Anomaly no more! “Muon g-2” puzzle resolved at last

When it comes to investigating the Universe, the ultimate goal is to uncover the closest approximation of the scientific truth about reality as possible. This takes the combination of two approaches, simultaneously.

• Over on the theory side, we have our best models, laws, and framework to represent reality and the rules that it obeys. We use what has already been established to make predictions about what we expect to see, experimentally and/or observationally, when we put the Universe itself to the test.
• Meanwhile, on the experiment side, we look at what theory predicts and attempt to test it to greater precision than ever before.

If it confirms what theory predicts, that’s an improvement in our understanding of the Universe: not a revolutionary change, but incrementally increasing what is known. If it disagrees with theoretical predictions, it’s an opportunity for advancement: either there’s a flaw with the theory or the experiment, or if not, there’s a chance that there’s more to reality than we presently understand. Or, if there are multiple different theoretical predictions, experiment gives us a chance to tell them apart.

In the case of the muon’s magnetic moment, often known as “g-2″ in physics circles, a longstanding disagreement between theory and experiment threatened to break the Standard Model. After years of work on all sides, the anomaly has finally disappeared. It’s much more than an incremental improvement in this case; it’s a showcase for exactly how science should work when we do it right.

This photo shows the Muon g-2 electromagnet departing its original home at Brookhaven, where it resided from 2001-2013. Workers from Emmert International affixed a red steel apparatus to the ring that kept it as flat as possible during transport., after which it was brought to Fermilab in July of 2013.

The idea of a magnetic moment is very old: it goes back to the fundamentals of electricity and magnetism. Although there are only electric charges in nature (+ and – charges) and not magnetic ones (i.e., you can have a magnet with both north and south poles, but not an isolated “north” or “south” pole by itself), electrically charged particles do indeed have magnetic properties.

• When electrically charged particles have a straight-line motion, they generate magnetic fields that curl around them.
• When electrically charged particles move in a circle-like fashion, like a current through a coil of wire or an electron orbiting a proton, they generate magnetic fields that are perpendicular to their circular motion.
• And when electrically charged particles spin on their axis — or, in the quantum mechanical case of an individual particle, have an intrinsic angular momentum to them — they exhibit a magnetic moment, or a north-south dipole magnet intrinsic to them.

In classical physics, the magnetic moment of a point particle, like an electron, muon, or tau lepton, would be easily calculable: ½ multiplied by its spin angular momentum multiplied by its charge-to-mass ratio. But the Universe isn’t classical; it’s quantum in nature, and that means that we have to include a whole slew of purely quantum effects as well.

Individual and composite particles can possess both orbital angular momentum and intrinsic (spin) angular momentum. When these particles have electric charges either within or intrinsic to them, they generate magnetic moments, causing them to be deflected by a particular amount in the presence of a magnetic field and to precess by a measurable amount.

These quantum effects mean, cumulatively, that you have to multiply the “classical” magnetic moment for a point particle by a prefactor, and that’s what g in the “g-2″ experiment is. If there were no quantum field theory and only basic quantum mechanics, g would equal 2, exactly. Therefore, the idea of measuring “g-2″ for a particle takes us deep into the weeds of quantum field theory, where we have to consider a whole slew of contributions to the results. These contributions include:

• the electromagnetic field, including radiative corrections from quantum electrodynamics,
• the weak field and the Higgs field, even though they have super-heavy bosons associated with them,
• and the contributions of the strong (quantum chromodynamic) field, including via vacuum polarization and also through what particle physicists call “light-by-light” contributions, where two photons interact through the virtual exchange of other particles, including through hadrons (particles made of quarks).

Experimentally, we can only measure the sum total of all contributions put together; we cannot separate them out piece-by-piece. That means, if we want to perform experiments to measure what “g-2″ actually is, we have three and only three options for measuring it that can be done to quality precision: measure it for the electron, measure it for the muon, or measure it for the tau lepton.

This infographic displays a number of essential properties, facts, and anecdotes about the muon: the first fundamental particle ever discovered that plays no role in the behavior of conventional matter found on Earth. Among physicists, the phrase “Who ordered that?” when it came to the muon, is now legendary.

We can do it for the electron quite easily, as its very low mass (just 1/1836^th the mass of a proton) for its electric charge (remember, charge-to-mass ratio is something that affects g, and hence g-2) ensures that the electromagnetic field’s contributions are far greater than all others. We’ve now both theoretically calculated and experimentally measured g-2 for the electron to 13 significant figures, making the electron’s magnetic moment the most accurately verified prediction in the history of physics.

The tau lepton is much heavier, but short-lived and very unstable, with extreme contributions from the field arising from the strong nuclear force. It’s a very messy calculation, but since it lacks high-precision data, it’s not a very good laboratory for testing the Standard Model.

But the muon hits the sweet spot. Even though the muon is unstable, it lives for relatively long periods of time 2.2 microseconds as its mean lifetime meaning that we can harness, control, and measure them exquisitely well. The muon is also much heavier than an electron: 206 times the electron’s mass, or 1/11^th the mass of a proton. Its magnetic moment is only a little bit smaller than the electron’s, but its heavier mass means that the contributions of the strong force show up in the 9th significant digit instead of the 14th or so. That’s why the muon is such an important testing ground for the Standard Model and our understanding of the strong nuclear force within it.

When a heavy lepton (like the tau, shown here) decays, it can decay to two pions through both real (b, c) and virtual (a) processes. There are also scattering (d) effects that can take place. Computing these decays requires understanding the contributions of the strong force (including all quarks and gluons) to the intermediate components of these diagrams, which is difficult for both the tau and the muon.

Over on the theory side, this led to a big initiative to calculate, as accurately as possible, the muon’s magnetic moment, as it would take a lot of effort to get down to the needed level of precision to test whether theory and experiment matched. The hadronic (vacuum and light-by-light) contributions, in particular, were a big challenge. This dovetailed nicely with an early-2000s experiment on the muon’s magnetic moment that was performed at Brookhaven National Laboratory: the E821 experiment. By 2006, the final report came out from Brookhaven’s E821 collaboration, and they were noticing a discrepancy between theory and experiment, although the errors on both were too large to declare a crisis.

This spurred a series of new endeavors on two different fronts.

• On the theory front, efforts to calculate the contributions of all the different channels to the muon’s magnetic moment. In many cases, direct calculations aren’t possible, so they require making extrapolations using experimental inputs of various known processes to estimate those contributions.
• On the experiment front, the magnetic moment experiment would get a huge upgrade: the magnetic apparatus would be moved to Fermilab, and they would reduce the errors and uncertainties on the measurement significantly, to a factor of ~5 greater in precision.

The top light blue line, which overlaps with the yellow line to the left of about 0.75 GeV on the x-axis, represents the combined full hadronic R ratio as determined by experimental inputs for the hadronic vacuum polarization contribution to the muon’s magnetic moment. The various experiments that compute the yellow portion of the curve are not all in mutual agreement with one another.

However, it’s worth pointing out the reason behind why these hadronic contributions are so difficult to calculate. When we have electromagnetic or weak nuclear interactions, we calculate progressively more and more complex interaction diagrams: the more diagrams we calculate (at higher loop-order), the more accurate our calculations get. But for the strong nuclear force, that doesn’t work at all; we have to resort to other means. That’s why the theory initiative resorted to the method of using “experimental inputs” to estimate what the contributions from the strong (hadronic) interactions are. These happen from a variety of channels and cover a range of energies, and are not trivial or straightforward to compute even from good experimental data, particularly for the hadronic vacuum polarization component on the theory side.

There were many who suspected that what appeared to be early hints of mismatch between theory and experiment arose from a faulty theoretical calculation in exactly this regard.

Fortunately, there is a superior method available, although it’s extraordinarily computationally intensive: the method of lattice QCD. In lattice QCD, instead of writing down interaction diagrams and calculating them to ever-greater precision and with ever-greater complexity, you formulate your gauge theory on a grid (or lattice) of points in both space and time. If you could calculate your theory on a lattice that was both infinitely large in size and with an infinitesimally small spacing between adjacent lattice points, you would be able to make theoretical calculations exactly, even for hadronic interactions.

This depiction of a lattice QCD method shows that space and time are discretized into a set of grid-like points on a lattice. As the spacing between points decreases and the overall size of the lattice tends toward infinity, the true value for QCD calculations is approached more and more accurately.

Starting in 2020, right as the Fermilab team began to report their first results and right as the Muon Theory Initiative published their first white paper calculating the theoretical (experimental-input-based) value for the muon’s magnetic moment, teams working on lattice QCD calculations of the hadronic contributions to the muon’s magnetic moment were also finally obtaining meaningful results. Whereas the experimental-input-based method was computing values that were significantly smaller than the Fermilab results, the lattice QCD calculations were much closer to the Fermilab results, and in fact when you included all the uncertainties, overlapped.

This led to a firestorm of speculation, and a number of divergent thoughts from members of the community. Were the lattice QCD results reliable, and would other lattice QCD groups wind up agreeing with these early computational results? Was there a significant mistake in the results coming from the theory groups that were using experimental inputs to compute the muon’s magnetic moment? Was the experiment really even necessary, or was the data it was providing simply going to cause the theory groups, whoever they were, to massage their results until they agreed with data?

This image, composed of two figures from the Muon Theory Initiative’s 2025 white paper, shows at top the differences between theory and experiment depending on which leading order hadronic vacuum polarization input is used. The green results are all r-ratio (experimental data input) inputs, while the blue lines are all lattice QCD inputs. The WP25 designation reflects what’s chosen in the 2025 white paper, with the lower table showing the differences between the 2020 and the 2025 white papers.

That last option — although it wasn’t necessarily an uncommon sentiment — was the opposite of what is actually true. If we hadn’t done the experiment, we would never have known where the root of the “problem” was when it comes to the calculation of the muon’s magnetic moment.

We would have been left with two theoretical “camps,” one using a data-driven approach to the hadronic vacuum polarization and the other using a lattice QCD-driven approach. They would have gotten different answers, and they would have each reported small errors and uncertainties, while the cumulative results, overall, didn’t even overlap. We would have had two predictions, inconsistent with one another, both claiming to be correct, and we would have no experimental data to tell us which one, if either, was correct.

Theory tells you what to expect, but if you have different methods of calculating your predictions and they don’t agree with one another, you have to perform the experiment. Without it, we would have had two results — a full 5-sigma apart from one another — and no way to know who was right. It’s by performing the critical experiment to the necessary precision that we can actually uncover what it is that nature is telling us.

This comparison plot shows the difference between theoretical predictions and experimental results for the muon’s magnetic moment. Data-driven (green) calculations of the hadronic vacuum polarization, as derived from various experiments, are not all consistent with one another, with CMD-3’s results (and results inferred from the tau lepton) differing from all others, being the only one marginally consistent with the experimental results. However, lattice QCD predictions, having improved dramatically here in the 2020s, lead to a great general agreement with the experimental data.

On May 27, 2025, the Muon Theory Initiative group released their second white paper, entitled The anomalous magnetic moment of the muon in the Standard Model: an update. They note that hadronic contributions remain the largest sources of uncertainty, but that three major advances have occurred.

1. For the light-by-light scattering contribution, both data-driven approaches and lattice QCD calculations have improved, cutting the uncertainty in half over the original 2020 white paper.
2. The CMD-3 collaboration, which itself is dedicated to measuring the cross section of electron-positrons converting to two (charged) pions at the VEPP-2000 collider, showed an inconsistency in the leading order contributions to hadronic vacuum polarization, pointing out precisely where the flaw in previous data-driven theoretical predictions were.
3. Meanwhile, the increased precision of lattice QCD calculations allowed that same leading order contribution to hadronic vacuum polarization to be known to incredible precision: to 99.1% accuracy, with an uncertainty of just 0.9%.

As the new theory initiative paper notes, “Adopting the [lattice QCD calculations for the leading order hadronic vacuum polarization] in this update has resulted in a major upward shift of the total SM prediction… which implies that there is no tension between the [Standrard Model] and experiment at the current level of precision.”

This combined plot shows the progression of the measurement of the muon’s anomalous magnetic moment, from Brookhaven’s E821 experiment through Fermilab’s six data runs, along with the amount of data gathered in each run, which has so significantly reduced the total uncertainty. This is a triumph of experimental precision.

And then, on June 3, 2025, it happened: the Muon g-2 collaboration at Fermilab announced their third and final result for their experiment. Remarkably, they got their uncertainties way, way down: down to 127 parts-per-billion, as detailed in their full paper. With the full results from all six runs of their data, we can now definitively see what the story is. Yes, the muon does have a magnetic moment that’s significantly different from the standard quantum mechanical prediction: because of quantum field theory. There are contributions from the electromagnetic, weak, and Higgs sectors that can be computed through conventional field theory means.

However, the hadronic (strong force) contributions are tricky, and inferring what they are by using data inputs is tricky, and has led to theorists fooling themselves in the recent past. Using lattice QCD contributions, where applicable, has greatly reduced those errors, and increased the precision and confidence of our predictions. At the same time, the final, most-precise experimental results ever have come in, and now we know for certain what’s going on: we know what the answer is and we also know how to calculate it. The muon g-2 puzzle has been resolved at last, and our understanding of physics, at not just a fundamental but a practical level, is deeper than ever before.