Nearly a decade ago, the value of the proton radius was unexpectedly thrown into doubt. Although the proton doesn’t have definite boundaries, the charge radius is still well-defined in terms of the root mean square of the range of cross sections seen by other charged particles. In 2010 Randolf Pohl of the Max Planck Institute of Quantum Optics in Garching, Germany, and his colleagues measured the proton radius as 0.84 femtometers, which is 5 standard deviations smaller than the accepted value of 0.88 fm.
In the flurry of activity since, studies have supported both the old value and the new, smaller one for the charge radius. The disagreement, known as the proton radius puzzle, opened the possibility of new physics to explain why and under what conditions the proton might behave differently. Now two different approaches have yielded consistent smaller values for the proton radius and possibly resolved the mystery.
The established proton charge radius was found through elastic electron–proton scattering experiments and hydrogen spectroscopy. The latter relies on the fact that lower-energy states don’t follow what we’d expect from Coulomb’s law, in part because the electron and proton can spatially overlap; however, only states with the electron and proton close together are affected. By measuring the energy difference between a highly affected state and a relatively unaltered one—for example, the so-called Lamb shift between the 2s and 2p states—a researcher can determine the proton radius. But both the shift and the component due to the proton’s spatial extent are small.
Back in 2010 Pohl and his colleagues measured the energy levels in muonic hydrogen, a proton orbited by a muon. With a mass 207 times as large as an electron, a muon has a tighter orbit around the proton. The Lamb shift for muonic hydrogen is thus much larger and easier to measure with a lower uncertainty. Pohl’s team measured a proton radius of about 0.84 fm, far less than the accepted value from scattering and spectroscopy measurements on normal (electronic) hydrogen. Muons and electrons have the same electrical charge and belong to the same lepton group, so the proton charge radius should be the same in the two systems. The results opened the possibility that protons interact differently with muons and electrons, an anomaly that would contradict the standard model of particle physics.
But a September 2019 study by Eric Hessels of York University in Canada and his colleagues confirmed that the proton radius is the same for muonic and electronic hydrogen in spectroscopy measurements. They measured the Lamb shift for electronic hydrogen—a measurement analogous to those of Pohl and his coauthors. But it required experimental strategies to reach a parts-per-million accuracy. The researchers drove energy transitions in hydrogen with a series of RF fields and added a phase difference between two of the fields. The result was a phase difference between the measured atomic signal and a reference signal from the interference of the RF fields. If the measured phase difference was not what they expected, the researchers could identify and reduce that source of systematic error. They obtained a proton radius of about 0.83 fm, consistent with the value from Pohl’s team.
Electron-scattering measurements still consistently yielded a larger value for the proton radius. Ashot Gasparian of North Carolina A&T State University and his colleagues in the Proton Radius (PRad) Experiment at Thomas Jefferson National Accelerator Facility in Virginia resolved that final dilemma with a new experiment in which electrons scatter off the protons in a hydrogen gas. The researchers reduced the background by not including target windows, which introduce unwanted scattering, and swapped the magnetic spectrometer typically used in electron–proton scattering experiments for an electromagnetic calorimeter, which measures a wider range of scattering angles as small as 0.7° and a larger range of momentum transfer. With those changes, the team could measure electron–electron scattering simultaneously and use that well-known process to normalize the scattering cross section and reduce the uncertainty.
The PRad radius result, about 0.83 fm, agrees with the smaller value from muonic and now electronic hydrogen spectroscopy measurements. With that, it seems the puzzle is resolved, and the discrepancy was likely due to measurement errors. Unfortunately, the conclusion requires no new physics.