The body text of a new paper quoted below sums up the myriad discrepancies between the "official" estimate of muon g-2 (a 4.2 sigma difference from the actual measurement) and data including the lattice QCD estimate by the BMW group which is consistent with the measurement at the 1.5 sigma level.
The bottom line is that the 2020 White Paper calculation of the Standard Model prediction for muon g-2 is wrong, and the BMW calculation is correct and confirms the experimentally measured value of muon g-2. There is not, in fact, a muon g-2 anomaly, just as there is also not, in fact, a W boson mass anomaly (a discrepancy arising from recent a misanalysis of old Tevatron data).
It isn't entirely clear if the problem with the 2020 White Paper arises from flaws in the e+e- data upon which its calculation relies. This was the problem in the case of the muonic hydrogen proton radius puzzle. But the problem could also be with the way that the e+e- data was incorporated into an otherwise first principles based calculation was incorrect in some way.
An unambiguous interpretation of the new measurement of the muon g−2 by the E989 experiment at Fermilab is impeded by several tensions that have been exposed since the publication of the 2020 White Paper: (1) There is a tension of 2.1σ between a single lattice calculation and the WP-recommended value for aµhvp,LO, based on e+e− cross section data published prior to 2023; (2) There is a tension of almost 4σ between several lattice calculations and the corresponding dispersive estimate based on the same e+e− data; (3) There is a tension of 2−3σ in the hadronic running of α, as estimated by two lattice calculations and e+e− data; (4) There is a slight tension of1−2σ in the Adler function determined from lattice and perturbative QCD on the one hand, and e+e−data on the other; (5) Finally, there is a tension of 2.7σ in the dominant π+π−channel between BaBar and KLOE, as well as a tension of about 4σ between CMD-3 and all other experiments.
In this context, it is important to realise that a larger SM prediction for aµ is not in contradiction with global electroweak constraints, at least at the current level of precision.
Obviously, an independent cross-check of the BMW lattice result for aµhvp,LO, with sub-percent precision is badly needed. Furthermore, the tension among e+e−data must be elucidated, a task for which the alternative determination of the R-ratio from τ decays could be useful. These activities are currently in progress. The Muon g−2 Theory Initiative is preparing an update of the original WP, which will thoroughly address the issues that have come to the fore since 2020.
The Larger Implications Of The Muon g-2 Results
The fact that the muon g-2 experiments confirm the Standard Model prediction, correctly calculated, to extreme precision, is an excellent global test of the completeness and correctness of the Standard Model up to a few orders of magnitude of energy scale above the electroweak scale. New physics at very high energies, however, effectively decouple from the lower energy physics involved in the muon g-2 observable and can't be detected even at the ultrahigh precision of these measurements and calculations.
Standing alone, the lack of a discrepancy between the muon g-2 measurement and the Standard Model prediction for it doesn't rule out all new physics in the experimentally testable energy range. But it does mean that any new physics in that energy range must either be tiny tweaks to the Standard Model (e.g. tweaks of the size involved in considering quantum gravity effects on the calculation) or must be tweaks that cancel out in the calculation of muon g-2, a calculation that involves all three of the Standard Model forces (the electromagnetic force, the weak force, and the strong force).
Ultimately, one can never rule out experimentally the possibility that beyond the Standard Model physics with effects smaller than the most precise experiments that are possible at any given time.
But one can rule out many possible beyond the Standard Model theories that would cancel out in a muon g-2 calculation at energy scales up to a few orders of magnitude above the electroweak scale with other experimental tests, direct and indirect.
Probably the most important class of new physics that wouldn't impact the muon g-2 calculation would be new "sterile" particles that interact via gravity but not via any of the three Standard Model forces, such as many dark matter particle candidates and the right handed neutrinos proposed in see-saw models of neutrino mass generation. Sterile dark matter candidates are disfavored by the tightness of the fit between inferred dark matter distributions and distributions of ordinary baryonic matter.
The neutrino physics experiments necessary to distinguish between right hand neutrinos in see-saw models of neutrino mass generation and alternative models of neutrino mass generation are a work in progress. Observing neutrinoless double beta decay would destroy the justification of these models, but if we determine from the non-observation of neutrinoless double beta decay that neutrinos do not have Majorana mass (and my educated guess is that this is precisely what will happen in the next decade or two), there isn't a clear agenda for experiments that would establish how neutrinos do get their mass.