Muons continue to confound physicists. These unstable subatomic particles are much like familiar electrons, only with 200 times the mass and a fleeting lifetime of just 2.2 microseconds. Unlike electrons, however, muons are at the center of a tangled inquiry into the prevailing theory of particle physics.
For decades, physicists have puzzled over tantalizing hints that muons are more sensitive to magnetic fields than theory says they should be: run muons in circles around a powerful magnet, and they “wobble,” decaying in a different direction than expected. This apparent discrepancy in the muon’s “magnetic moment” has been significant to physicists because it could arise via nudges from undiscovered particles that are unaccounted for by current theory. But the discrepancy could just as well have been a statistical fluke, an experimental uncertainty or a product of various potential errors in theorists’ arcane calculations. Making progress on this vexing problem boils down to better calculations and more precise measurements of the muon’s magnetic moment.
On Thursday researchers announced the latest measurement milestone, which pins down the muon’s magnetic moment to an error of just one part in five million. The paper reporting their results, which has been submitted to the journal Physical Review Letters, was based on two years of data taken at the Muon g−2 experiment, a 50-foot-wide magnetic ring of circulating muons located at Fermi National Accelerator Laboratory in Batavia, Ill. (Disclosure: The writer of this story is related to Robert Garisto, managing editor of Physical Review Letters. They had no communications about the story.) The new result confirms and doubles the precision of a previous experimental measurement in 2021, banishing doubts about the Muon g−2 experiment’s reliability.
“The experiment has really done its job,” says Dominik Stöckinger, a theorist at the Dresden University of Technology in Germany, who is also part of the Muon g−2 collaboration. He praises his colleagues for the increase in precision, and other scientists agree.
“The g−2 measurement is a fantastic achievement.... It’s very difficult stuff with very high precision,” says Patrick Koppenburg, an experimental physicist at the Dutch National Institute for Subatomic Physics, who was not involved in the research.
Despite the recent experimental success, theory-based problems remain. In the subatomic realm, the Standard Model reigns as the current theory of fundamental particles and their interactions. But the Standard Model leaves physicists unsatisfied; it doesn’t explain phenomena such as dark matter or mysteries such as the surprisingly low mass of the Higgs boson. Such limitations have pushed researchers to hunt for as-yet-undescribed new particles within the Standard Model—ones that could subtly influence the muon’s behavior in ways theory does not predict.
Spotting disagreements between theoretical predictions and the results of experiments like Muon g−2 requires extraordinary precision on both sides. But right now theorists can’t agree on a sufficiently precise prediction for the muon’s magnetic moment because of conflicting (but equally plausible) results from disparate ways to calculate it. And without a consensus, high-precision theoretical prediction, a meaningful comparison with the Muon g−2 experiment’s results is effectively impossible.
“You can only call it an anomaly once there is an agreement on what the Standard Model prediction is,” Koppenburg says. “And presently that seems not to be the case.”
Muon Math
Nearly a century ago the theorist Paul Dirac calculated a value, called g, for how much a charged particle should be affected by a magnetic field. Dirac said g should be exactly 2. (This is where “g−2” comes from.) But over the next two decades, experiments found that the electron’s so-called g-factor was not quite 2—it was off by about a tenth of a percent. The small difference would change the way physicists understood the universe.
In 1947 another eminent theorist, Julian Schwinger, worked out what was happening: the electron was being jostled by the photon. This photon was “virtual”—it was not really there but affected the electron with the photon’s potential to pop into existence, nudge the electron and disappear. The realization transformed particle physics. No longer could the vacuum of space be considered truly empty; instead it was brimming with a dizzying assortment of virtual particles, all of which conveyed a slight influence.
“As they pop into existence, [virtual particles] bounce off the muon. They cause it to wobble a bit more, and then they disappear again,” says Alex Keshavarzi, a theorist and experimentalist at the University of Manchester in England, who is part of the Muon g−2 experiment. “And you basically sum them all up.”
This is easier said than done. Physicists must calculate the remote possibility that the muon interacts not with one but up to five photons popping in and out of existence before continuing on its way. Diagrams of these unlikely events require onerous calculations and resemble abstract art, with arcane loops and squiggles representing hosts of virtual interactions.
Not all calculations of virtual particles can be exactly solved. Although it’s relatively straightforward to compute the influence of virtual photons, muons are also affected by a class of particles called hadrons—clumps of quarks bound together by gluons. Hadrons interact recursively with themselves such that they create what physicists call a “hadronic blob,” which in simulations resemble less abstract art and look more like a tangled ball of yarn. Hadronic blobs defy precise, clean modeling. Stymied researchers have instead tried to refine their models of virtual hadronic blobs with data harvested from real ones produced by collisions of electrons in other experiments. For decades, this data-driven approach has allowed theorists to make predictions about otherwise intractable contributions to the muon’s behavior.
More recently, theorists have begun using a new tool to calculate hadronic blobs: lattice quantum chromodynamics (QCD). Essentially, by plugging the equations of the Standard Model into powerful computers, researchers can numerically approximate the mess of hadronic blobs, cutting through the subatomic Gordian knot. In 2020 about 130 theorists pooled their efforts into the Muon g−2 Theory Initiative and combined parts of both techniques to make the most precise prediction of the muon’s magnetic moment to date—just in time for an experimental update.
Clashing Calculations
To measure the muon’s magnetic moment, physicists at the Muon g−2 experiment begin by funneling a beam of muons into a storage ring around the 50-foot magnet. There, a muon does thousands of laps in the span of a few microseconds before it decays. Recording when and where the decay takes place gave the researchers an experimental answer to how much the muon wobbled because of its interactions with virtual particles such as photons and hadronic blobs.
In 2021 the collaboration measured the muon’s magnetic moment to a precision of one part in two million. At the time, the discrepancy between theory and experiment was, in particle-physics parlance, 4.2 sigma. This means that in one out of every 30,000 runs of the experiment, an effect so large should show up from random chance (assuming it is not caused by “new physics” beyond the Standard Model). That’s roughly equivalent to getting 15 heads in a row on tosses of a fair coin. (This does not mean the result has 30,000-to-one odds of being true; it’s simply a way for physicists to keep track of how much their measurements are ruled by uncertainty.)
Since then the ever shifting landscape of theoretical predictions has been roiled by clashing results and updates. First came a lattice QCD result from the Budapest–Marseille–Wuppertal (BMW) collaboration. Using an enormous amount of computational resources, the BMW team made the most precise calculation of the muon’s magnetic moment—and found it disagreed with all other theoretical predictions. Instead it agreed with the experimental value measured by Muon g−2. If BMW is correct, there’s no real disagreement between theory and experiment, and that anomaly would essentially vanish.
None of the half-dozen other lattice QCD groups have fully corroborated the BMW prediction, but initial signs suggest that they will, according to Aida X. El-Khadra, a physicist at the University of Illinois at Urbana-Champaign and chair of the Muon g−2 Theory Initiative. “The lattice QCD community is now in agreement on a small piece of the calculation, and I’m confident we’ll get there for the entire calculation,” she says.*
But if it has solved one discrepancy—between theory and experiment—BMW may have created another. There is now a sizable difference between lattice QCD predictions and the data-driven ones derived from empirical experiments.
“A lot of people would look at that and say, ‘Okay, that weakens the new physics case.’ I don't see that at all,” Keshavarzi says. He believes the discrepancy within the theory result—between the lattice and data-driven methods—could be linked to new physics, such as an as-yet-undetected low-mass particle. Other researchers are less gung ho about such heady prospects. Christoph Lehner, a theorist at the University of Regensburg in Germany and a co-chair of the Muon g−2 Theory Initiative, says it is much more likely that the theoretical discrepancy is caused by problems in the data-driven method.
In February another curveball hit the community, this time from the data-driven side: A new analysis of data from an experiment called CMD-3 that is based in Novosibirsk, Russia, agreed with the BMW result and the experimental value. “No one expected that,” Keshavarzi says. If CMD-3 were found to be correct, there would be no discrepancy in theory—or between theory and experiment. But CMD-3 doesn’t agree with any of the previous results, including those of its predecessor, CMD-2. “There is no good understanding for why CMD-3 is so different,” El-Khadra says. Within a year or two, she expects more data-driven and lattice results, which she and her peers hope will sort out some of this increasingly unwieldy mess.
What began a century ago as a nice, even number—g=2—has now spiraled into a task of monstrous precision and fractal complexity. There is not even a clear anomaly between theory and experiment. Instead there is disagreement between the lattice and data-driven theoretical methods. And with the BMW and CMD-3 results, there is further conflict within each method.
For better or worse, this is what a frontier of 21st-century particle physics looks like: a messy back-and-forth as physicists desperately searching for breakthroughs compete to see who can most meticulously measure muons.
*Editor’s Note (8/10/23): This paragraph was edited after posting to better clarify Aida X. El-Khadra’s comments about the Budapest–Marseille–Wuppertal (BMW) collaboration’s lattice quantum chromodynamics (QCD) result.