Measurement of the positive muon anomalous magnetic moment to 127 ppb
Muon g-2 Collaboration
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In one page
The Muon g-2 Collaboration at Fermilab has finished a six-year measurement of how magnetic a muon is, and the answer is now known to 127 parts per billion — about one part in eight million. A muon is a heavy cousin of the electron, and its magnetism is not quite the value a lone particle would have, because the muon is never alone: the vacuum around it is full of particles flickering in and out of existence, and every one of them nudges the number. Measuring the anomaly is therefore a direct reading of what empty space contains. The team stored polarised muons in a 7-metre ring at 1.45 tesla and compared how fast their spin turns against how fast protons in a water sample precess in the same field, with the analysis kept blind until the very end. Their result improves the world average by more than a factor of four, and it now sits on top of the Standard Model’s own prediction.
Why it matters hereChapter 2 treats the vacuum as a real, structured medium rather than an absence, and this is the most precise reading anyone has ever taken of it: the muon’s magnetism is the sum of everything that flickers around it, measured to eight significant figures. Chapter 1 is about what a claim has to look like before it counts, and this measurement is the standard to hold anything else against — five independent analysis teams, a hardware blind set by outsiders, and every correction published with its own uncertainty.
What it claims
01The muon’s magnetic anomaly is now measured to 127 parts per billion. Combining the 2020-2023 data with the collaboration’s earlier runs gives a value of 1.165920705 for the anomaly, with an uncertainty of 148 in the last three digits. The new experimental world average, including the Brookhaven E821 result but dominated by Fermilab, is 1.165920715 with an uncertainty of 145 in the last three digits, or 124 parts per billion. The Fermilab measurements have improved the precision of the world average by more than a factor of four over the previous best.Abstract; Calculation of the muon anomaly, closing paragraphs; Figure 3
Settled physics02The quantity being measured is a direct reading of the vacuum. A lone Dirac particle would have a g factor of exactly 2; the anomaly is the fractional excess over that, and it arises entirely from radiative corrections from virtual particles — the flicker of the field around the muon. Schwinger’s 1948 calculation of the first term, the fine-structure constant divided by two pi, founded modern relativistic field theory. The electron’s anomaly is measured a thousand times more precisely, but the muon is roughly two hundred times heavier and that makes its anomaly about forty thousand times more sensitive to physics the Standard Model does not yet contain.Introduction, paragraphs 1 and 2
Settled physics03The measurement is a comparison of two frequencies and nothing else. Polarised muons are injected into a storage ring of 7.112 metre radius carrying a homogeneous 1.45 tesla vertical field, at a momentum chosen so that the electric focusing fields make no net contribution to the spin precession. The experiment then measures the difference between the muon’s spin-precession and cyclotron frequencies, and divides it by the nuclear magnetic resonance frequency of shielded protons in a spherical water sample sitting in the same field. Parity violation in muon decay, combined with the beam’s relativistic boost, makes the number of high-energy positrons hitting the twenty-four lead-fluoride calorimeters oscillate at exactly the frequency wanted. Everything else in the answer is a ratio of separately known fundamental constants.Experimental principle, paragraphs 1 to 3; equations 1 and 2
Settled physics04The analysis was blinded twice over, and cross-checked twenty times. Each analysis group worked with an unknown, fixed, pseudorandom offset in the precession frequency, on top of a hardware blind on the digitisation frequency that was set and monitored by Fermilab physicists outside the collaboration. Five groups using three different positron-reconstruction and pileup-correction algorithms produced seven asymmetry-weighted results plus thirteen further crosschecks; the twenty correlated values agreed within the statistical variation assessed from two hundred bootstrap samples. The hardware blind was removed only after every aspect of the determination was complete and frozen. The best-fit chi-squared for the largest dataset was 4007 for 4097 degrees of freedom.Anomalous precession frequency section, paragraphs 4 and 5 (omitted from the reproduced text for length)
Settled physics05Every correction is published with its own uncertainty, and they total 572 parts per billion. The measured frequency ratio is adjusted for the residual effect of the electric focusing fields, for vertical betatron motion, for three separate ways the observed muon ensemble’s average phase drifts with time, and for two magnetic-field transients — eddy currents from the fast injection kickers and vibration of the electrostatic quadrupole plates. The statistical uncertainty of 114 parts per billion still dominates: the total systematic uncertainty on the frequency ratio is 76 parts per billion, and the total on the anomaly for the new datasets is 139.Experimental principle, final paragraph; Table I; Beam-dynamics corrections and Muon-weighted magnetic field sections (omitted for length)
Settled physics06What to watch: the theory side of the comparison has moved, and it is not settled. The Muon g-2 Theory Initiative’s 2025 White Paper prediction agrees with this measurement — but it shifted considerably from the 2020 White Paper, almost entirely because it now uses lattice-QCD calculations alone for the leading-order hadronic vacuum polarisation, the contribution in which the muon’s field briefly becomes a cloud of quarks. The older route, a dispersion integral over measured electron-positron annihilation into hadrons, had shown a discrepancy with experiment, and a recent cross-section measurement increased the tension among the experimental inputs, so that route was left out of the 2025 prediction entirely. Work on both evaluations continues, and which one converges is the open question this measurement now waits on.Calculation of the muon anomaly, penultimate paragraph
What to watch
Read it
Muon g-2 Collaboration, Measurement of the Positive Muon Anomalous Magnetic Moment to 127 ppb, Physical Review Letters 135, 101802 (2025), doi.org/10.1103/7clf-sm2v. Received 3 June 2025, accepted 9 July 2025, published 2 September 2025. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.
Abstract
A new measurement of the magnetic anomaly a-mu of the positive muon is presented based on data taken from 2020 to 2023 by the Muon g − 2 Experiment at Fermi National Accelerator Laboratory (FNAL). This dataset contains over 2.5 times the total statistics of our previous results. From the ratio of the precession frequencies for muons and protons in our storage ring magnetic field, together with precisely known ratios of fundamental constants, we determine a-mu = 1 165 920 710(162) × 10 to the −12 (139 ppb) for the new datasets, and a-mu = 1 165 920 705(148) × 10 to the −12 (127 ppb) when combined with our previous results. The new experimental world average, dominated by the measurements at FNAL, is a-mu (exp) = 1 165 920 715(145) × 10 to the −12 (124 ppb). The measurements at FNAL have improved the precision on the world average by over a factor of 4.
Introduction
Precise measurements of magnetic moments of charged leptons serve as precision probes of the Standard Model (SM) due to their sensitivity to particles and interactions within the SM and potentially beyond the Standard Model (BSM). The Dirac equation predicted g-e = 2 for the g factor g-e that relates the electron magnetic moment to its spin. Schwinger’s radiative correction, inspired by contemporaneous experimental data, refined this result and introduced the anomaly a-e = alpha / 2 pi. This Letter laid the foundation for modern relativistic field theory and the development of the SM.
The magnetic anomaly a = (g − 2) / 2 arises from radiative corrections from virtual particles and can be calculated precisely within the SM. While a-e is measured 1000 times more precisely than a-mu, the muon’s greater mass makes a-mu about 4 × 10 to the 4 times more sensitive to much BSM physics. Precision measurements of g-mu span decades of advances, beginning with early experiments at Columbia University Nevis Laboratory and the University of Liverpool. Direct measurement of a-mu started with the CERN-I, CERN-II, and CERN-III experiments, which the Brookhaven National Laboratory (BNL) E821 experiment further improved. The E821 results revealed a statistically significant tension with SM predictions at the time. The Muon g − 2 Experiment at Fermi National Accelerator Laboratory (FNAL) confirmed the E821 result with the 2018 Run-1 data, and then refined a-mu with over twice the precision with the Run-2/3 data.
This Letter presents a measurement of a-mu from the Muon g − 2 Experiment using data collected in three runs spanning 2020 to 2023 (designated as Run-4, Run-5, and Run-6). The Run-4/5/6 positron statistics, over 2.5 times that of our previous measurements, improve our final Run-1 to Run-6 statistical precision by more than 1.8 compared to the Run-1/2/3 result. Our final result surpasses our original statistical and systematic goals and establishes a stringent benchmark for future theoretical BSM extensions.
Experimental principle
Our Run-1 and Run-2/3 publications detail the experiment. Polarized muon beams are injected into a 7.112 m radius storage ring with a design storage momentum of 3.1 GeV/c. A superferric magnet generates a homogeneous vertical 1.45 T dipole field that provides weak horizontal focusing of the beam and drives the muon spin precession. Two critical components for beam storage are a fast kicker that redirects muons onto the central orbit and an electrostatic quadrupole (ESQ) system for vertical focusing. At the design momentum, the contributions to the muon spin precession from the electric fields in the ESQ cancel.
The experiment determines the ratio of two frequencies, R-prime-mu = omega-a divided by omega-p-prime at the reference temperature, where omega-a is the difference between the spin precession and cyclotron frequencies of the muon, and omega-p-prime at the reference temperature is the nuclear magnetic resonance (NMR) precession frequency of shielded protons in a spherical water sample (corrected to a reference temperature), averaged over the muon distribution, which expresses the magnetic field strength. The omega-a measurement utilizes 24 lead-fluoride electromagnetic (EM) calorimeters that record the energy and time of incident positrons. Parity violation in muon decay and the Lorentz boost of the beam couple to provide an oscillation in the rate of high-energy positrons at a frequency of omega-a. A laser system continuously monitors the gain of each crystal in the calorimeters. A chain of magnetic field measurements yields the muon-weighted proton frequency. The chain begins with a periodic mapping of the magnetic field by a movable mapper with 17 NMR probes. These probes are calibrated in situ against a water-based cylindrical probe that transfers the absolute calibration of shielded protons in a spherical water sample. Additional NMR probes, embedded in the experiment’s vacuum chambers, track the field while muons are stored between mappings. The mapped and tracked magnetic field is weighted by the muon distribution, measured using two straw tracker stations.
The narrow aperture through which the beam enters the storage ring produces a mismatch between the phase space of the incoming beam and the storage ring acceptance that leads to coherent betatron oscillation (CBO). This coherent beam motion introduces a time variation into the positron detection efficiency. After Run-4, an additional radio frequency (rf) system added a small modulation of the ESQ high voltage during the first 6 microseconds after muon injection. The rf system generates dipole fields tuned to the CBO frequency, resonantly damping the CBO by applying forces out of phase. For analysis purposes, the data are divided into four distinct datasets based on rf configurations: noRF (no rf system), xRF (horizontal rf fields only), and xyRF5/xyRF6 (both horizontal and vertical rf fields in Run-5 and Run-6, respectively).
The measured anomalous spin frequency, and the muon-weighted magnetic field, must be corrected for several effects, collectively shifting the frequency ratio by 572 ppb. The corrections to the measured anomalous spin frequency address the residual contribution to the muon spin precession rate from electric fields; the contribution to the muon spin precession from the vertical betatron motion; time-dependent changes in the mean phase of the observed muon ensemble caused by (i) detector acceptance, and by phase-momentum correlations coupled to (ii) momentum-dependent muon lifetimes and (iii) momentum-dependent muon storage losses. Corrections to the muon-weighted magnetic field accommodate the fast transient fields not captured in the NMR-based field maps, specifically from eddy currents generated by the fast injection kickers and from vibrations of the ESQ plates, both synchronous with muon injection.
(Sections omitted for length: the anomalous precession frequency analysis, the beam-dynamics corrections, the muon-weighted magnetic field, and the magnetic field transients, together with Tables I and II and Figures 1 and 2. The complete Letter is at the source.)
Calculation of the muon anomaly
The values of the two frequencies for the four fit datasets, along with the corrections, form the frequency ratio. The ratio measurements are statistically uncorrelated, while nearly all systematic uncertainties are fully correlated. The four fit datasets show good consistency with a chi-squared of 0.96 for 3 degrees of freedom, which has a probability of 80%. No statistically significant correlations with magnet current, magnetic field, field gradients, or time of day were observed.
We report the frequency ratio at a reference temperature of 25 degrees Celsius. This change from the reference temperature used in previous publications aligns with both the Committee on Data of the International Science Council (CODATA) standard and our actual measurement conditions. Our previous ratio values were adjusted by −101 ppb to reflect this change in reference temperature and external constants. The anomaly values do not change as a result of the temperature shift, though the external CODATA constants have been updated.
The superior statistical power of this larger dataset, along with additional dedicated measurements, enabled further cross-checks of the Run-2/3 results. Three corrections with corresponding uncertainty adjustments were identified and applied when combined with the latest dataset: the sensitivity of the anomalous precession frequency to small, slow gain shifts noted earlier; improved understanding of spatial dependencies in the transient magnetic fields from kicker system eddy currents; and a sign error correction in one component of the differential decay correction. These corrections, determined independently, happened to have the same sign and combine to shift the Run-1/2/3 results by 50-98 ppb, respectively, and result in a total systematic uncertainty of 78 ppb for the adjusted Run-2/3 result. The corrections were finalized before unblinding the Run-4/5/6 results. The latest result agrees well with the previous measurements.
The combined FNAL average, Run-1 to Run-6, with a total uncertainty of 127 ppb, assumes fully correlated systematic uncertainties between the results.
We determine the muon anomaly to be 1 165 920 710(162) × 10 to the −12 (139 ppb) for Run-4/5/6, and 1 165 920 705(148) × 10 to the −12 (127 ppb) for the full dataset, with the statistical, systematic, and external parameter uncertainties combined in quadrature. The combined experimental average, from BNL E821 and Run-1 to Run-6, becomes 1 165 920 715(145) × 10 to the −12 (124 ppb).
The Muon g − 2 Theory Initiative has released an updated SM value of a-mu in their 2025 White Paper (WP2025), which agrees with the measured average. The value shifts considerably compared to their 2020 White Paper (WP2020), which is almost entirely due to the exclusive use of new, published leading-order hadronic vacuum polarization estimates based on lattice-QCD calculations. The previous value in their WP2020 used experimental electron-positron to hadron cross section measurements from multiple experiments to evaluate this contribution based on a dispersion integral and showed a discrepancy with the experimental value. However, a recent cross section measurement has increased the tension among the experimental inputs; thus a prediction based on the dispersion integral was not included in their WP2025. Efforts are continuing toward an evaluation of this leading-order hadronic contribution using both lattice QCD and dispersion integral calculations.
In summary, we report the measurement of the muon magnetic anomaly to a precision of 127 ppb using our full six years of data. With over a fourfold improvement in precision over the BNL E821 measurement, this result represents the most precise determination of the muon magnetic anomaly and provides a powerful benchmark for extensions of the SM.
Companion sheets on this site: the calculation that founded this whole line of measurement, and the vacuum-polarization formalism behind it, is Julian Schwinger’s On gauge invariance and vacuum polarization. For what else the vacuum’s virtual content does when a field is strong enough to see it, see Extremely high-intensity laser interactions with fundamental quantum systems and Advances in QED with intense background fields. For the standing puzzle of how much energy that same vacuum carries, see The quantum vacuum and the cosmological constant problem.
The way in
https://doi.org/10.1103/7clf-sm2vPROMOTED FROM ABSTRACT-ONLY. The harvested record listed this paper under the APS default licence, which would have limited the sheet to the abstract. The published article was fetched and read directly on 2026-09-08, and its licence footer on page 101802-2 states Creative Commons Attribution 4.0 International, funded by SCOAP3, so key passages are reproduced here under that licence with the required attribution. AUTHOR. The harvested record named the author ‘Anonymous’. The author is the Muon g-2 Collaboration; the published author list runs to about 180 named physicists from 37 institutions, led alphabetically by D. P. Aguillard, and the paper is signed ‘(Muon g-2 Collaboration)’. Two of the authors, G. Pauletta and L. Santi, are marked deceased in the published list. TEXT. Reproduced below are the abstract, the Introduction, the Experimental principle section, and the closing Calculation of the muon anomaly and summary passages, verbatim under CC BY. The intervening sections on the anomalous precession frequency, the beam-dynamics corrections, the muon-weighted magnetic field and the magnetic-field transients, the two data tables, the three figures and the reference list are omitted for length; the complete Letter is at the source. SYMBOLS AND EQUATIONS. The PDF extraction flattened subscripts, primes, tildes and Greek letters, and rendered the equals sign and parentheses of the typeset mathematics as other characters. Symbols are therefore written out in plain form in the reproduced passages — the muon anomaly as a-mu, the anomalous precession frequency as omega-a, the muon-weighted shielded-proton NMR frequency as omega-p-prime at the reference temperature, and the frequency ratio as R-prime-mu — and the two displayed equations, which define that ratio and convert it into the anomaly, are not reproduced; the sentences around them are kept unaltered. Bracketed reference numbers are dropped as page furniture. No wording is otherwise changed.
How to cite it
Muon g-2 Collaboration (2025) Measurement of the positive muon anomalous magnetic moment to 127 ppb. doi:10.1103/7clf-sm2v
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