Abstract
In April 2021, Physicists at Fermilab made an ultra-precise measurement of the magnetic properties of an ephemeral particle called a muon. This long-awaited announcement could well be crucial evidence that our understanding of the subatomic world is incomplete. This article takes a deep dive into what we know and what we don’t about this very exciting Muon g-2 (pronounced "gee minus two") experiment.
The g factor
So, what was the measurement? Scientists used a fifty-foot diameter ring of very uniform magnets to measure the magnetic properties of the muon, which is a heavy cousin of the electron. Basically, they were trying to measure how strong a magnet the muon is. The key measurement is a factor called g, short for gyromagnetic ratio, which, if you ignore a few constants, is basically the ratio of the magnetic strength of a muon compared to how much it’s spinning.
Using a version of the state-of-the-art 1930s quantum mechanics, g should equal exactly two. But in 1948, scientists announced a super-precise measurement of g for electrons and found that it wasn’t exactly two. Instead, g was equal to 2.00238 or 0.1 percent higher. Shortly after the measurement was reported, physicists devised a more complete theory of quantum mechanics that agreed with this measurement. This theory is called quantum electrodynamics, or QED.
The tiny extra bit of magnetic strength comes from an amazing source. It turns out that the strength of the electric field close to an electron is so strong and it contains enough energy that the energy converts into pairs of matter and antimatter particles. Those pairs then convert back into energy quickly. And more than one pair appears at a time. At a subatomic level, space near a particle like an electron or a muon looks like a swarm of fireflies, blinking into and out of existence. This cloud is important because the interactions between the electron and the cloud slightly enhance the magnetic properties of the electron. What scientists actually measure is a combination of the electron and the cloud. And the cloud is the source of that extra tenth of a percent.
Over the last five decades, scientists have measured g for muons with increasing precision. By 2006, researchers at Brookhaven National Laboratory had measured the muon’s g to be :
Meanwhile, theorists were calculating the same quantity, and they achieved a similar number with a similar level of precision. Their result back in 2006 was:
Alpha
So the first thing you’ll notice is that both the measurement and the theory have lots of digits and very small uncertainties, meaning that they're very precise. In addition, the measurement and the theory agree with each other, digit by digit for eight digits and disagree only in the ninth. For two things to disagree in the ninth digit means they agree to a couple of parts per billion. That’s like someone predicting the circumference of the Earth with a precision of ten centimetres. This level of agreement is pretty impressive, but all those digits can be distracting when we’re trying to understand the recent Fermilab measurement. So, let’s change how I’m presenting them to make them easier to understand. We just want to concentrate on what fraction of the muon's g comes from that cloud of particles surrounding it. That means we need to get rid of the part from ordinary quantum mechanics and we also need to divide our multi-digit numbers by the old-school quantum number, which I remind you is two. You can see how that works below.
That’s a lot of words, so I can say it a bunch easier if we remember that the gyromagnetic ratio is written as g and the old-school prediction is just 2. When you do that, you get the equation:
Now you might wonder why exactly are we subtracting g with 2 and dividing again with two well here is the reason. The Dirac equation predicts the value of 'g' (g-factor, a kind of magnetic moment) for the electron to be exactly 2. However, the measured value differs ever so slightly from 2, due to the contribution of quantum fluctuations in a vacuum which the Dirac equation doesn't consider. So this anomaly in the g-factor can be measured by subtracting the predicted value from the actual value, hence the (g-2) term. Then, this is divided by 2 as a convention in statistics where any difference in similar values is treated as a class interval and halved to find the class mark.
For no good reason, scientists call that quantity alpha or anomalous magnetic moment of a particle. If we do all of that, we can rewrite our measurement and prediction for the part of the muon’s gyromagnetic ratio caused by modern physics for both the experiment and the prediction. In the year 2006, the state-of-the-art numbers for alpha was:
We’re getting somewhere, but to understand the scientific situation, we need to focus on where the two disagree. So, the easiest way to do that is to simply erase the number that are the same in experiment and theory and just keep the ones where they disagree. If we do that, we’re left with a way more manageable set of numbers:
Why do the two measurements disagree
The image above shows the prediction and measurement as dots, with lines representing their uncertainties. The first thing we notice is that the two dots are far apart, and the uncertainties are small compared to how far apart they are. Basically, the lines don’t overlap. This sort of situation can mean a couple of things:
First, the measurement or prediction can simply be wrong. That happens all too often, even when researchers try to make sure that they get the numbers correctly.
The second and more interesting explanation is that there are some physical phenomena that the prediction just doesn’t include. If that’s true, then the discrepancy means discovery and that we need to come up with an improved theory. That’s really exciting.
What I’ve talked about so far is the situation back in 2006. What's happened since then? Well, theoretical physicists have revisited the prediction and found that it’s basically sound. They made some tiny changes, but nothing of substance. So, eyes turned to the measurement. Could have the researchers at Brookhaven Lab made a mistake? Well, in 2021, researchers at Fermilab repeated the measurement. So, what’s the answer? It turns out that the Brookhaven scientists did a good job. The Fermilab and Brookhaven measurement agreed pretty well. And now we’re in an exciting place. If the theoretical calculation is sound and the measurement is accurate, we could be looking at a discovery.
Now, what do we do? Well, we can combine the Fermilab and Brookhaven measurements into a single experimental result. That should get both a more accurate and precise measurement. Okay, now we’ve come to the place where we can start discussing the bottom line.
What does the g-2 experiment tell us?
Well, first - let’s be honest, we don’t know. Nobody has a definitive answer. All we know are the possibilities. There are two big classes, one exciting and one humbling.
Lattice QCD
On the same day that Fermilab scientists announced their amazing measurement, a paper was published in the prestigious journal Nature. It made a different prediction for the gyromagnetic ratio of the muon. Let’s take a step back and consider how the prediction is done. Let’s briefly return to the full theoretical number for the muon’s gyromagnetic ratio and sort of unpack it.
Basically, it’s a series of numbers that get smaller and smaller as you go to the right. The first 2 is handled by old-time quantum mechanics. The zeros mean that nothing is contributed by effects that are 10% or 1% of what the 1930s quantum predictions cover. The second 2 says that a 0.1% size effect matters and the first 3 says there is a contribution from a 0.01% effect, and so it goes.
When one considers what contributes to the correction to the gyromagnetic ratio due to the cloud of particles surrounding the muon, it’s easy to calculate bigger effects, like photons and electrons and antimatter electrons. These are the things that cause that tenth of a percent addition. But smaller effects include the case when the muon’s electric field creates a photon that then makes a quark and antimatter quark. It’s a tiny contribution to the gyromagnetic ratio, and furthermore, it’s also hard to calculate precisely. In traditional calculation, physicists estimate the result using other measurements from other experiments. The resulting prediction gives a discrepancy between measurements and predictions. However, the new paper published in Nature takes a fresh approach. Rather than estimating the effect on the muon’s magnetic moment from quarks, these researchers try to calculate it by a brute force method called lattice QCD. Basically, they set up a three-dimensional grid and use supercomputers to calculate how the equations governing the strong force predict how these grid points interact. It takes a lot of computer power, but the approach has had some success in other areas of physics.
The lattice QCD calculation doesn’t agree with the earlier theory calculations. In fact, it agrees better with the experimental measurement recently released by Fermilab scientists. So that could be a big letdown. Maybe there never was tension between data and prediction. Maybe the early prediction was just wrong. But it’s too soon to conclude that. For instance, the uncertainty quoted by the lattice QCD researchers originates from how certain they are of their methodology. In short, they’re not entirely certain that they’ve approached the problem completely correctly. Now, this doesn’t mean that they made a mistake. After all, they are excellent scientists. But it shows you how hard it is to do calculations involving quarks. The bottom line is we need to be careful about drawing conclusions. Luckily, there are theorists and computer professionals in the world who can also try to reproduce the lattice QCD calculation. It will probably take a year for them to announce their results.
A new Physics
So, putting aside the lattice QCD calculation for a moment, and assuming that the new measurement and the old way of predicting the muon’s gyromagnetic ratio are correct, what could explain that? Well, that’s where things can be exciting. It means that new physics is required.
What might that new physics be?
Unfortunately, the data can’t answer that. Maybe there are low mass particles that interact very rarely with muons. Or maybe there are high mass particles that interact more frequently with muons. Such a range of behaviours is a bit frustrating, but it’s similar to being in the situation where somebody made a measurement of an object’s density, but people want to know how big it is and what its mass is. Density is just mass divided by volume, so any specific density could be a low mass object with a small size or a high mass object with a big size. A density measurement alone won’t tell you either the size of the mass. But it would tell you that a huge mass and a small size is forbidden by the data. So, you know something. Similarly, if the prediction and measurement of the muon’s gyromagnetic ratio continue to disagree, it doesn’t give us an exact prediction of what the new physics will be.
Conclusion
So, where do we stand? Well, we stand where scientists often stand where we know a lot, but not enough, and we have more and more questions. If the discrepancy persists, we know that there is some sort of new physics and we have some information about what is possible. On the other hand, if the new theoretical prediction from lattice QCD is validated, it may be that what we’ve found is that the existing theory does a good job predicting this property of the muon. Now you might ask when we’ll know the answer, and for that, you’re going to just have to wait. The experimenters at Fermilab will be recording something like 16 times more data than they’ve reported so far. So, the measurement will improve, and we’ll learn more in a year or two. And getting other groups to reproduce the lattice QCD calculations will take a similar amount of time. And that means we just have to wait to see what the future brings. But there is no question that there is enormous interest in what the Fermilab Muon g-2 experiment will tell us about the laws of nature. It’s incredibly, incredibly exciting.
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