Solving the proton-radius puzzle

On target

How big is a proton? Experiments during the past decade have called well-established measurements of the proton’s radius into question – even prompting somewhat outlandish suggestions that new physics might be at play. Soon-to-be-published results promise to settle the proton-radius puzzle once and for all.

Contrary to popular depictions, the proton does not have a hard physical boundary like a snooker ball. Its radius was traditionally deduced from its charge distribution via electron-scattering experiments. Scattering from a charge distribution is different from scattering from a point-like charge: the extended charge distribution modifies the differential cross section by a form factor (the Fourier transform of the charge distribution). For a proton this takes the form of a dipole with respect to the scale of the interaction, and an exponentially decaying charge distribution as a function of the distance from the centre of the proton. Scattering experiments found the root mean square (RMS) radius to be about 0.88 fm.

Since the turn of the millennium, a modest increase in precision on the proton radius was made possible by comparing measurements of transitions in hydrogen with quantum electrodynamics (QED) calculations. Since atomic energy levels need to be corrected due to overlapping electron clouds in the extended charge distribution of the proton, precise measurements of the transition frequencies provide a handle on the proton’s radius. A combination of these measurements yielded the most recent CODATA value of 0.8751(61) fm.

The surprise came in 2010, when the CREMA collaboration at the Paul Scherrer Institute (PSI) in Switzerland achieved a 10-fold improvement in precision via the Lamb shift (the 2S–2P transition) in muonic hydrogen, the bound state of a muon orbiting a proton. As the muon is 200 times heavier than the electron, its Bohr radius is 200 times smaller, and the QED correction due to overlapping electron clouds is more substantial. CREMA observed an RMS proton radius of 0.8418(7) fm, which was five sigma below the world average, giving rise to the so-called “proton radius puzzle”. The team confirmed the measurement in 2013, reporting a radius of 0.8409(4) fm. These observations appeared to call into question the cherished principle of lepton universality.

More recent measurements have reinforced the proton’s slimmed-down nature. In 2016 CREMA reported a radius of 0.8356(20) fm by measuring the Lamb shift in muonic deuterium (the bound state of a muon orbiting a proton and a neutron). Most interestingly, in 2017 Axel Beyer of the Max Planck Institute of Quantum Optics in Garching and collaborators reported a similarly lithe radius of 0.8335(95) fm from observations of the 2S–4P transition in ordinary hydrogen. This low value is confirmed by soon-to-be-published measurements of the 1S–3S transition by the same group, and of the 2S–2P transition by Eric Hessels of York University, Canada, and colleagues. “We can no longer speak about a discrepancy between measurements of the proton radius in muonic and electronic spectroscopy,” says Krzysztof Pachucki of CODATA TGFC and the University of Warsaw.

But what of the discrepancy between spectroscopic and scattering experiments? The calculation of the RMS proton radius using scattering data is tricky due to the proton’s recoil, and analyses must extrapolate the form factor to a scale of Q2 = 0. Model uncertainties can therefore be reduced by performing scattering experiments at increasingly low scales. Measurements may now be aligning with a lower value consistent with the latest results in electronic and muonic spectroscopy. In 2017 Miha Mihovilovic of the University of Mainz and colleagues reported an interestingly low value of 0.810(82) fm using the Mainz Microtron, and results due from the Proton Radius Experiment (pRad) at Jefferson Lab will access a similarly low scale with even smaller uncertainties. Preliminary pRad results presented in October 2018 at the 5th Joint Meeting of the APS Division of Nuclear Physics and the Physical Society of Japan in Hawaii indicate a proton radius of 0.830(20) fm. These electron-scattering results will be complemented by muon-scattering results from the COMPASS experiment at CERN, and the MUSE experiment at PSI.

For now, says Pachucki, the latest CODATA recommendations published in 2016 list the higher value obtained from electron scattering and pre-2015 hydrogen-spectroscopy experiments. If the latest experiments continue to line up with the slimmed-down radius of CREMA’s 2010 result, however, the proton radius puzzle may soon be solved, and the world average revised downwards.

About the author

Mark Rayner, CERN.