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Progress on the proton-radius puzzle

Atomic physicists and nuclear physicists have each made a refined measurement of the radius of the proton. Both values agree with a hotly debated result obtained by spectroscopy of an exotic form of hydrogen called muonic hydrogen.

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The proton, discovered 100 years ago1, is an essential building block of visible matter. The nucleus of a hydrogen atom consists of a single proton, making this atom a suitable platform for determining the proton’s intrinsic properties. One such property is the proton charge radius, which corresponds to the spatial extent of the distribution of the proton’s charge. In 2010, a highly accurate measurement of the proton radius was made using spectroscopy of muonic hydrogen — an exotic form of hydrogen in which the electron is replaced by a heavier version called a muon2. However, the value obtained was almost 4% smaller than the previously accepted one3. Bezginov et al.4, writing in Science, and Xiong et al.5, writing in Nature, report experiments that could represent a decisive step towards solving this proton-radius puzzle.

Atomic physicists determine the proton radius by measuring the energy difference between two electronic states of a hydrogen atom using spectroscopy. According to quantum mechanics, there is a non-zero probability that the electron will be found inside the proton if the electron is in a rotationless state (an S state). When inside, the electron is less strongly influenced by the proton’s electric charge than it would otherwise be. This effect slightly weakens the binding of the electron and proton, and causes a tiny shift in the energy of the S state with respect to other states. The high precision achieved both by experiments and by the theory of quantum electrodynamics allows this energy shift and, in turn, the proton radius, to be extracted from measurements.

A muon is about 200 times heavier than an electron. As a result, there is a much higher probability that the muon in a muonic-hydrogen atom will be found inside the proton than would the electron in an ordinary hydrogen atom. Consequently, the associated energy shift is about 8 million (2003) times larger for muonic hydrogen than for regular hydrogen6. Muonic hydrogen is therefore a highly sensitive probe of the proton radius.

Bezginov and colleagues’ work concerns the Lamb shift of ordinary hydrogen — the energy difference between the 2S and 2P excited states. This shift was investigated previously in muonic hydrogen2,7. To measure the Lamb shift, the authors developed an experimental method8 that derives from a technique known as Ramsey interferometry, which is used in atomic clocks.

This experimental method has many technical advantages over other approaches with regard to eliminating systematic uncertainties, filtering environmental noise, and simplicity in the shape of the spectral signal. A key feature of the set-up is the ability to measure a full spectrum in only a few hours. This allowed Bezginov et al. to carry out a meticulous study of systematic uncertainties and to extract a precise value for the proton radius: 0.833 ± 0.010 femtometres (1 fm is 10–15 metres).

Nuclear physicists measure the proton radius using the ‘elastic’ scattering of electrons from protons. In this interaction, the incident electron transfers energy to the targeted proton through the exchange of a virtual (transient) photon. In a similar way to microscopy, short-wavelength photons (which transfer a lot of energy) reveal details at small scales. To determine the full extent of the proton’s charge distribution, one should, in principle, use photons of infinite wavelength (that transfer zero energy), but no scattering at all would occur in this situation. Experiments therefore aim to achieve the lowest-possible energy transfer and then to extrapolate down to zero. This extrapolation, which relies on a parameterization of experimental data, is one of the main challenges in precisely determining the proton radius.

Xiong and colleagues implemented several key improvements over previous studies in their experiment, the Proton Radius experiment at Jefferson Laboratory in Virginia. Crucially, this investigation explores extremely low energy transfers (ten times closer to zero than previous data) while also probing larger energy transfers, to ensure consistency with existing data. The scattered electrons were detected through their energy loss in a detector called an electromagnetic calorimeter. This set-up avoided the need to use a magnetic spectrometer, the multiple settings of which induce systematic errors.

Furthermore, rather than making absolute measurements, Xiong et al. advantageously relied on relative measurements. Specifically, they determined the ratio between the number of events corresponding to elastic electron–proton scattering and the number related to Møller scattering — a well-understood and calculable quantum-electrodynamics process in which electrons are scattered from atomic electrons. This strategy led to the cancellation of many systematic effects that are associated with absolute measurements.

In addition, the protons were in a hydrogen gas that was kept inside a chamber that did not have entrance and exit windows as used in previous similar experiments. This arrangement avoided background noise that would have been produced by the interaction of particles with window materials. Overall, Xiong and colleagues’ chosen set-up, careful systematic-uncertainty checks at each step and exhaustive study of several parameterizations to extrapolate the data to zero energy transfer lend support to their value for the proton radius: 0.831 ± 0.014 fm.

The independent measurements of the proton radius made by Bezginov et al. and Xiong et al. are precise and consistent (Fig. 1). They tip the scales in favour of a small proton radius, in agreement with the highly accurate results from muonic-hydrogen experiments2,7.

Figure 1 | Values for the proton radius. A key property of the proton is its charge radius — the spatial extent of its charge distribution. This quantity is expressed in femtometres (1 fm is 10–15 metres). The data points are values for the proton radius obtained over the past decade, including the latest results, from Bezginov et al.4 and Xiong et al.5, with uncertainties indicated by the error bars. The data were obtained using three different measurement techniques: electron–proton scattering5,10, spectroscopy of ordinary hydrogen4,9,13 and spectroscopy of an exotic type of hydrogen called muonic hydrogen2,7. The error bars for the two data points associated with muonic-hydrogen spectroscopy are too small to be depicted in this figure. The bands denote the values adopted by the Committee on Data for Science and Technology (CODATA) in 201411 and in 2018 (see

But to conclusively solve the proton-radius puzzle, one still needs to understand why there are discrepancies between the latest results and the data from previous hydrogen-spectroscopy9 and electron–proton scattering10 experiments. For instance, the value of the proton radius11 adopted by the Committee on Data for Science and Technology in 2014 was 0.8751 ± 0.0061 fm. Because no convincing explanation for these discrepancies has been proposed, worldwide efforts must be pursued to validate the latest results and to critically assess the different measurement techniques.

Next-generation experiments will provide innovative approaches to this task. For example, the Muon Scattering Experiment12 at the Paul Scherrer Institute in Switzerland is simultaneously investigating muon–proton and electron–proton scattering. This experiment is testing for possible differences in the behaviour of electrons and muons — an observation that would imply the existence of physics beyond that of the standard model of particle physics. On the spectroscopy side, high-precision measurements will be extended to other nuclei such as helium, and to molecules. It is highly probable that the harvest of results from future experiments will not only definitely solve, but might also explain, the proton-radius puzzle.

Nature 575, 61-62 (2019)


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