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Top Comments: The Standard Model Survives the Muon Kerfuffle [1]
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Date: 2025-06-08
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Way back in 2021, I wrote a TC diary (Muon Kerfuffle Edition) about an effort to measure what’s called the magnetic moment anomaly of an elementary particle called the muon. The reason it was interesting is that the experimental measurement of this property did not agree with the theoretical calculation of it. Physicists viewed this disagreement as a possible opportunity to find a crack in the Standard Model, the theory used to explain particle physics, hence revealing new physics. Now, four years later, comes the announcement that the discrepancy between experiment and theory has disappeared.
Here’s a brief overview so nobody will have to go back and read the old diary. The muon is an unstable elementary particle (with a half-life of 2.2 microseconds) that is like the electron in its physical behavior except that it is significantly more massive. (The muon is approximately 207 times heavier than the electron.) Fast-traveling muons are present in the environment as they are created by collisions of cosmic rays with the nuclei of atoms in the upper atmosphere.
Like electrons, muons have a unit negative charge, as well as a magnetic moment associated with its spin. The first-order value one obtains for the magnetic moment of the electron (symbol g), in units of the Bohr magneton, is equal to -2 exactly. However, careful experimental measurement reveals that these magnetic moments are ever so slightly larger (i. e. more negative) than -2. The difference is the magnetic moment anomaly, given the symbol g — 2,* and it is caused by interaction of the particle with virtual photons, resulting in a slight “wobble” in its motion, which in turn increases the value of the magnetic moment very slightly. How slightly? The current best value of the electron’s magnetic moment is −2.00231930436092. The great triumph of the theory of quantum electrodynamics (QED) was its ability to reproduce this value to within one part in a trillion.
It’s relatively easy to make this measurement on electrons because they don’t decay. Performing this measurement on muons is much trickier due to their short lifetimes. However, over the course of twenty-plus years, starting at Brookhaven in New York, and then moving to Fermilab in Illinois, a precise value was determined. As a matter of fact, the latest value for the muon’s magnetic moment anomaly was published on June 3 of this year, just last week, with an uncertainty of 127 parts per billion.
Unlike for the electron, however, the theoretical value (from 2021) did not agree with the experimental value, by a tiny but persistent, and fairly precise 2.5 parts per billion. If this difference was real, it suggested that there might be a flaw in the Standard Model, which meant there is a previously unknown physical process, possible new particles, that the Standard Model does not account for. Given that the Standard Model, from its inception, was regarded as provisional and temporary when it was devised more than 40 years ago, theorists had been hoping that some phenomenon would violate it so that a better and more complete theory could replace it. Perhaps this small difference was the door to a new theory?
Or, perhaps there was a mistake in the calculation? A second calculation of the muon magnetic moment anomaly was also published in 2021 shortly after the newly measured value of g — 2 for the muon appeared, and that new theoretical value actually agreed with experiment.
So how did the two theoretical calculations differ? It had to do with the treatment of one specific contribution to the anomaly involving the strong nuclear force, called the hadronic vacuum polarization (HVP). Hadrons are particles that respond to the strong nuclear force, such as protons and neutrons. These particles are capable of winking in and out of existence for short periods, just like virtual photons, but the contribution is small compared to other forces because the masses of these virtual particles is so large. Calculations involving the strong force from first principles are extremely hard to do because the strong force is so strong. The standard technique applied for such calculations, called perturbation theory, is to start with the result from a simplified, but exactly solvable model of the particle, and then add in the missing interactions in order of the number of virtual particles involved in the interaction. For the electromagnetic force (virtual photons), for example, each successive correction gets progressively smaller, so when you’ve added enough of them, the number eventually converges.
This is not what happens when perturbation theory is applied to the strong force. The terms don’t necessarily get smaller, and if they do, they get smaller at a very slow rate, and are unlikely to converge on a single value in a reasonable amount of time. So some other strategy must be used to calculate the HVP. What was used in the original calculation of g — 2 for the muon was to base the contribution from HVP on experimental data. That strategy worked for the electron, but as the contribution of HVP to the muon’s anomaly is larger, it may not work in that case.
The second calculation applied a totally new strategy to calculate the HVP from first principles using a lattice of points to represent the space in which the effect occurs. This approach has only been possible to apply since the rise of the supercomputer. While the result of this calculation agreed with the experimental measurement, it was not viewed as trustworthy by the physics establishment in 2021 because it was the first time this strategy had ever been applied, and so it wasn’t clear how reliable it was. Four years later, the researchers have tested, refined, and corrected their program, and their result for the muon g — 2 continues to agree with the experimental result. Most physicists have relented, recognizing that the Standard Model still works to accurately describe the muon magnetic moment anomaly, so there is no new physics here. We’re still stuck with the Standard Model. The one remaining question is why the first calculation strategy, based on experimental results, didn’t work.
*A note on notation: g is a negative number, due to the fact that both the electron and the muon are negatively charged. However, in the expression for the anomaly, g — 2, both g and the anomaly are treated as though they are positive. I don’t know why they do this. It’s inconsistent and confusing.
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