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 radius was measured via two independent methods, which converged to a value of about 0.8768 femtometres (1 fm = 10-15 m). This value was challenged by a 2010 experiment utilizing a third method, which produced a radius about 5% smaller than this.[2] The discrepancy remains unresolved, and is a topic of ongoing research.


Prior to 2010, the proton radius was measured using one of two methods: one relying on spectroscopy, and one relying on nuclear scattering.[3]

Spectroscopy method[edit]

This method uses the energy levels of electrons orbiting the nucleus. The exact values of the energy levels is sensitive to the nuclear radius (see Lamb Shift). For hydrogen, whose nuclei 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]

This 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. As with the spectroscopy method, this produces a proton radius of about (8.768±0.069)×10−16 m.

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][4] 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.)[5]

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),[6] 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.[7][8] 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.[3][9]

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.[3] The immediate concern is for other groups to reproduce the anomaly.[3] 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,[10] interactions between gravity and the weak force or a flavour-dependent interaction,[11][4] higher dimension gravity,[12] a new boson,[13] 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 new 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.[15]


  1. ^ Krauth, J. J.; Schuhmann, K.; et al. (2 June 2017). The proton radius puzzle. 52nd Rencontres de Moriond EW 2017. La Thuile, Aosta Valley. Bibcode:2017arXiv170600696K. arXiv:1706.00696Freely accessible [physics.atom-ph].  Presentation slides (19 March 2917).
  2. ^ a b c Pohl R, et al. (July 2010). "The size of the proton" (PDF). Nature. 466 (7303): 213–216. Bibcode:2010Natur.466..213P. PMID 20613837. doi:10.1038/nature09250. 
  3. ^ a b c d Davide Castelvecchi (5 October 2017). "Proton-size puzzle deepens". Nature. 
  4. ^ a b Zyga, Lisa (November 26, 2013). "Proton radius puzzle may be solved by quantum gravity". Retrieved September 2, 2016. 
  5. ^ Carlson CE (May 2015). "The proton radius puzzle". Progress in Particle and Nuclear Physics. 82: 59–77. Bibcode:2015PrPNP..82...59C. arXiv:1502.05314Freely accessible [hep-ph]. doi:10.1016/j.ppnp.2015.01.002. 
  6. ^ "CODATA Internationally recommended 2014 values of the Fundamental Physical Constants: Proton RMS charge radius rp". 
  7. ^ Pohl R, et al. (2016). "Laser spectroscopy of muonic deuterium" (PDF). Science. 353 (6300): 669–673. Bibcode:2016Sci...353..669P. PMID 27516595. doi:10.1126/science.aaf2468. 
  8. ^ "Proton-radius puzzle deepens". CERN Courier. 16 September 2016. After our first study came out in 2010, I was afraid some veteran physicist would get in touch with us and point out our great blunder. But the years have passed, and so far nothing of the kind has happened. 
  9. ^ The Rydberg constant and proton size from atomic hydrogen,
  10. ^ Karr J, Hilico L (2012). "Why Three-Body Physics Does Not Solve the Proton-Radius Puzzle". Physical Review Letters. 109 (10): 103401. Bibcode:2012PhRvL.109j3401K. PMID 23005286. arXiv:1205.0633Freely accessible [physics.atom-ph]. doi:10.1103/PhysRevLett.109.103401. 
  11. ^ Onofrio R (2013). "Proton radius puzzle and quantum gravity at the Fermi scale". EPL. 104 (2): 20002. Bibcode:2013EL....10420002O. arXiv:1312.3469Freely accessible [hep-ph]. doi:10.1209/0295-5075/104/20002. 
  12. ^ Dahia F, Lemos AS (2016). "Is the proton radius puzzle evidence of extra dimensions?". European Physical Journal. 76 (8): 435. Bibcode:2016EPJC...76..435D. arXiv:1509.08735Freely accessible [hep-ph]. doi:10.1140/epjc/s10052-016-4266-7. 
  13. ^ Liu Y, McKeen D, Miller GA (2016). "Electrophobic Scalar Boson and Muonic Puzzles". Physical Review Letters. 117 (10): 101801. Bibcode:2016PhRvL.117j1801L. PMID 27636468. arXiv:1605.04612Freely accessible [hep-ph]. doi:10.1103/PhysRevLett.117.101801. 
  14. ^ Lestone, JP (4 October 2017). "“Muonic atom Lamb shift via simple means”, Los Alamos Report LA-UR-17-29148". Los Alamos National Laboratory.  According to the report, “Muonic hydrogen (μp) and muonic deuterium (μd) Lamb shifts can be obtained to better than 1% via simple methods. The smallness of the muon fuzziness suggests that the associated Lamb shifts need to be calculated including some aspects of the internal degrees of freedom of the proton. If the charge of the proton is assumed to be contained within a quasi-free
    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.”
  15. ^ Wolchover, Natalie (August 11, 2016). "New Measurement Deepens Proton Puzzle". Quanta Magazine. Retrieved September 2, 2016.