For more than twenty years, the anomalous magnetic moment of the muon was the most promising crack in the Standard Model of particle physics. Every measurement of the muon’s magnetic wobble seemed slightly larger than the theory predicted, and the gap held steady at around 3.7 standard deviations, tantalizingly short of the 5-sigma threshold required to claim a discovery but too persistent to ignore. If the discrepancy was real, it meant that unknown particles were contributing quantum fluctuations that the Standard Model could not account for. The search for those particles became one of the highest priorities in fundamental physics.
In April 2021, a collaboration known as BMW published a paper in Nature that changed the landscape. Using lattice quantum chromodynamics (QCD), a method that simulates the strong nuclear force from first principles on a supercomputer, the group calculated the leading-order hadronic contribution to the muon’s magnetic moment with unprecedented precision. Their result was 707.5 × 10^-10, with a total uncertainty of 5.5 × 10^-10. When combined with other Standard Model contributions, this value agreed with the experimental measurement, closing the gap that had driven the search for new physics.
The BMW result did something else. It introduced a disagreement with the alternative method for calculating the same quantity, a technique known as the R-ratio or dispersive method that relies on decades of experimental data from electron-positron colliders. The R-ratio determinations, produced by the groups of Davier, Keshavarzi, and Colangelo and Hoferichter, converge on values around 692 to 694 × 10^-10, with uncertainties of 2.4 to 4.0. The BMW lattice result is 2.0 to 2.5 sigma larger than these R-ratio numbers.
This tension is the real story. It is not a conflict between theory and experiment, but between two ways of knowing within theory itself.
The BMW calculation, led by Szabolcs Borsanyi and Kalman Szabo at the University of Wuppertal and collaborators at Eotvos University in Budapest and Aix-Marseille University, represents the culmination of a decade of advances in lattice QCD. The method works by discretizing spacetime into a four-dimensional grid of points, placing quark fields on the grid points and gluon fields on the links between them, and solving the equations of quantum chromodynamics numerically. The calculation requires enormous computing power: the group accumulated over 20,000 configurations on lattices with a spatial extent of approximately 6 femtometers, and developed sophisticated noise-reduction techniques based on the lowest eigenmodes of the Dirac operator to achieve the necessary precision.
The result was a breakthrough in lattice methodology. The team included quantum electrodynamics effects and allowed up and down quarks to have different masses, computed all isospin-symmetry-breaking contributions explicitly rather than estimating them, performed dedicated finite-size studies, and applied a taste-improvement procedure to reduce lattice artifacts in the continuum extrapolation. The total relative accuracy of 0.8 percent represents a substantial advance over earlier lattice calculations.
The R-ratio method works differently. It takes the experimentally measured cross-section of electron-positron annihilation into hadrons, integrates it with a kernel function derived from quantum electrodynamics, and extracts the hadronic vacuum polarization contribution directly. The method is conceptually simpler and has been refined over decades, with multiple independent experiments, including BABAR at SLAC in California, KLOE at Frascati in Italy, and BESIII in Beijing, providing cross-checks on the data. The R-ratio results are highly correlated with each other because they draw from the same pool of experimental measurements, but those measurements themselves were produced by different detectors in different laboratories using different techniques, making systematic bias unlikely.
The BMW paper itself acknowledges the tension cautiously. The authors note that the discrepancy between their lattice result and the R-ratio values is smaller than what would constitute evidence for a new phenomenon (3 sigma) and much smaller than what would constitute a discovery (5 sigma). But they also point to a more troubling specific comparison: when they restrict the calculation to a window observable known as a_mu_win, which probes shorter distance scales and is easier to compute reliably on the lattice, the tension grows to 3.7 sigma. And in this window, independent lattice groups have published results that partially diverge from each other as well, with the BMW result differing by 2.2 sigma from the lattice calculation of Blum and collaborators and by only 0.2 sigma from that of Aubin and collaborators.
The question that follows is not whether new particles exist. It is whether the Standard Model can be trusted as a predictive framework when its own internal methods disagree. The lattice calculation says the muon’s wobble is consistent with known physics. The R-ratio method says it is not. Both cannot be right. The choice between them will determine whether particle physics continues the search for new particles through the muon channel or accepts that the anomaly was an artifact of incomplete computation.
Since 2021, independent lattice groups have published their own calculations, and most have moved in the direction of the BMW result, though with varying levels of agreement. The Fermilab Muon g-2 experiment has continued collecting data, aiming to reduce the experimental uncertainty by a factor of four. The VEPP-2000 collider in Novosibirsk has published a new measurement of the pion production cross-section that shifts the R-ratio prediction upward, toward the lattice result, but that measurement itself disagrees with older, well-established experiments, creating a secondary conflict within the experimental tradition.
The muon has not revealed new particles. It has revealed something perhaps more unsettling: that the tools physicists use to understand the world do not always agree, and that resolving their disagreement requires not better experiments but a better understanding of what each method actually knows.
References
Borsanyi, Sz., Fodor, Z., Guenther, J. N., et al. Leading hadronic contribution to the muon magnetic moment from lattice QCD. Nature 593, 51-55 (2021). DOI: 10.1038/s41586-021-03418-1
Davier, M., Hoecker, A., Malaescu, B., & Zhang, Z. A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to alpha(mZ2). European Physical Journal C 80, 241 (2020). DOI: 10.1140/epjc/s10052-020-7792-2
Keshavarzi, A., Nomura, D., & Teubner, T. g-2 of charged leptons, alpha(MZ2), and the hyperfine splitting of muonium. Physical Review D 101, 014029 (2020). DOI: 10.1103/PhysRevD.101.014029
Muon g-2 Collaboration. Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm. Physical Review Letters 126, 141801 (2021). DOI: 10.1103/PhysRevLett.126.141801