This past Wednesday, I attended the webinar that presented the first results of the muon g-2 experiment since it was transferred to Fermilab in Illinois. I had worked on this experiment when it was at Brookhaven National Laboratory on Long Island. The collective results of the Brookhaven experiments contradicted the theoretically predicted value for the muon’s magnetic moment or g-factor, suggesting that physics beyond the Standard Model may be needed to explain this phenomenon. Experiment deviated from theory by over 3 standard errors, enough to suggest a new phenomenon, but not enough to meet the accepted threshold of 5 sigma (standard errors) for a new discovery. The team at Fermilab hope to meet this threshold by reducing the systematic error (e.g., by improving the uniformity of the magnetic field by a factor of 3) and by having large statistics, using the lab’s accelerator as high-intensity source of muons (decaying from pions).
In the 15 years since the last published results from Brookhaven, the computed theoretical value had been fine-tuned, only amplifying the discrepancy with experiment. Dirac’s relativistic quantum mechanics predicts that the dimensionless magnetic moment of all charged leptons (electrons, muons, tau particles) should be exactly 2. Under quantum field theory, however, there should be small effects by self-interaction mediated by virtual particles. These virtual interactions should span all possible combinations within the Standard Model. The largest of these corrections, discovered by Schwinger, is a virtual photon interaction resulting in a deviation of alpha/2pi, where alpha is the fine structure constant, approximately equal to 1/137, so this modifies g to 2.00116. There are other, smaller scale corrections by other types of interactions. When we consider all of these, the consensus theoretical value for the muon’s anomalous magnetic moment, i.e., its deviation from 2, or g – 2, published in 2020 is:
116,591,810(43) x 10-11
where the parenthetic figure is the error in the value. The reason this theoretical computation has an error is that some of the contributions, notably those of quantum chromodynamics (QCD), cannot be computed exactly, due to the analytic unsolvability of the integrals. Instead, numerical approaches must be used. The largest source of error is the leading-order hadronic vacuum polarization (LO-HVP) contribution. There are two major approaches to modeling this hadronic contribution. The more purely computational approach is lattice QCD, where we approximate space-time as a discrete lattice with finite volume, and use Monte Carlo sampling to select points for computation. (We must sample points randomly so that the error is not proportionate to the large number of variables.) The other approach is to use dispersive methods, combined with experimental data on electron-positron cross-sections. The latter approach, though it is more data-driven and less purely computational, has the advantage of a smaller error. Dispersive techniques tend to have lower values for the hadronic contribution, and thus more strongly deviate from the experimentally measured muon g-2, which at Brookhaven was:
116,592,089(63)x 10-11
The theoretical consensus value differs from the Brookhaven results by 279 x 10-11, or 3.7 sigma, where the standard error sigma is the theoretical and experimental errors added in quadrature.
On April 7, 2021, the Fermilab team announced the results of its first run. This was not expected to meet the 5-sigma threshold, since there are not yet enough statistics. That problem should be surmounted when the next runs are analyzed, which will take at least another two years. Even with the smaller statistics, the reductions in systematic error already resulted in a smaller total error than Brookhaven. The investigators were blinded to their high-precision clock’s time scale, and they agreed in their by-laws to publish the results no matter what, once unblinded (to eliminate bias by cherry-picking results). The dramatic unblinding took place at the webinar, as two non-investigators revealed the handwritten clock scale in a sealed envelope: 39997844. This allowed the instant computation of the anomalous magnetic moment:
116,592,040(54)x 10-11
This was slightly lower than Brookhaven’s result, though within the error of both experiments. It was still different from theory by 230 x 10-11, or 3.3 sigma. When the Fermilab results are combined with those of Brookhaven, reducing the statistical error, we get an experimental result of:
116,592,061(41)x 10-11
This is a difference of 251 x 10-11, or 4.2 sigma, from the 2020 consensus value. The probability of this discrepancy being due to chance sampling choice is 1/40,000. (The 5-sigma threshold would be 1 in 3.5 million.) This result is already strongly indicative of the likelihood of physics beyond the Standard Model.
Or is it? Recall we have the unusual situation where a theoretical value has significant error. The contributions to the 2020 value, as identified by Aida al-Khadra at the Fermilab webinar, are:
- 116,584,718.9(1) x 10-11 quantum electrodynamic (QED) contribution
- 153.6 (1.0) x 10-11 weak interaction contribution
- 6845(40) x 10-11 hadronic vacuum polarization (HVP) contribution
- 92(18) x 10-11 hadronic light-by-light (HLbL) contribution
The hadronic contribution has the largest error and the second-largest value, so improving this calculation has the most importance for confirming if the muon g-2 result really does contradict the standard model. Over the last 15 years, theoreticians reduced the error in the computed muon g-2, yet its value remained within the range of previous calculations.
On the same day as the result announcement, Chao et al. published an improved value of the hadronic light-by-light scattering contribution: 106.8(14.7) x 10-11. They used lattice QCD. Although this slightly increases the HLbL contribution, it is not enough to account for the discrepancy between theory and experiment.
More notably, Borsanyi, Fodor et al. published in Nature (again on April 7) a significant result on the leading hadronic contribution to the muon magnetic moment. While its error is larger than that of dispersive techniques, it is by far the smallest error yet achieved by ab initio QCD. Their value for the LO-HVP contribution is 7075(55) x 10-11. If their value is used instead of the consensus, we get a muon g-2 that is close to the Brookhaven result, and in near-perfect agreement with the combined Brookhaven-Fermilab result! Factoring in the HLbL by Chao et al., I get a revised theoretical value of:
116,592,055(57) x 10-11
which is within 0.1 sigma of the experimental value (BNL-Fermilab).
This would seem almost too good to be true, were it not for the fact that Borsanyi, Fodor et al. certainly could not have known the Fermilab value in advance, as this was not known even to the investigators themselves! Nonetheless, we must be wary of arbitrarily selecting computations that agree with experiment, even though these particular computations (Borsanyi; Chao) happen to be the best in their respective classes (Lattice QCD for HVP; HLbL). We would need grounds for preferring the improved lattice QCD models over the dispersion techniques, before deciding this issue. While Borsanyi et al. agree with experiment, they now have a 2-sigma discrepancy with other theoretical calculations, so this disagreement must be resolved.
If the Borsanyi, Fodor et al. (2021) result is confirmed, then there would indeed be no new physics indicated by the muon g-2 experiment, and the Standard Model would have withstood its most ultra-precise test yet. This would strongly suggest that our inventory of the fundamental particle and interaction types in nature is in fact complete.