The proton size


The proton charge radius has been measured since the 1950s using elastic electron–proton scattering and ordinary hydrogen atomic spectroscopy. In 2010, a highly precise measurement of the proton charge radius using, for the first time, muonic hydrogen spectroscopy unexpectedly led to controversy, as the value disagreed with the previously accepted one. Since then, atomic and nuclear physicists have been trying to understand this discrepancy by checking theories, questioning experimental methods and performing new experiments. Recently, two measurements from electron scattering and ordinary hydrogen spectroscopy were found to agree with the results from muonic atom spectroscopy. Is the ‘proton-radius puzzle’ now resolved? In this Review, we scrutinize the experimental studies of the proton radius to gain insight on this issue. We provide a brief history of the proton before describing the techniques used to measure its radius and the current status of the field. We assess the precision and reliability of available experimental data, with particular focus on the most recent results. Finally, we discuss the forthcoming new generation of refined experiments and theoretical calculations that aim to definitely end the debate on the proton size.

Key points

  • The charge radius of the proton can be determined using two different experimental techniques: measurements of electron–proton elastic scattering cross sections and high-resolution spectroscopy of the hydrogen atom.

  • A decade ago, the precision of the atomic spectroscopy method was greatly improved using muonic hydrogen atoms, wherein the electron is replaced by a muon. However, the measured value of the proton radius was in disagreement with previous determinations, giving rise to the ‘proton-radius puzzle’.

  • The latest results from refined scattering and spectroscopy experiments agree with the muonic value, leading to questions regarding the estimation of systematic uncertainties in previous scattering and ordinary hydrogen experiments.

  • More measurements are needed to confirm or disprove this trend. Various projects are in progress, aiming to improve some aspects of existing techniques or using new approaches.

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Fig. 1: Transitions in electronic and muonic hydrogen.
Fig. 2: Proton-radius values from electron scattering experiments.
Fig. 3: Proton-radius values from hydrogen spectroscopy experiments.
Fig. 4: Measurements of the electric form factor.
Fig. 5: Precision spectroscopy of light atoms and molecules.


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This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant number 824093. The authors thank R. Pohl for discussions and advice on the section on atomic spectroscopy.

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Karr, JP., Marchand, D. & Voutier, E. The proton size. Nat Rev Phys 2, 601–614 (2020).

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