Proton radius puzzle

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. 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]

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]

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]

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 a paper by Belushkin et al. (2007)[21] including different constraints and pertubative quantum chromodynamics predicted a smaller proton radius than the 0.877 femtometres which was the accepted value at the time.[21]

Papers from 2016 suggested that the problem was with the extrapolations that had typically been used to extract the proton radius from the electron scattering data[22][23][24] though these explanation would require that there was also a problem with the atomic Lamb shift measurements.

In one of the most recent attempts to resolve the puzzle without new physics, Alarcón, et al. (2018),[25] at Jefferson Labs, have proposed that a different technique to fit the experimental scattering data in a theoretically as well as analytically justified manner produces a proton charge radius from the existing electron scattering data that is consistent with the muonic hydrogen measurement.[25] Effectively, this approach attributes the cause of the proton radius puzzle to a failure to use a theoretically motivated function for the extraction of the proton charge radius from the experimental data. Another recent paper has pointed out how a simple, yet theory motivated change to previous fits will also give the smaller radius.[26]

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.[27][28] Polarization factors in muonic hydrogen which are not material in ordinary hydrogen have also been proposed as a possible solution.[29]

Yet another paper in April 2019 suggested that scale relativity may provide an answer based on the relativistic sizes of protons and muons.[30]

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.[31] In November W. Xiong et al. reported a similar result using extremely low momentum transfer electron scattering.[32]

Their results support the smaller proton charge radius, but do not explain why the results before 2010 came out larger. It is likely future experiments will be able to both explain and settle the proton radius puzzle.[33]