Muon G-2 Experiment Results Explained (Patreon)

When a theory makes a prediction that disagrees with an experimental test, sometimes it means we should throw the theory away. But what if that theory has otherwise produced the most successful predictions in all of physics? Then, that little glitch may be pointing the way to layers of physics deeper than we've yet imagined. Well, FermiLabs Muon G-2 experiment has been chasing the most promising glitch of all, and they've just announced their results.

The standard model of particle physics describes the elementary building blocks of nature with incredible success. At some level it’s “right” in a very fundamental way. But it’s not the whole picture - for one thing, it doesn’t explain gravity. In fact it doesn’t play nice with Einstein’s entire theory of gravity, which itself is clearly right in its own way. Our search for a theory of everything which will bring these theories together is perhaps the next great quest in physics.

To find the path forward, we need to find glitches in these theories - loose threads that might lead us to deeper layers of physics. We don’t have many leads - these theories are woven pretty tight. But there is one glitch, one stray thread, that is just begging to be tugged - that’s the anomalous magnetic moment of the muon. And the scientists at FermiLab have just tugged it hard with the g-2 experiment. Today we’re going to see what has been unraveled, and what might lie beneath.

But first what’s all this technobabble? What’s an anomalous magnetic dipole moment? What’s a muon? What’s g-2? Let’s take them one at a time. We’ve actually talked about the anomalous magnetic moment before - in terms of the electron. That’s an episode well worth checking out for a deeper dive. But briefly - of all the incredible successes of the standard model, this seems the most miraculous. It comes from the part of the standard model that describes how particles with electric charge interact via the electromagnetic force - quantum electrodynamics. One of the interactions that QED describes is how a charged particle will tend to rotate to align with a magnetic field. The strength of that interaction is

defined by something called the g-factor for the particle - that’s the g in our g-2, and we’ll come back to it later. QED predicts a value for the electron’s g-factor that matches experimental measurements to 10 decimal places - by far the most accurate prediction in all of physics.

If this works so well for electrons, surely it works for other particles. Well, actually not so much. The muon is a close cousin to the electron - identical in all properties besides its heavier mass. Starting 20 years ago, experimental measurements of the muon g-factor do NOT agree with the QED calculation. And that’s not because QED is just wrong. Rather, it tells us that the

calculations have missed something - and that something may be physics beyond the standard model.

Let’s start by talking about quantum spin. We’re actually going to be doing a deep dive into this topic soon. Today we’ll just do what we need to get by. Every particle with electric charge also has quantum spin. This isn’t the same thing as simple rotation, but particles with quantum spin do generate a magnetic field. Same as if you send an electric charge around a looped wire, or have electrical currents in the Earth’s spinning core. The result is a dipole magnetic field, with a north and south pole. Place an object with such a field inside a second magnetic field and the object will tend to rotate to align with that field. The strength of that rotational pull, or torque, is defined by the object’s dipole moment. For a rotating charge, that depends on the object’s rate of rotation or angular momentum, its charge, and its mass. Here’s the equation for the classical dipole moment - for a non-quantum rotating charge.

Yes, this will be on the test, and it’s also useful for the next thing I’m going to say.

An electron also has a dipole field and a dipole moment, which depends on the electron’s spin, charge and mass. But the electron dipole moment is different to the classical one by this factor g. For the electron, g is around 2 - so the electron responds to a magnetic field twice as strongly compared to what you’d expect for an equivalent classical rotating charge.

Quantum electrodynamics tells us exactly where this difference comes from. To understand this, let’s look at the QED picture of the world. In this theory, electromagnetic interactions result from charged particles communicating by exchanging virtual photons. In QED you figure out the strength of an interaction by counting up all the ways that this interaction could occur. For example, a pair of electrons could repel each other by exchange one virtual photon, or two, three, et

Or those virtual photons could do weird things like momentarily becoming an electron-positron pair

We depict these interactions in Feynman diagrams. Each Feynman

diagram represents one family of ways that the interaction could proceed, and the sum of all possible Feynman diagrams gives you the interaction strength.

For a deeper dive into Feynman diagrams, virtual particles, and quantum electrodynamics, we have you covered - episode list in the description.

We can represent an electron interacting with a magnetic field with the simplest possible Feynman diagram - really not even a full Feyman diagram. We have an electron being deflected by a single photon from that field

If you calculate the g-factor from just this simplest case, you get a value of exactly 2. Paul Dirac first calculated g=2 from his eponymous equation. But there are other ways this interaction can happen. The next simplest is for the electron to emit a virtual photon just prior to absorbing the magnetic field photon, and then reabsorbing the virtual photon. Same particles in and out, but a slightly more complicated sequence of events

Adding this interaction allowed Julian Schwinger to calculate a slightly higher value of g=2.0011614  Over time, more and more complicated interactions were added. Each layer of complication involved many more Feynman diagrams, but also added less and less to the interaction. The latest calculations rely on powerful computers to add many thousands of Feynman diagrams and get us our g-factor to 12 significant figures, and a latest calculated electron g-factor of 2.001159652181643

By the way, the “anomalous” in the anomalous magnetic dipole moment refers to that little bit extra above 2.

So the g-2 in the Fermilab  experiment’s name refers to that leftover bit. Measure that leftover bit, and you’re testing the subtlest interactions of the particle. For the electron the current measurement is precise to 1 part in a billion - and to that degree it agrees perfectly with the theoretical number.

An obvious next step is to do the same test for other particles. The electron is the lightest and most common of the lepton family. It has two heavier cousins - the muon and the tau particle. Muons are nice because they’re easily produced in radioactive decay. They live only a few microseconds, but that’s long enough to work with them. During their brief existence they’re very similar to electrons: they have the same exact charge, interact with the same forces, and have the same quantum spin. They have a different g-factor because there are slightly different ways that the muon can interact with the quantum fields.

So you add up the Feynman diagrams for all possible electromagnetic interactions of the muon. But you don’t stop there - the quantum vacuum is seething with an incredible variety of possible virtual particles. There can be very subtle interactions that involve the other forces - weak, strong, and even the Higgs field. All of the tweaking the muon’s g-factor by tiny degrees. And when we include every possibility encompassed by the standard model of particle physics, we get a g-factor that’s ever so slightly off the experimental results.

So why would we get the wrong value for the muon but not the electron? Well, I can tell you what physicists HOPE is the reason. The muon is 200 times more massive than the electron. The probability of interaction between a particle and a massive virtual particle is proportional to mass squared - so the muon is 40,000 times more likely to be perturbed in this waIt’s 40,000 times more likely than the electron to encounter virtual Higgs bosons, virtual protons or other hadrons, and 40,000 times more likely to encounter any completely unknown virtual particles that might be hiding out there. Accounting for all the known particles still gives a muon g-factor that’s off - so the rising hope is that an as-yet unknown particle is at work here.

So finally we get to the FermiLab g-2 experiment. Prior to this experiment, various labs over the past 20 years have refined the muon g-factor measurement. Until now, it had been measured precisely enough to claim a 3.7-sigma difference compared to theory - that was at Brookhaven National Laboratories in 2001. There’s roughly a 1 in 10000 chance that random fluctuations could lead to that degree of difference just by chance. Physicists prefer a 5-sigma signal before declaring a discovery. For 5-sigma, there’s only a 1 in 3.5 million chance of random noise resulting in the same signal.

The Muon G-2 experiment at Fermilab hopes to push closer to this level of confidence, and is designed to achieve 4 times the sensitivity of the Brookhaven experiment. At the Fermilab experiment, physicists send muons flying at nearly the speed of light around a 50 foot diameter magnetic tube. The muons interact with the magnetic field and their own magnetic dipole axes rotate - like a top just before it falls. We call this Larmor precession, and its frequency depends on the g-factor. The frequency of precession also governs the energy of the particles that these muons decay into. So by measuring the energies of those particles - positrons in particular, the researchers can determine the precession rate, and so measure the g-factor. So what did they find?

So there you have it. That’s how we peer beneath the hood of reality. We scratch our heads and scrawl on chalkboards for about a hundred years. Then we build a giant magnet and watch the muons dance. And that dance may just have revealed to us the next step on our path to a more complete understanding of this quantum spacetime.