The anomalous magnetic moment of the muon: status and perspectives

TL;DR

This review assesses the muon anomalous magnetic moment $a_$ after the FNAL Muon $g-2$ results and the Muon $g-2$ Theory Initiative White Paper, emphasizing that the experimental precision currently supersedes the SM prediction by roughly a factor of four. It details the FNAL E989 experiment’s innovations in muon storage, beam delivery, precession measurement, and magnetic-field mapping, which together achieved a total uncertainty of about 127 ppb. The SM prediction is dissected into QED, EW, HVP, and HLbL components, with 2025 WP25 results indicating lattice-HVP plays a central role amid tensions among $e^+e^-$ data, and a final SM value around $a_^ ext{SM}=116592033(62) imes10^{-11}$ and a world-average tension $ riangle a_=38(63) imes10^{-11}$. The paper outlines a coordinated path to reach 124 ppb precision via improved hadronic inputs (data-driven and lattice), MUonE, and complementary experiments like J-PARC E34, while also considering strategies to push beyond 124 ppb at FNAL. The discussion emphasizes the continued synergy between high-precision experiments and SM theory as the primary avenue to probe beyond-Standard-Model physics.

Abstract

We review the status of the anomalous magnetic moment of the muon as a precision probe of physics beyond the Standard Model (SM) after the release of the final results from the Fermi National Accelerator Laboratory (FNAL) Muon $g-2$ experiment and the second White Paper of the Muon $g-2$ Theory Initiative. While the SM prediction requires further improvements by a factor of four to fully leverage the sensitivity achieved in experiment, the FNAL measurement will set the standard for many years to come, and we discuss a variety of features of the experimental campaign that made this achievement possible. In going forward, we discuss current efforts to improve the SM prediction, and imagine how an experiment would have to be devised to surpass 124 ppb in precision.

Figures (6)

• Figure 1: A: Experimental results from BNL (blue triangles) Muong-2:2001kxuBennett:2002jbBennett:2004pv,FNAL (red squares) Muong-2:2021ojoMuong-2:2023cdqMuong-2:2025xyk, and the world average (black diamond). The uncertainties combine statistics and systematics. BNL values are rounded to the nearest $10\,\text{ppb}$. All measurements employed positive muons except BNL-01, which ran with negative muons (inverted triangle). The CERN III result CERN-Mainz-Daresbury:1978ccd, with a precision of $7\,300\,\text{ppb}$, is not shown. B: Final relatively balanced systematic uncertainties at FNAL, see Sec. \ref{['sec:FNAL']} for a detailed discussion of the individual contributions.
• Figure 2: A: Muon Campus beamline overview and B: schematic of key elements in the SR. C: The beam intensity (orange) and kicker strength (blue) profiles vs. time at injection. The resulting storage efficiency, plotted with respect to mean momentum vs. time is shown in the lower panel. D: The intensity of events striking a calorimeter at early (blue) and later (red) times. The peaks align with $\omega_c$ but the width evolves owing to the finite momentum spread in the bunch. Note, $\omega_{a}$ and the exponential decay are divided out in this figure. E: Momentum distribution (upper axis) of the stored beam overlaid with the ESQ plate positions and their electric field lines. The collimator is shown as a black ring.
• Figure 3: A: A zoom-in on a portion of the precession data plot with an inset to show the relationship of spin and momentum vectors to the peaks and troughs of the data. B: The light blue trace shows the FFT of the residuals to a 5-parameter fit of the positron decay spectrum $N(t)$. The red dashed line stands at the $\omega_a$ frequency, which is well fit. The largest peaks are associated with CBO and the vertical betatron motion. The flat purple trace is the same procedure applied after additional terms are included in the fit function.
• Figure 4: A: Cross section of the C-shaped SR magnet illustrating the physical adjustments to develop a uniform field and the locations of the fixed NMR probes above and below the muon storage region. B: The NMR trolley probe locations, above which is a station with six fixed probes that are embedded just outside of the vacuum chamber. The small figures represent the multipoles of the field beyond the dipole that are determined by combinations of the NMR probes. C: The azimuthally averaged field contours and the stored muon relative intensity from Run-5.
• Figure 5: A: Summary of various determinations of $a_\mu^\text{HVP, LO}$, propagated to $a_\mu^\text{SM}$. The first two panels refer to data-driven determinations, where the three points for each $e^+e^-$ experiment reflect the CHKLS Colangelo:2018mtwColangelo:2022przStoffer:2023gbaLeplumey:2025kvv (triangle), DHMZ Davier:2017zfyDavier:2019canDavier:2023fpl (diamond), and KNTW Keshavarzi:2018mgvKeshavarzi:2019abfKeshavarzi:2024wow (square) methods, see WP25 for details. The gray band indicates the WP20 result, based on the $e^+e^-$ experiments above the first dashed line. The second panel represents $e^+e^-$ experiments that became available afterwards as well as the $\tau$-based estimate from WP25. The last panel summarizes lattice-QCD determinations, including the hybrid evaluation by BMW/DMZ-24 Boccaletti:2024guq, three individual lattice-QCD calculations by RBC/UKQCD-24+18 RBC:2024fic, Mainz/CLS-24 Djukanovic:2024cmq, BMW-20 Borsanyi:2020mff, and five lattice HVP averages from WP25. The blue band refers to the final WP25 result, which coincides with "Avg. 1." In all cases, except for the gray WP20 band, the remaining contributions to $a_\mu^\text{SM}$ beyond $a_\mu^\text{HVP, LO}$ are taken from WP25. The red band denotes the experimental world average. Adapted from Ref. Aliberti:2025beg. B: Sample diagrams for $a_\mu$ in the SM: solid lines refer to the muon, wiggly lines to photons, and the red blobs to hadronic matrix elements. The first diagram gives the leading QED contribution by Schwinger, the second one represents a one-loop EW diagram with $Z$ exchange, and the last two diagrams correspond to HVP and HLbL topologies, respectively. C: Summary of HLbL evaluations, from data-driven methods (green), lattice QCD (blue), and combinations (black). The averages are from WP20 and WP25, respectively, the other points refer to HSZ-24 Hoferichter:2024vbuHoferichter:2024bae, RBC/UKQCD-19 Blum:2019ugy, Mainz/CLS-21+22 Chao:2021tvpChao:2022xzg, RBC/UKQCD-23 Blum:2023vlm, and BMW-24 Fodor:2024jyn. Adapted from Ref. Aliberti:2025beg.