How big is a proton? The puzzle that began with muons at PSI

What the size of a proton means

A proton has no sharp edge, like a billiard ball. Its electric charge is spread out, and physicists describe the size of that cloud with a single number, the charge radius. It is measured in femtometres: one femtometre is 10⁻¹⁵ metres.

Until 2010 the radius came from two kinds of experiment. One fired electrons at protons and looked at how they scattered. The other measured the energy levels of ordinary hydrogen with lasers, because an electron that spends time close to the proton feels its size, and the energy levels shift very slightly. Both gave about 0.88 femtometres. The value recommended in 2006 by CODATA, the international task group that sets the fundamental constants, was 0.8768 femtometres.

Why a muon helps

The effect of the proton’s size on hydrogen is tiny, because the electron spends little time near the nucleus. A muon is a heavier relative of the electron, about 200 times heavier. If it takes the electron’s place, it orbits much closer to the proton, and the energy levels become far more sensitive to the proton’s size.

The difficulty is that muons do not last. A muon decays within about two millionths of a second, so the hydrogen atom has to be formed and measured within that time. The experiment needed a very intense muon beam, and PSI writes that its muon source is the only place in the world able to supply one intense enough. The team built all the equipment from scratch, including a specially developed infrared laser that could be fired within moments of a muon arriving. According to PSI the idea had been put forward there some 30 years earlier, and the project took more than ten years from start to result.

2010: a smaller proton

The result was published in Nature on 8 July 2010 by a collaboration of institutions in Switzerland, Germany, France and Portugal. The proton came out at 0.84184 femtometres, about 4 per cent smaller than the accepted value. According to PSI, the gap was too large to be explained by measurement inaccuracies.

The team members said they had not expected any discrepancy. If the new value was right, then either a piece of quantum electrodynamics, the most precisely tested theory in physics, or the Rydberg constant, one of the most accurately determined constants, would need correcting. The result is also described on PSI.

The collaboration brought together PSI, ETH Zurich and the University of Fribourg in Switzerland, the Max Planck Institute of Quantum Optics in Garching and the University of Stuttgart in Germany, the Laboratoire Kastler Brossel in Paris and the University of Coimbra in Portugal. The Nobel laureate Theodor Hänsch, one of the project’s scientists, has called hydrogen the “Rosetta Stone of quantum physics”: because it is the simplest atom, its energy levels can be both calculated and measured with extreme precision, and any disagreement between the two is a signal worth chasing.

2013 and after: a puzzle, not a mistake

In January 2013 the same team reported in Science a second measurement, using a different transition in muonic hydrogen. It gave 0.84087 femtometres, agreeing with 2010 but 1.7 times as precise, and also yielded the proton’s magnetic radius. The discrepancy with the electron-based values remained. PSI listed the possible explanations at the time: unrecognised errors affecting all measurements, a proton structure more complex than assumed that shows up only with the heavy muon, or physics beyond the Standard Model.

Electron scattering did not settle it either. A large scattering experiment in Mainz published in 2010 gave 0.879 femtometres. Reanalyses of older scattering data gave values from about 0.84 to over 0.9. For several years CODATA left the muonic results out of its adjustment; its 2014 value was still 0.8751 femtometres.

New measurements of ordinary hydrogen

The turn came from ordinary hydrogen. A laser measurement published in 2017 gave 0.8335 femtometres; one published in 2019, of the same 2S–2P transition that the muon experiment used, gave 0.833. A measurement of a different transition, published in 2018, gave 0.877 instead, so even the hydrogen results did not all agree. A new electron-scattering experiment published in 2019 found 0.831.

In its 2018 adjustment CODATA decided that the muonic hydrogen data should no longer be excluded, and noted that the reduced disagreement was mainly due to two new hydrogen measurements pointing to a smaller radius. It also stated that the tension between the methods had not been fully resolved. The CODATA 2022 recommended value is 0.84075 femtometres, with an uncertainty of 0.00064, close to the muonic results.

What the story shows

The proton did not shrink. What changed was the measurement, and with it the accepted number. The story shows how a precise result from a single facility can overturn a value that several methods seemed to agree on, and how long it takes – more than a decade here – before independent experiments confirm or refute it.

The work continues at PSI. In 2013 PSI named laser spectroscopy of muonic helium as a next step, and it is preparing an upgrade of its muon beams, HIMB, which is to raise the available muon rate by up to a factor of 100. Questions such as whether the remaining small tensions are errors or a hint of new physics will be decided by measurements, not by preference. How PSI’s other big machines, its two X-ray sources, differ from each other is explained in Synchrotron or X-ray laser?