We present the final report from a series of precision measurements of the muon anomalous magnetic moment, a µ = (g − 2)/2. The details of the experimental method, apparatus, data taking, and analysis are summarized. Data obtained at Brookhaven National Laboratory, using nearly equal samples of positive and negative muons, were used to deduce a µ (Expt) = 11 659 208.0(5.4)(3.3) × 10 −10 , where the statistical and systematic uncertainties are given, respectively. The combined uncertainty of 0.54 ppm represents a 14-fold improvement compared to previous measurements at CERN. The standard model value for a µ includes contributions from virtual QED, weak, and hadronic processes. While the QED processes account for most of the anomaly, the largest theoretical uncertainty, ≈ 0.55 ppm, is associated with first-order hadronic vacuum polarization. Present standard model evaluations, based on e + e − hadronic cross sections, lie 2.2 -2.7 standard deviations below the experimental result.

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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The anomalous magnetic moment of the negative muon has been measured to a precision of 0.7 ppm (ppm) at the Brookhaven Alternating Gradient Synchrotron. This result is based on data collected in 2001, and is over an order of magnitude more precise than the previous measurement for the negative muon. The result a(mu(-))=11 659 214(8)(3) x 10(-10) (0.7 ppm), where the first uncertainty is statistical and the second is systematic, is consistent with previous measurements of the anomaly for the positive and the negative muon. The average of the measurements of the muon anomaly is a(mu)(exp)=11 659 208(6) x 10(-10) (0.5 ppm).

A precise measurement of the anomalous g value, aµ = (g − 2)/2, for the positive muon has been made at the Brookhaven Alternating Gradient Synchrotron. The result a µ + = 11 659 202(14)(6) × 10 −10 (1.3 ppm) is in good agreement with previous measurements and has an error one third that of the combined previous data. The current theoretical value from the standard model is aµ(SM)= 11 659 159.6(6.7) × 10 −10 (0.57 ppm) and aµ(exp)−aµ(SM) = 43(16) × 10 −10 in which aµ(exp) is the world average experimental value.PACS number: 14.60.Ef 13.40.EmPrecise measurement of the anomalous g value, a µ = (g−2)/2, of the muon provides a sensitive test of the standard model of particle physics and new information on speculative theories beyond it. Compared to the electron, the muon g value is more sensitive to standard model extensions, typically by a factor of (m µ /m e ) 2 . In this Letter we report a measurement of a µ for the positive muon from Brookhaven AGS experiment 821, based on data collected in 1999.The principle of the experiment, previous results, and many experimental details have been given in earlier publications [1,2]. Briefly, highly polarized µ + of 3.09 GeV/c from a secondary beamline are injected through a superconducting inflector [3] into a storage ring 14.2 m in diameter with an effective circular aperture 9 cm in diameter. The superferric storage ring [4] has a homogeneous magnetic field of 1.45 T, which is measured by an NMR system relative to the free proton NMR frequency [5,6]. Electrostatic quadrupoles provide vertical focusing. A pulsed magnetic kicker gives a 10 mrad deflection which places the muons onto stored orbits. The muons start in 50 ns bunches and debunch with a decay time of about 20 µs due to their 0.6% momentum spread. Positrons are detected using 24 lead/scintillating fiber electromagnetic calorimeters [7] read out by waveform digitizers. The waveform digitizer and NMR clocks were phase-locked to the Loran C frequency signal.The muon spin precesses faster than its momentum rotates by an angular frequency ω a in the magnetic field B weighted over the muon distribution in space and time. The quantity a µ iswhere ω a is unaffected by the electrostatic field for muons with γ = 29.3. Parity violation in the decay µ + → e +ν µ ν e causes positrons to be emitted with an angular and energy asymmetry. Because of the Lorentz boost, the positron emission angle with respect to the muon spin direction in the muon rest frame is strongly coupled to its energy in the laboratory frame. The number of decay positrons with energy greater than E is described byin which the time dilated lifetime γτ ≈ 64.4 µs. Some 140 g − 2 periods of 4.37 µs were observed. Most experimental aspects of the data taking in 1999 were the same as in 1998 [1]. However, some improvements were made. Care was taken in tuning the AGS ejection system to minimize background from any extraneous proton beam extracted during the muon storage time. Scintillating fiber detectors which could be moved in and out of the storage region were u...