In a press conference held at Fermilab on 10 August, the Muon g-2 collaboration presented its latest measurement of the anomalous magnetic moment of the muon (a_{μ}). Combining the data from Run 1 to Run 3, the collaboration obtained a value for a_{μ} of 116 592 055 (24) × 10^{-11} (0.20 ppm) improving the precision of their previous result by a factor of 2, going down to 0.20 ppm. This results in the new experimental world average for a_{μ} of 116 592 059 (22) × 10^{-11 }(0.19 ppm), which deviates about 5.1 σ compared to the Standard Model (SM) prediction published by the Muon g-2 Theory Initiative. The discrepancy is likely to be lowered, once the Muon g-2 Theory Initiative will provide an update, say the researchers. The result of the Muon g-2 collaboration is described in a preprint submitted to Physical Review Letters.

The anomalous magnetic moment of the muon is the difference between the observed value of the particle’s magnetic moment and Dirac’s value 2. This difference is associated with quantum corrections arising from contributions of virtual SM particles and also possibly undiscovered particles. This makes measurements of a_{μ}, which is one of the most precisely calculated and measured quantities in particle physics, an ideal testbed for physics beyond the SM.

Getting the systematic uncertainty down to this level is a big deal and is something we didn’t expect to achieve so soon

Peter Winter

In order to measure g-2, a muon beam is sent into the superconducting storage ring (see image “Work done”), which has been reused from the former g-2 experiment at Brookhaven National Laboratory. The muons come in with aligned spins and once they enter the magnetic field, their spin axes precess around the ring’s magnetic field. Detectors located along the ring’s inner circumference gather data that allow the precession rate and thus a_{μ} to be determined. As the precession depends on the magnetic field, its field strength is measured as well.

“We improved a lot of things between our first year of data taking and our second and third year,” says Brendan Casey (Fermilab), who recently finished his term as co-spokesperson. The upgrades consist of better running conditions, more stable beams and an improved knowledge of the magnetic field weighted by the muon distribution and of the anomalous precession frequency corrected for beam-dynamics effects.

The new result is based on the data taking from 2019 and 2020 and has 4 times the statistics compared to the first result, enhancing it by a factor of 2. With this updated measurement, the collaboration also decreased the systematic uncertainty, which already surpassed the experiment’s goals. “This measurement is an incredible experimental achievement,” says Peter Winter (Argonne National Lab), co-spokesperson of the Muon g-2 collaboration. “Getting the systematic uncertainty down to this level is a big deal and is something we didn’t expect to achieve so soon.” Currently, about 25% of the total data (Run 1 – Run 6) is analysed. On 9 July the collaboration finished their final run; in total their collected data set is 21 times the size obtained by Brookhaven. The collaboration plans to publish their final results in 2025, targeting a precision of 0.14 ppm. By then data from all runs will be analysed and the final statistical uncertainty will be reduced to a minimum as well.

“We have moved the accuracy bar of this experiment one step further and now we are waiting for the theory to complete the calculations and cross-checks necessary to match the experimental accuracy”, explains Graziano Venanzoni (INFN Pisa & University of Liverpool). “A huge experimental and theoretical effort is going on, which makes us confident that theory prediction will be in time for the final experimental result from FNAL in a few years from now.“

**Theory puzzle
**Theoretical predictions for a

_{μ}play a crucial role in the interpretation of this much anticipated result. The SM prediction for the anomalous magnetic moment receives contributions from the electromagnetic, electroweak and strong interactions. While the former two can be computed to high precision in perturbation theory, it is only possible to compute the latter analytically in certain kinematic regimes. Contributions from hadronic vacuum polarisation (HVP) and hadronic light-by-light scattering dominate the overall theoretical uncertainty on a

_{μ}with 83 % and 17 %, respectively. The HVP describes how the electromagnetic properties of the muon are modified by strong interaction quantum effects.

To date the experimental results are confronted with the two theory predictions, which are the one by the Muon g-2 Theory Initiative based on the R-ratio method, and the one by the Budapest-Marseille-Wuppertal (BMW) collaboration based on simulations of lattice QCD + QED.

This new result by the Fermilab Muon g-2 experiment is a true milestone in the precision study of the Standard Model

Andreas Jüttner

In 2020 the Muon g-2 Theory Initiative published their consensus prediction evaluating HVP contributions with dispersion integrals. These calculations rely on input data from hadronic cross-section measurements, known as “R-ratio”, as perturbation theory cannot be used to determine the strong coupling constant close to the pion threshold, where it receives the dominant contributions. As this prediction is a combination of theory and experiment, it only works if the experimental input is assumed to be free of BSM contributions. Contrasting with all other experiments and puzzling the theory community, a recently published value of hadronic cross-section measurements by the CMD-3 collaboration would narrow the gap between the Muon g-2 Theory Initiative and the BMW collaboration.

The BMW collaboration employs lattice QCD, which mostly does not rely on experimental input data, except for a small number of hadronic inputs to tune the quark masses and fix the lattice scale in physical units.

Since 2018, so-called window quantities have been devised from lattice QCD. These quantities are a subset of the full HVP contributions that remove the regions with large uncertainties introduced by lattice QCD and led to an agreement across the lattice community about the HVP contributions. If the experimental R-ratio data in the same window are considered, then a similar tension between lattice calculations and R-ratio is found.

“This new result by the Fermilab Muon g-2 experiment is a true milestone in the precision study of the Standard Model,” says lattice gauge theorist Andreas Jüttner (CERN & University of Southampton). “This is really exciting — we are now faced with getting to the roots of various tensions between experimental and theoretical findings.”

### Further reading

Muon g-2 Collab. 2023 https://muon-g-2.fnal.gov/result2023.pdf

Muon g-2 Collab. 2021 *Phys. Rev. Lett.* **126** 141801

BMW Collab. 2021 *Nature* **593** 51

Muon g-2 Theory Initiative 2020 *Phys. Rept.* **887** 1