This work presents a complete re-evaluation of the hadronic vacuum polarisation contributions to the anomalous magnetic moment of the muon, a had, VP µ and the hadronic contributions to the effective QED coupling at the mass of the Z boson, ∆α had (M 2 Z ), from the combination of e + e − → hadrons cross section data. Focus has been placed on the development of a new data combination method, which fully incorporates all correlated statistical and systematic uncertainties in a bias free approach. All available e + e − → hadrons cross section data have been analysed and included, where the new data compilation has yielded the full hadronic R-ratio and its covariance matrix in the energy range m π ≤ √ s ≤ 11.2 GeV. Using these combined data and perturbative QCD above that range results in estimates of the hadronic vacuum polarisation contributions to g − 2 of the muon of a had, LO VP µ = (693.26 ± 2.46) × 10 −10 and a had, NLO VP µ = (−9.82 ± 0.04) × 10 −10 . The new estimate for the Standard Model prediction is found to be a SM µ = (11 659 182.04±3.56)×10 −10 , which is 3.7σ below the current experimental measurement. The prediction for the five-flavour hadronic contribution to the QED coupling at the Z boson mass is ∆α (5) had (M 2 Z ) = (276.11±1.11)×10 −4 , resulting in α −1 (M 2 Z ) = 128.946±0.015. Detailed comparisons with results from similar related works are given. arXiv:1802.02995v2 [hep-ph] 13 Jul 2018 4 Conclusions and future prospects for the muon g − 2 39

Following updates in the compilation of e þ e − → hadrons data, this work presents reevaluations of the hadronic vacuum polarization contributions to the anomalous magnetic moments of the electron (a e), muon (a μ) and tau lepton (a τ), to the ground-state hyperfine splitting of muonium and also updates the hadronic contributions to the running of the QED coupling at the mass scale of the Z boson, αðM 2 Z Þ. Combining the results for the hadronic vacuum polarization contributions with recent updates for the hadronic light-bylight corrections, the electromagnetic and the weak contributions, the deviation between the measured value of a μ and its Standard Model prediction amounts to Δa μ ¼ ð28.02 AE 7.37Þ × 10 −10 , corresponding to a muon g − 2 discrepancy of 3.8σ.