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 autumn of 2019 agree with the smaller measurement. While some believe that this difference has been resolved,[3] this opinion is not yet universally held.[4][5]

## Problem

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

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]

### Nuclear scattering

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

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

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
π+
hypothesis.[a]

### Measurement artefact

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]

### Quantum chromodynamic calculation

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.

### Data analysis method

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]

### Relativistic reference frame issues

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]

### 2019 measurements

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]

## Footnotes

1. ^ According to a report by Lestone (2017),[19] “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
π+
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
π+
with an inertia of approximately 140 MeV.”[19]

## References

1. ^ Krauth, J. J.; Schuhmann, K.; Abdou Ahmed, M.; Amaro, F. D.; Amaro, P.; et al. (2 June 2017). The proton radius puzzle. 52nd Rencontres de Moriond EW 2017. La Thuile, Aosta Valley. arXiv:1706.00696. Bibcode:2017arXiv170600696K. Presentation slides (19 March 2017).
2. ^ a b c Pohl R, Antognini A, Nez F, Amaro FD, Biraben F, et al. (July 2010). "The size of the proton" (PDF). Nature. 466 (7303): 213–216. Bibcode:2010Natur.466..213P. doi:10.1038/nature09250. PMID 20613837. S2CID 4424731.
3. ^ Hammer, Hans-Werner; Meißner, Ulf-G. (2020). "The proton radius: From a puzzle to precision". Science Bulletin. 65 (4): 257–258. arXiv:1912.03881. doi:10.1016/j.scib.2019.12.012. S2CID 208909979.
4. ^ Karr, Jean-Philippe; Marchand, Dominique (2019). "Progress on the proton-radius puzzle". Nature. 575 (7781): 61–62. Bibcode:2019Natur.575...61K. doi:10.1038/d41586-019-03364-z. PMID 31695215.
5. ^ Hill, Heather (6 November 2019). "Proton radius puzzle may be solved". Physics Today. doi:10.1063/PT.6.1.20191106a. ISSN 1945-0699.
6. ^ a b c d Davide Castelvecchi (5 October 2017). "Proton-size puzzle deepens". Nature.
7. ^ Sick I, Trautmann D (2014). "Proton root-mean-square radii and electron scattering". Physical Review C. 89 (1): 012201. arXiv:1407.1676. Bibcode:2014PhRvC..89a2201S. doi:10.1103/PhysRevC.89.012201. S2CID 118615444.
8. ^ a b Zyga, Lisa (November 26, 2013). "Proton radius puzzle may be solved by quantum gravity". Phys.org. Retrieved September 2, 2016.
9. ^ Carlson CE (May 2015). "The proton radius puzzle". Progress in Particle and Nuclear Physics. 82: 59–77. arXiv:1502.05314. Bibcode:2015PrPNP..82...59C. doi:10.1016/j.ppnp.2015.01.002. S2CID 54915587.
10. ^
11. ^ Pohl R, et al. (2016). "Laser spectroscopy of muonic deuterium" (PDF). Science. 353 (6300): 669–673. Bibcode:2016Sci...353..669P. doi:10.1126/science.aaf2468. hdl:10316/80061. PMID 27516595. S2CID 206647315.
12. ^ "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.
13. ^ Beyer, Axel; Maisenbacher, Lothar; Matveev, Arthur; Pohl, Randolf; Khabarova, Ksenia; Grinin, Alexey; Lamour, Tobias; Yost, Dylan C.; Hänsch, Theodor W.; Kolachevsky, Nikolai; Udem, Thomas (2017). "The Rydberg constant and proton size from atomic hydrogen". Science. 358 (6359): 79–85. Bibcode:2017Sci...358...79B. doi:10.1126/science.aah6677. PMID 28983046. S2CID 206652697.
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. Bibcode:2019Sci...365.1007B. doi:10.1126/science.aau7807. PMID 31488684. S2CID 201845158.
15. ^ Karr, J.; Hilico, L. (2012). "Why three-body physics does not solve the proton-radius puzzle". Physical Review Letters. 109 (10): 103401. arXiv:1205.0633. Bibcode:2012PhRvL.109j3401K. doi:10.1103/PhysRevLett.109.103401. PMID 23005286. S2CID 12752418.
16. ^ Onofrio, R. (2013). "Proton radius puzzle and quantum gravity at the Fermi scale". EPL. 104 (2): 20002. arXiv:1312.3469. Bibcode:2013EL....10420002O. doi:10.1209/0295-5075/104/20002. S2CID 119243594.
17. ^ Dahia, F.; Lemos, A.S. (2016). "Is the proton radius puzzle evidence of extra dimensions?". European Physical Journal. 76 (8): 435. arXiv:1509.08735. Bibcode:2016EPJC...76..435D. doi:10.1140/epjc/s10052-016-4266-7. S2CID 118672005.
18. ^ Liu Y, McKeen D, Miller GA (2016). "Electrophobic Scalar Boson and Muonic Puzzles". Physical Review Letters. 117 (10): 101801. arXiv:1605.04612. Bibcode:2016PhRvL.117j1801L. doi:10.1103/PhysRevLett.117.101801. PMID 27636468. S2CID 20961564.
19. ^ a b Lestone, J.P. (4 October 2017). Muonic atom Lamb shift via simple means (Report). Los Alamos Report. Los Alamos National Laboratory. LA-UR-17-29148.
20. ^ Wolchover, Natalie (11 August 2016). "New measurement deepens proton puzzle". Quanta Magazine. Retrieved 2 September 2016.
21. ^ a b Belushkin, M.A.; Hammer, H.-W.; Meißner, Ulf-G. (2007). "Dispersion analysis of the nucleon form factors including meson continua". Physical Review C. 75 (3): 035202. arXiv:hep-ph/0608337. Bibcode:2007PhRvC..75c5202B. doi:10.1103/PhysRevC.75.035202. ISSN 0556-2813. S2CID 42995123.
22. ^ Higinbotham, Douglas W.; Kabir, Al Amin; Lin, Vincent; Meekins, David; Norum, Blaine; Sawatzky, Brad (31 May 2016). "Proton radius from electron scattering data". Physical Review C. 93 (5): 055207. doi:10.1103/PhysRevC.93.055207.
23. ^ Griffioen, Keith; Carlson, Carl; Maddox, Sarah (17 June 2016). "Consistency of electron scattering data with a small proton radius". Physical Review C. 93 (6): 065207. doi:10.1103/PhysRevC.93.065207.
24. ^ Horbatsch, Marko; Hessels, Eric A.; Pineda, Antonio (13 March 2017). "Proton radius from electron-proton scattering and chiral perturbation theory". Physical Review C. 95 (3): 035203. arXiv:1610.09760. doi:10.1103/PhysRevC.95.035203. S2CID 119232774.
25. ^ a b Alarcón, J.M.; Higinbotham, D.W.; Weiss, C.; Ye, Zhihong (5 April 2019). "Proton charge radius extraction from electron scattering data using dispersively improved chiral effective field theory". Physical Review C. 99 (4): 044303. arXiv:1809.06373. Bibcode:2019PhRvC..99d4303A. doi:10.1103/PhysRevC.99.044303.
26. ^ Barcus, Scott K.; Higinbotham, Douglas W.; McClellan, Randall E. (10 July 2020). "How analytic choices can affect the extraction of electromagnetic form factors from elastic electron scattering cross section data". Physical Review C. 102 (1): 015205. arXiv:1902.08185. doi:10.1103/PhysRevC.102.015205. S2CID 146808413.
27. ^ Giannini, M.M.; Santopinto, E. (2013). "On the proton radius problem". arXiv:1311.0319. Cite journal requires `|journal=` (help)
28. ^ Robson, D. (2013). "Solution to the proton radius problem". arXiv:1305.4552. doi:10.1142/S0218301314500906. Cite journal requires `|journal=` (help)
29. ^ Pineda, Antonio (2011). "Brief review of the theory of the muonic hydrogen Lamb shift and the proton radius". arXiv:1108.1263. Cite journal requires `|journal=` (help)
30. ^ Nottale, Laurent (2019). "Scale relativity of the proton radius: Solving the puzzle". arXiv:1904.05772. Cite journal requires `|journal=` (help)
31. ^ Bezginov, N.; Valdez, T.; Horbatsch, M.; Marsman, A.; Vutha, A. C.; Hessels, E. A. (2019). "A measurement of the atomic hydrogen Lamb shift and the proton charge radius". Science. 365 (6457): 1007–1012. Bibcode:2019Sci...365.1007B. doi:10.1126/science.aau7807. ISSN 0036-8075. PMID 31488684. S2CID 201845158.
32. ^ Xiong, W.; Gasparian, A.; Gao, H.; Dutta, D.; Khandaker, M.; et al. (2019). "A small proton charge radius from an electron–proton scattering experiment". Nature. 575 (7781): 147–150. Bibcode:2019Natur.575..147X. doi:10.1038/s41586-019-1721-2. ISSN 0028-0836. OSTI 1575200. PMID 31695211. S2CID 207831686.
33. ^ Karr, Jean-Philippe; Marchand, Dominique (2019). "Progress on the proton-radius puzzle". Nature. 575 (7781): 61–62. Bibcode:2019Natur.575...61K. doi:10.1038/d41586-019-03364-z. ISSN 0028-0836. PMID 31695215.