Proton radius puzzle

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The proton radius puzzle is an unanswered problem in physics relating to the size of the proton.[1] Historically the proton charge radius was measured by two independent methods, which converged to a value of about 0.877 femtometres (1 fm = 10−15 m). This value was challenged by a 2010 experiment using a third method, which produced a radius about 4% smaller than this, at 0.842 femtometres.[2] New experimental results reported in the fall of 2019 agree with the smaller measurement, and it has been proposed that the puzzle is now solved,[3] though this opinion is not yet universally held.[4][5]


Prior to 2010, the proton charge radius was measured using one of two methods: one relying on spectroscopy, and one relying on nuclear scattering.[6]

Spectroscopy method[edit]

The spectroscopy method uses the energy levels of electrons orbiting the nucleus. The exact values of the energy levels are sensitive to the nuclear radius (see Lamb Shift). For hydrogen, whose nucleus consists only of one proton, this indirectly measures the proton radius. Measurements of hydrogen's energy levels are now so precise that the proton radius is the limiting factor when comparing experimental results to theoretical calculations. This method produces a proton radius of about (8.768±0.069)×10−16 m (or 0.8768±0.0069 fm), with approximately 1% relative uncertainty.[2]

Nuclear scattering[edit]

The nuclear method is similar to Rutherford's scattering experiments that established the existence of the nucleus. Small particles such as electrons can be fired at a proton, and by measuring how the electrons are scattered, the size of the proton can be inferred. Consistent with the spectroscopy method, this produces a proton radius of about (8.775±0.005)×10−16 m (or 0.8775 fm), .[7]

2010 experiment[edit]

In 2010, Pohl et al. published the results of an experiment relying on muonic hydrogen as opposed to normal hydrogen. Conceptually, this is similar to the spectroscopy method. However, the much higher mass of a muon causes it to orbit 207 times closer than an electron to the hydrogen nucleus, where it is consequently much more sensitive to the size of the proton. The resulting radius was recorded as 0.842±0.001 fm, 5 standard deviations (5σ) smaller than the prior measurements.[2][8] The newly measured radius is 4% smaller than the prior measurements, which were believed to be accurate within 1%. (The new measurement's uncertainty limit of only 0.1% makes a negligible contribution to the discrepancy.)[9]

Since 2010, additional measurements using electrons have slightly reduced the estimated radius to (8.751±0.061)×10−16 m (0.8751±0.0061 fm),[10] but by reducing the uncertainty even more the disagreement has worsened to over 7σ.

A follow-up experiment by Pohl et al. in August 2016 used a deuterium atom to create muonic deuterium and measured the deuteron radius. This experiment allowed the measurements to be 2.7 times more accurate, but also found a discrepancy of 7.5 standard deviations smaller than the expected value.[11][12] In 2017 Pohl's group performed yet another experiment, this time using hydrogen atoms that had been excited by two different lasers. By measuring the energy released when the excited electrons fell back to lower-energy states, the Rydberg constant could be calculated, and from this the proton radius inferred. The result is again ~5% smaller than the generally-accepted proton radius.[6][13] In 2019, another experiment reported a measurement of the proton size using a method that was independent of the Rydberg constant -- its result, 0.833 femtometers, agreed with the smaller 2010 value once more.[14]

Proposed resolutions[edit]

The anomaly remains unresolved and is an active area of research. There is as yet no conclusive reason to doubt the validity of the old data.[6] The immediate concern is for other groups to reproduce the anomaly.[6]

The uncertain nature of the experimental evidence has not stopped theorists from attempting to explain the conflicting results. Among the postulated explanations are the three-body force,[15] interactions between gravity and the weak force or a flavour-dependent interaction,[16][8] higher dimension gravity,[17] a new boson,[18] and the quasi-free

Randolf Pohl, the original investigator of the puzzle, stated that while it would be "fantastic" if the puzzle led to a discovery, the most likely explanation is not new physics but some measurement artefact. His personal assumption is that past measurements have misgauged the Rydberg constant and that the current official proton size is inaccurate.[20]

In one of the most recent attempts to resolve the puzzle without new physics, Alarcón, et al. (2018), at Jefferson Labs, have proposed that a different technique to fit the experimental scattering data to a conclusion regarding the proton charge radius that gives more weight to a larger number of higher energy data points in an analytically well justified manner produces a proton charge radius determination from the existing electron scattering data that is consistent with the muonic hydrogen measurement.[clarification needed][long and non-grammatical sentence whose meaning cannot be deduced][21] Effectively, this approach attributes the cause of the proton radius puzzle to a failure to account for significant theoretical uncertainty introduced in previous determinations by the process used to fit the experimental observations made to a determination of the proton charge radius.

Other investigators have suggested that the analysis used for the electron-based proton charge radius may not properly consider the rest frames of the different components of the experiments in the light of special relativity.[22][23] Polarization factors in muonic hydrogen which are not material in ordinary hydrogen have also been proposed as a possible solution.[24] Yet another paper in April 2019 suggested that scale relativity may provide an answer based on the relativistic sizes of protons and muons.[25]

2019 measurements[edit]

In September 2019, Bezginov et al. reported the remeasurement of the proton's charge radius for electronic hydrogen and found a result consistent with Pohl's value for muonic hydrogen.[26] In November W. Xiong et al. reported a similar result using extremely low momentum transfer electron scattering.[27] These results support the smaller proton radius but do not explain why there are differences from the pre-2010 results. It is likely future experiments will be able to both solve and explain the proton radius puzzle.[28]


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  13. ^ The Rydberg constant and proton size from atomic hydrogen,
  14. ^ Bezginov, N.; Valdez, T.; Horbatsch, M.; Marsman, A.; Vutha, A. C.; Hessels, E. A. (5 September 2019). "A measurement of the atomic hydrogen Lamb shift and the proton charge radius". Science. 365 (6457): 1007–1012. doi:10.1126/science.aau7807. PMID 31488684.
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    for half of the time, then the calculated μp and μd Lamb shifts are consistent with experiment without any need for a change in the proton radius. ... As a simple approximation, we here assume that the proton can be thought of as spending approximately half its time as a neutron with a nearby quasi-free
    with an inertia of approximately 140 MeV.”
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