Shifting sands for muon g–2

The Dirac equation predicts the magnetic moment of the muon (g) to be precisely two in units of the Bohr magneton. Virtual lines and loops add roughly 0.1% to this value, giving rise to a so-called anomalous contribution often quantified by a_μ = (g–2)/2. Countless electromagnetic loops dominate the calculation, spontaneous symmetry breaking is evident in the effect of weak interactions, and contributions from the strong force are non-perturbative. Despite this formidable complexity, theoretical calculations of a_μ have been experimentally verified to nine significant figures.

The devil is in the 10th digit. The experimental world average for a_μ currently stands more than 5σ above the Standard Model (SM) prediction published by the Muon g-2 Theory Initiative in a 2020 white paper. But two recent results may ease this tension in advance of a new showdown with experiment next year.

The first new input is data from the CMD-3 experiment at the Budker Institute of Nuclear Physics, which yields a_μconsistent with experimental data. Comparable electron–positron (e⁺e^–) collider data from the KLOE experiment at the National Laboratory of Frascati, the BaBar experiment at SLAC, the BESIII experiment at IHEP Beijing and CMD-3’s predecessor CMD-2, were the backbone of the 2020 theory white paper. With KLOE and CMD-3 now incompatible at the level of 5σ, theorists are exploring alternative bases for the theoretical prediction, such as an ab-initio approach based on lattice QCD and a data-driven approach using tau–lepton decays.

The second new result is an updated theory calculation of a_μ by the Budapest–Marseille–Wuppertal (BMW) collaboration. BMW’s ab-initio lattice–QCD calculation of 2020 was the first to challenge the data-driven consensus expressed in the 2020 white paper. The recent update now claims a superior precision, driven in part by the pragmatic implementation of a data-driven approach in the low-mass region, where experiments are in good agreement. Though only accounting for 5% of the hadronic contribution to a_μ, this “long distance” region is often the largest source of error in lattice–QCD calculations, and relatively insensitive to the use of finer lattices.

The new BMW result is fully compatible with the experimental world average, and incompatible with the 2020 white paper at the level of 4σ.

“It seems to me that the 0.9σ agreement between the direct experimental measurement of the magnetic moment of the muon and the ab-initio calculation of BMW has most probably postponed the possible discovery of new physics in this process,” says BMW spokesperson Zoltán Fodor (Wuppertal). “It is important to mention that other groups have partial results, too, so-called window results, and they all agree with us and in several cases disagree with the result of the data-driven method.”

These two analyses were among the many discussed at the seventh plenary workshop of the Muon g-2 Theory Initiative held in Tsukuba, Japan from 9 to 13 September. The theory initiative is planning to release an updated prediction in a white paper due to be published in early 2025. With multiple mature e⁺e^– and lattice–QCD analyses underway for several years, attention now turns to tau decays – the subject of a soon-to-be-announced mini-workshop to ensure their full availability for consideration as a possible basis for the 2025 white paper. Input data would likely originate from tau decays recorded by the Belle experiment at KEK and the ALEPH experiment at CERN, both now decommissioned.

I am hopeful we will be able to establish consolidation between independent lattice calculations at the sub-percent level

“From a theoretical point of view, the challenge for including the tau data is the isospin rotation that is needed to convert the weak hadronic tau decay to the desired input for hadronic vacuum polarisation,” explains theory-initiative chair Aida X El-Khadra (University of Illinois). Hadronic vacuum polarisation (HVP) is the most challenging part of the calculation of a_μ, accounting for the effect of a muon emitting a virtual photon that briefly transforms into a flurry of quarks and gluons just before it absorbs the photon representing the magnetic field (CERN Courier May/June 2021 p25).

Lattice QCD offers the possibility of a purely theoretical calculation of HVP. While BMW remains the only group to have published a full lattice-QCD calculation, multiple groups are zeroing in on its most sensitive aspects (CERN CourierSeptember/October 2024 p21).

“The main challenge in lattice-QCD calculations of HVP is improving the precision to the desired sub-percent level, especially at long distances,” continues El-Khadra. “With the new results for the long-distance contribution by the RBC/UKQCD and Mainz collaborations that were already reported this year, and the results that are still expected to be released this fall, I am hopeful that we will be able to establish consolidation between independent lattice calculations at the sub-percent level. In this case we will provide a lattice-only determination of HVP in the second white paper.”