Unstable helium adds a limit on the ongoing saga of the proton’s size

Physicists, who dedicate their lives to studying the topic, don’t actually seem to like physics very much since they’re always hoping it’s broken. But we’ll have to forgive them; finding out that a bit of theory can’t possibly explain experimental results is a sign that we probably need a new theory, which is something that would excite any physicist.

In recent years, one of the things that has looked the most broken is a seemingly simple measurement: the charge radius of the proton, which is a measure of its physical size. Measurements made with hydrogen atoms, which have a single electron orbiting a proton, gave us one answer. Measurements in which the electron was replaced by a heavier particle, called a muon, gave us a different answer—and the two results were incompatible. A lot of effort has gone into eliminating this discrepancy, and it has gotten smaller, but it hasn’t gone away.

That has theorists salivating. The Standard Model has no space for these kind of differences between electrons and muons, so could this be a sign that the Standard Model is wrong? The team behind some of the earlier measurements is now back with a new one, this one tracking the behavior of a muon orbiting a helium nucleus. The results are consistent with other measurements of helium’s charge radius, suggesting there’s nothing funny about the muon. So the Standard Model can breathe a sigh of relief.

Measuring muons?

The measurement involved is, to put it simply, pretty insane. Muons are essentially heavy versions of electrons, so substituting one for another in an atom is relatively simple. And a muon’s mass provides some advantages for these sorts of measurements. The mass ensures that the muon’s orbitals end up so compact that its wave function overlaps with the wave function of the nucleus. As a result, the muon’s behavior when it is orbiting a nucleus is very sensitive to the nucleus’ charge radius.

All of this would be great if it weren’t for the fact that muons are unstable and typically decay in under two microseconds. Putting one in orbit around a helium nucleus adds to the complications, since helium typically has two electrons in orbit, and they can interact with each other. The expected three-way interactions of a nucleus-muon-electron are currently beyond our ability to calculate, meaning we would have no idea if the actual behavior differed from theory.

So the researchers solved this problem by creating a positively charged ion composed of a helium nucleus and a single muon orbiting it. Making one of these—or, more correctly, making hundreds of them—is where the insanity starts.

The researchers had access to a beam of muons created by a particle collider, and they decided to direct the beam into some helium gas. In this process, as the muons enter, they have too much energy to stay in orbit around a helium nucleus, so they bounce around, losing energy with each collision. Once the muons slow down enough, they can enter into a high-energy orbit in a helium atom, bumping out one of its electrons in the process. But the second electron is still around, messing up any potential measurements.

But the muon has a lot of momentum because of its mass, and energy transfers within an atom are faster than losing the energy to the environment. So as the muon transfers some of its energy to the electron, the electron’s smaller mass ensures that this is enough to boot the electron out of the atom, and we’re left with a muonic helium ion. Fortunately, all of this happens quickly enough that the muon hasn’t had a chance to decay.

Let the insanity begin

By this point, the muon is typically in an orbital that is lower energy but has more energy than the ground state. The researchers set up a trigger sensitive to the appearance of muons in the experiment. After a delay to allow the muons to boot out the two electrons, the trigger causes a laser to hit the sample with the right amount of energy to boost the muon from the 2S orbital to the 2P orbital. From there, it will decay into the ground state, releasing an X-ray in the process.

Many of the muons won’t be in a 2S orbital, and the laser will have no effect on them. The researchers were willing to sacrifice much of the muonic helium they made in order to get precision measurements of the ones that were in the right state. Their presence was signaled by the detection of an X-ray with the right energy. To further ensure they were looking at the right thing, the researchers only took data that was associated with a high-energy electron produced by the decay of the muon.

And remember, all of this had to take place fast enough to happen within the microsecond time window before the muon decayed.

The first step involved tuning the laser used to the right frequency to boost the muon into the 2P orbit, since this is the value that we need to measure. This was done by adjusting a tunable laser across a frequency range until the helium started producing X-rays. Once the frequency was identified, the researchers took data for 10 days, which was enough for precision measurements of the frequency. During this time, the researchers observed 582 muonic helium ions.

Based on calculations using the laser frequency, the researchers found that the helium nucleus’ charge radius is 1.6782 femtometers. Measurements made by bouncing electrons off the nucleus indicate it is 1.681. These two values are within experimental errors, so they’re in strong agreement.

We’re sorry, it’s not broken

On the simplest level, the fact that the muon measurements agree with measurements made independently indicates that there’s nothing special about muons. Consequently, the Standard Model, which says the same thing, is intact down to fairly small limits allowed by the experimental errors here. (That is not to say it’s not broken in some other way, of course.) So theorists everywhere will be disappointed.

As an amusing aside, the researchers compared their value to one generated decades ago in the particle accelerators at CERN. It turns out this value is similar, but only by accident, since the earlier work had two offsetting errors. “Their quoted charge radius is not very far from our value,” the researchers note, “but this can be traced back to an awkward coincidence of a wrong experiment combined with an incomplete 2P–2S theory prediction, by chance yielding a not-so-wrong value.” So in this case, two wrongs did make an almost-right.

In any case, this work will focus researchers’ attention back onto trying to figure out why different experiments with protons keep producing results that don’t quite agree, since we can’t blame things on the muon being weird. In the meantime, we can all appreciate how amazing it is that we can manage to do so much with muons within the tiny fraction of a second in which they exist.

Nature, 2021. DOI: 10.1038/s41586-021-03183-1  (About DOIs).