Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Review Article
  • Published:

The proton size

Nature Reviews Physics volume 2, pages 601–614 (2020)Cite this article

Subjects

Abstract

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.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

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.

Similar content being viewed by others

References

  1. Rutherford, E. LXXIX. The scattering of α and β particles by matter and the structure of the atom. Lond. Edinb. Dubl. Phil. Mag. 21, 669–688 (1911).

    MATH  Google Scholar 

  2. Rutherford, E. LIV. Collision of α particles with light atoms. IV. An anomalous effect in nitrogen. Lond. Edinb. Dubl. Phil. Mag. 37, 537–587 (1919).

    Google Scholar 

  3. Frisch, R. & Stern, O. Über die magnetische Ablenkung von Wasserstoffmolekülen und das magnetische Moment des Protons. Z. Phys. 85, 4–16 (1933).

    ADS  Google Scholar 

  4. Estermann, I. & Stern, O. Über die magnetische Ablenkung von Wasserstoffmolekülen und das magnetische Moment des Protons. II. Z. Phys. 85, 17–24 (1933).

    ADS  Google Scholar 

  5. Hofstadter, R. & McAllister, R. W. Electron scattering from the proton. Phys. Rev. 98, 217–218 (1955).

    ADS  Google Scholar 

  6. Gell-Mann, M. A schematic model of baryons and mesons. Phys. Lett. 8, 214–215 (1964).

    ADS  Google Scholar 

  7. Zweig, G. An SU(3) Model for Strong Interaction Symmetry and its Breaking Version 2, 22–101 (Hadronic Press, 1980).

  8. Breidenbach, M. et al. Observed behavior of highly inelastic electron-proton scattering. Phys. Rev. Lett. 23, 935–939 (1969).

    ADS  Google Scholar 

  9. Müller, D., Robaschik, D., Geyer, B., Dittes, F.-M. & Hořejši, J. Wave functions, evolution equations and evolution kernels from light-ray operators of QCD. Fortschr. Phys. 42, 101–141 (1994).

    Google Scholar 

  10. Ji, X. Gauge-invariant decomposition of nucleon spin. Phys. Rev. Lett. 78, 610–613 (1997).

    ADS  Google Scholar 

  11. Radyushkin, A. V. Nonforward parton distributions. Phys. Rev. D 56, 5524–5557 (1997).

    ADS  Google Scholar 

  12. Airapetian, A. et al. Measurement of the beam-spin azimuthal asymmetry associated with deeply-virtual Compton scattering. Phys. Rev. Lett. 87, 182001 (2001).

    ADS  Google Scholar 

  13. Stepanyan, S. et al. Observation of exclusive deeply virtual Compton scattering in polarized electron beam asymmetry measurements. Phys. Rev. Lett. 87, 182002 (2001).

    ADS  Google Scholar 

  14. Mohr, P. J., Taylor, B. N. & Newell, D. B. CODATA recommended values of the fundamental physical constants: 2006. Rev. Mod. Phys. 80, 633–730 (2008).

    ADS  Google Scholar 

  15. Sick, I. On the rms-radius of the proton. Phys. Lett. B 576, 62–67 (2003).

    ADS  Google Scholar 

  16. Bernauer, J. C. et al. High-precision determination of the electric and magnetic form factors of the proton. Phys. Rev. Lett. 105, 242001 (2010).

    ADS  Google Scholar 

  17. Pohl, R. et al. The size of the proton. Nature 466, 213–216 (2010).

    ADS  Google Scholar 

  18. Antognini, A. et al. Proton structure from the measurement of 2S–2P transition frequencies of muonic hydrogen. Science 339, 417–420 (2013).

    ADS  Google Scholar 

  19. Pohl, R. et al. Laser spectroscopy of muonic deuterium. Science 353, 669–673 (2016).

    ADS  Google Scholar 

  20. Pohl, R., Gilman, R., Miller, G. A. & Pachucki, K. Muonic hydrogen and the proton radius puzzle. Ann. Rev. Nucl. Part. Sci. 63, 175–204 (2013).

    ADS  Google Scholar 

  21. Bernauer, J. & Pohl, R. The proton radius problem. Sci. Am. 310, 32–39 (2014).

    Google Scholar 

  22. Carlson, C. E. The proton radius puzzle. Prog. Part. Nucl. Phys. 82, 59–77 (2015).

    ADS  Google Scholar 

  23. Hill, R. J. Review of experimental and theoretical status of the proton radius puzzle. EPJ Web Conf. 137, 01023 (2017).

    Google Scholar 

  24. Beyer, A. et al. The Rydberg constant and proton size from atomic hydrogen. Science 358, 79–85 (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  25. Fleurbaey, H. et al. New measurement of the 1S–3S transition frequency of hydrogen: contribution to the proton charge radius puzzle. Phys. Rev. Lett. 120, 183001 (2018).

    ADS  Google Scholar 

  26. Bezginov, N. et al. A measurement of the atomic hydrogen Lamb shift and the proton charge radius. Science 365, 1007–1012 (2019).

    ADS  Google Scholar 

  27. Xiong, W. et al. A small proton charge radius from an electron–proton scattering experiment. Nature 575, 147–150 (2019).

    ADS  Google Scholar 

  28. Punjabi, V., Perdrisat, C., Jones, M., Brash, E. & Carlson, C. The structure of the nucleon: elastic electromagnetic form factors. Eur. Phys. J. A 51, 79–122 (2015).

    ADS  Google Scholar 

  29. Rosenbluth, M. N. High energy elastic scattering of electrons on protons. Phys. Rev. 79, 615–619 (1950).

    ADS  MATH  Google Scholar 

  30. Akhiezer, A. & Rekalo, M. Polarization phenomena in electron scattering by protons in the high energy region. Sov. Phys. Dokl. 13, 572 (1968).

    ADS  Google Scholar 

  31. Akhiezer, A. & Rekalo, M. Polarization effects in the scattering of leptons by hadrons. Sov. J. Part. Nucl. 4, 277–286 (1974).

    Google Scholar 

  32. Arnold, R. G., Carlson, C. E. & Gross, F. Polarization transfer in elastic electron scattering from nucleons and deuterons. Phys. Rev. C 23, 363–374 (1981).

    ADS  Google Scholar 

  33. Dombey, N. Scattering of polarized leptons at high energy. Rev. Mod. Phys. 41, 236–246 (1969).

    ADS  Google Scholar 

  34. Guichon, P. A. M. & Vanderhaeghen, M. How to reconcile the Rosenbluth and the polarization transfer methods in the measurement of the proton form factors. Phys. Rev. Lett. 91, 142303 (2003).

    ADS  Google Scholar 

  35. Blunden, P. G., Melnitchouk, W. & Tjon, J. A. Two-photon exchange and elastic electron–proton scattering. Phys. Rev. Lett. 91, 142304 (2003).

    ADS  Google Scholar 

  36. Adikaram, D. et al. Towards a resolution of the proton form factor problem: new electron and positron scattering data. Phys. Rev. Lett. 114, 062003 (2015).

    ADS  Google Scholar 

  37. Rachek, I. A. et al. Measurement of the two-photon exchange contribution to the elastic e±p scattering cross sections at the VEPP-3 storage ring. Phys. Rev. Lett. 114, 062005 (2015).

    ADS  Google Scholar 

  38. Henderson, B. S. et al. Hard two-photon contribution to elastic lepton-proton scattering determined by the OLYMPUS experiment. Phys. Rev. Lett. 118, 092501 (2017).

    ADS  Google Scholar 

  39. Gorchtein, M. Forward sum rule for the 2γ-exchange correction to the charge-radius extraction from elastic electron scattering. Phys. Rev. C 90, 052201 (2014).

    ADS  Google Scholar 

  40. Tomalak, O., Pasquini, B. & Vanderhaeghen, M. Two-photon exchange contribution to elastic e−–proton scattering: full dispersive treatment of πN states and comparison with data. Phys. Rev. D 96, 096001 (2017).

    ADS  Google Scholar 

  41. Miller, G. A. Defining the proton radius: a unified treatment. Phys. Rev. C 99, 035202 (2019).

    ADS  Google Scholar 

  42. Karplus, R., Klein, A. & Schwinger, J. Electrodynamic displacement of atomic energy levels. II. Lamb shift. Phys. Rev. 86, 288–301 (1952).

    ADS  MATH  Google Scholar 

  43. Lamb, W. E. & Retherford, R. C. Fine structure of the hydrogen atom by a microwave method. Phys. Rev. 72, 241–243 (1947).

    ADS  Google Scholar 

  44. Eides, M. I., Grotch, H. & Shelyuto, V. A. Theory of Light Hydrogenic Bound States (Springer Tracts in Modern Physics Vol. 222, Springer-Verlag, 2007).

  45. Mohr, P. J., Newell, D. B. & Taylor, B. N. CODATA recommended values of the fundamental physical constants: 2014. Rev. Mod. Phys. 88, 035009 (2016).

    ADS  Google Scholar 

  46. Parthey, C. G. et al. Improved measurement of the hydrogen 1S–2S transition frequency. Phys. Rev. Lett. 107, 203001 (2011).

    ADS  Google Scholar 

  47. Matveev, A. et al. Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link. Phys. Rev. Lett. 110, 230801 (2013).

    ADS  Google Scholar 

  48. Antognini, A. et al. Theory of the 2S–2P Lamb shift and 2S hyperfine splitting in muonic hydrogen. Ann. Phys. 331, 127–145 (2013).

    ADS  MATH  Google Scholar 

  49. Pachucki, K. Theory of the Lamb shift in muonic hydrogen. Phys. Rev. A 53, 2092–2100 (1996).

    ADS  Google Scholar 

  50. Pachucki, K. Proton structure effects in muonic hydrogen. Phys. Rev. A 60, 3593–3598 (1999).

    ADS  Google Scholar 

  51. Karshenboim, S. G., Korzinin, E. Y., Ivanov, V. G. & Shelyuto, V. A. Contribution of light-by-light scattering to energy levels of light muonic atoms. JETP Lett. 92, 8–14 (2010).

    ADS  Google Scholar 

  52. Jentschura, U. D. Lamb shift in muonic hydrogen — I. Verification and update of theoretical predictions. Ann. Phys. 326, 500–515 (2011).

    ADS  MathSciNet  MATH  Google Scholar 

  53. Jentschura, U. D. Relativistic reduced-mass and recoil corrections to vacuum polarization in muonic hydrogen, muonic deuterium, and muonic helium ions. Phys. Rev. A 84, 012505 (2011).

    ADS  Google Scholar 

  54. Borie, E. Lamb shift in light muonic atoms — revisited. Ann. Phys. 327, 733–763 (2012).

    ADS  MATH  Google Scholar 

  55. Karshenboim, S. G., Ivanov, V. G. & Korzinin, E. Y. Relativistic recoil corrections to the electron-vacuum-polarization contribution in light muonic atoms. Phys. Rev. A 85, 032509 (2012).

    ADS  Google Scholar 

  56. Indelicato, P. Nonperturbative evaluation of some QED contributions to the muonic hydrogen n = 2 Lamb shift and hyperfine structure. Phys. Rev. A 87, 022501 (2013).

    ADS  Google Scholar 

  57. Pachucki, K., Patkóš, V. & Yerokhin, V. A. Three-photon-exchange nuclear structure correction in hydrogenic systems. Phys. Rev. A 97, 062511 (2018).

    ADS  Google Scholar 

  58. Miller G. A. Proton polarizability contribution: muonic hydrogen Lamb shift and elastic scattering. Phys. Lett. B 718, 1078–1082 (2013).

  59. Birse, M. C. & McGovern, J. A. Proton polarisability contribution to the Lamb shift in muonic hydrogen at fourth order in chiral perturbation theory. Eur. Phys. J. A 48, 120 (2012).

    ADS  Google Scholar 

  60. Alarcón, J. M., Lensky, V. & Pascalutsa, V. Chiral perturbation theory of muonic-hydrogen Lamb shift: polarizability contribution. Eur. Phys. J. C 74, 2852–2861 (2014).

    ADS  Google Scholar 

  61. Chambers, E. E. & Hofstadter, R. Structure of the proton. Phys. Rev. 103, 1454–1463 (1956).

    ADS  Google Scholar 

  62. Simon, G., Schmitt, C., Borkowski, F. & Walther, V. Absolute electron–proton cross sections at low momentum transfer measured with a high pressure gas target system. Nucl. Phys. A 333, 381–391 (1980).

    ADS  Google Scholar 

  63. Bernauer, J. C. et al. Electric and magnetic form factors of the proton. Phys. Rev. C 90, 015206 (2014).

    ADS  Google Scholar 

  64. Hill, R. J. Effective field theory for large logarithms in radiative corrections to electron proton scattering. Phys. Rev. D 95, 013001 (2017).

    ADS  Google Scholar 

  65. Barcus, S. K., Higinbotham, D. W. & McClellan, R. E. How analytic choices can affect the extraction of electromagnetic form factors from elastic electron scattering cross section data. Phys. Rev. C 102, 015205 (2020).

    ADS  Google Scholar 

  66. Hill, R. J. & Paz, G. Model-independent extraction of the proton charge radius from electron scattering. Phys. Rev. D 82, 113005 (2010).

    ADS  Google Scholar 

  67. Lee, G., Arrington, J. R. & Hill, R. J. Extraction of the proton radius from electron–proton scattering data. Phys. Rev. D 92, 013013 (2015).

    ADS  Google Scholar 

  68. Kraus, E., Mesick, K. E., White, A., Gilman, R. & Strauch, S. Polynomial fits and the proton radius puzzle. Phys. Rev. C 90, 045206 (2014).

    ADS  Google Scholar 

  69. Higinbotham, D. W. et al. Proton radius from electron scattering data. Phys. Rev. C 93, 055207 (2016).

    ADS  Google Scholar 

  70. Griffioen, K., Carlson, C. & Maddox, S. Consistency of electron scattering data with a small proton radius. Phys. Rev. C 93, 065207 (2016).

    ADS  Google Scholar 

  71. Lorenz, I. & Meißner, U.-G. Reduction of the proton radius discrepancy by 3σ. Phys. Lett. B 737, 57–59 (2014).

    ADS  Google Scholar 

  72. Horbatsch, M. & Hessels, E. A. Evaluation of the strength of electron–proton scattering data for determining the proton charge radius. Phys. Rev. C 93, 015204 (2016).

    ADS  Google Scholar 

  73. Sick, I. & Trautmann, D. Reexamination of proton rms radii from low-q power expansions. Phys. Rev. C 95, 012501 (2017).

    ADS  Google Scholar 

  74. Yan, X. et al. Robust extraction of the proton charge radius from electron–proton scattering data. Phys. Rev. C 98, 025204 (2018).

    ADS  Google Scholar 

  75. Höhler, G. et al. Analysis of electromagnetic nucleon form factors. Nucl. Phys. B 114, 505–534 (1976).

    ADS  Google Scholar 

  76. Mergell, P., Meißner, U.-G. & Drechsel, D. Dispersion-theoretical analysis of the nucleon electromagnetic form factors. Nucl. Phys. A 596, 367–396 (1996).

    ADS  Google Scholar 

  77. Belushkin, M. A., Hammer, H.-W. & Meißner, U.-G. Dispersion analysis of the nucleon form factors including meson continua. Phys. Rev. C 75, 035202 (2007).

    ADS  Google Scholar 

  78. Adamuscin, C., Dubnicka, S. & Dubnickova, A. New value of the proton charge root mean square radius. Prog. Part. Nucl. Phys. 67, 479–485 (2012).

    ADS  Google Scholar 

  79. Lorenz, I. T., Hammer, H.-W. & Meißner, U.-G. The size of the proton — closing in on the radius puzzle. Eur. Phys. J. A 48, 151 (2012).

    ADS  Google Scholar 

  80. Lorenz, I. T., Meißner, U.-G., Hammer, H.-W. & Dong, Y.-B. Theoretical constraints and systematic effects in the determination of the proton form factors. Phys. Rev. D 91, 014023 (2015).

    ADS  Google Scholar 

  81. Alarcón, J. M., Higinbotham, D. W., Weiss, C. & Zhihong, Y. Proton charge radius extraction from electron scattering data using dispersively improved chiral effective field theory. Phys. Rev. C 99, 044303 (2019).

    ADS  Google Scholar 

  82. Yerokhin, V. A., Pachucki, K. & Patkóš, V. Theory of the Lamb shift in hydrogen and light hydrogen-like ions. Ann. Phys. 531, 1800324 (2019).

    Google Scholar 

  83. Karshenboim, S. G. & Shelyuto, V. A. Three-loop radiative corrections to the 1s Lamb shift in hydrogen. Phys. Rev. A 100, 032513 (2019).

    ADS  MathSciNet  Google Scholar 

  84. Karshenboim, S. G., Ozawa, A. & Ivanov, V. G. Higher-order logarithmic corrections and the two-loop self-energy of a 1s electron in hydrogen. Phys. Rev. A 100, 032515 (2019).

    ADS  Google Scholar 

  85. Thomas, S. et al. High-resolution hydrogen spectroscopy and the proton radius puzzle. Ann. Phys. 531, 1800363 (2019).

    Google Scholar 

  86. Arnoult, O., Nez, F., Julien, L. & Biraben, F. Optical frequency measurement of the 1S–3S two-photon transition in hydrogen. Eur. Phys. J. D 60, 243–256 (2010).

    ADS  Google Scholar 

  87. Yost, D. C. et al. Spectroscopy of the hydrogen 1S–3S transition with chirped laser pulses. Phys. Rev. A 93, 042509 (2016).

    ADS  Google Scholar 

  88. Lundeen, S. R. & Pipkin, F. M. Separated oscillatory field measurement of the Lamb shift in H, n = 2. Metrologia 22, 9–54 (1986).

    ADS  Google Scholar 

  89. Marsman, A., Horbatsch, M., Corriveau, Z. A. & Hessels, E. A. Systematic effects important to separated-oscillatory-field measurements of the n = 2 Lamb shift in atomic hydrogen. Phys. Rev. A 98, 012509 (2018).

    ADS  Google Scholar 

  90. Hoballah, M. et al. Merits and constraints of low-K2 experimental data for the proton radius determination. Eur. Phys. J. A 55, 112 (2019).

    ADS  Google Scholar 

  91. Mihovilovič, M. et al. First measurement of proton’s charge form factor at very low Q2 with initial state radiation. Phys. Lett. B 771, 194–198 (2017).

    ADS  Google Scholar 

  92. Mihovilovič, M. et al. The proton charge radius extracted from the initial state radiation experiment at MAMI. Preprint at arXiv https://arxiv.org/abs/1905.11182 (2019).

  93. Akimov, Y. K. et al. Electron scattering by protons at small angles. Sov. J. Exp. Theor. Phys. 35, 651–654 (1972).

    ADS  Google Scholar 

  94. Murphy, J. J., Shin, Y. M. & Skopik, D. M. Proton form factor from 0.15 to 0.79 fm−2. Phys. Rev. C 9, 2125–2129 (1974).

    ADS  Google Scholar 

  95. Merkel, H. in Proc. 54th International Winter Meeting on Nuclear Physics — PoS(BORMIO2016) Vol. 272, 037 (Sissa Medialab, 2016).

  96. Suda, T. Measurement of proton charge radius by low-energy electron scattering. J. Part. Acc. Soc. Jpn 15, 52–59 (2018).

    Google Scholar 

  97. Barger, V., Chiang, C.-W., Keung, W.-Y. & Marfatia, D. Proton size anomaly. Phys. Rev. Lett. 106, 153001 (2011).

    ADS  Google Scholar 

  98. Tucker-Smith, D. & Yavin, I. Muonic hydrogen and MeV forces. Phys. Rev. D 83, 101702 (2011).

    ADS  Google Scholar 

  99. Batell, B., McKeen, D. & Pospelov, M. New parity-violating muonic forces and the proton charge radius. Phys. Rev. Lett. 107, 011803 (2011).

    ADS  Google Scholar 

  100. Carlson, C. E. & Rislow, B. C. New physics and the proton radius problem. Phys. Rev. D 86, 035013 (2012).

    ADS  Google Scholar 

  101. Pauk, V. & Vanderhaeghen, M. Lepton universality test in the photoproduction of e−e+ versus μ−μ+ pairs on a proton target. Phys. Rev. Lett. 115, 221804 (2015).

    ADS  Google Scholar 

  102. Liu, Y.-S. & Miller, G. A. Polarized lepton-nucleon elastic scattering and a search for a light scalar boson. Phys. Rev. C 92, 035209 (2015).

    ADS  Google Scholar 

  103. Bennett, G. W. et al. Final report of the E821 muon anomalous magnetic moment measurement at BNL. Phys. Rev. D 73, 072003 (2006).

    ADS  Google Scholar 

  104. Jegerlehner, F. & Nyffeler, A. The muon g − 2. Phys. Rep. 477, 1–110 (2009).

    ADS  Google Scholar 

  105. Krasznahorkay, A. J. et al. Observation of anomalous internal pair creation in 8Be: a possible indication of a light, neutral boson. Phys. Rev. Lett. 116, 042501 (2016).

    ADS  Google Scholar 

  106. Krasznahorkay, A. J. et al. New evidence supporting the existence of the hypothetic X17 particle. Preprint at arXiv https://arxiv.org/abs/1910.10459 (2019).

  107. Liu, Y.-S., McKeen, D. & Miller, G. A. Electrophobic scalar boson and muonic puzzles. Phys. Rev. Lett. 117, 101801 (2016).

    ADS  Google Scholar 

  108. Gilman, R. et al. Technical design report for the Paul Scherrer Institute experiment R-12-01.1: studying the proton “radius” puzzle with μp elastic scattering. Preprint at arXiv https://arxiv.org/abs/1709.09753 (2017).

  109. Adams, B. et al. Letter of intent: a new QCD facility at the M2 beam line of the CERN SPS (COMPASS++/AMBER). Preprint at arXiv https://arxiv.org/abs/1808.00848 (2018).

  110. Beyer, A. et al. Precision spectroscopy of 2S–nP transitions in atomic hydrogen for a new determination of the Rydberg constant and the proton charge radius. Phys. Scr. 2015, 014030 (2015).

    Google Scholar 

  111. Galtier, S., Nez, F., Julien, L. & Biraben, F. Ultraviolet continuous-wave laser source at 205 nm for hydrogen spectroscopy. Opt. Commun. 324, 34–37 (2014).

    ADS  Google Scholar 

  112. Karshenboim, S. G. Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants. Phys. Rep. 422, 1–63 (2005).

    ADS  Google Scholar 

  113. Cooke, D. A. et al. Observation of positronium annihilation in the 2S state: towards a new measurement of the 1S–2S transition frequency. Hyperfine Interact. 233, 67–73 (2015).

    ADS  Google Scholar 

  114. Mills, A. in Advances in Atomic, Molecular, and Optical Physics Vol. 65 (eds Arimondo, E., Lin, C. C. & Yelin, S. F.) 265–290 (Elsevier, 2016).

  115. Cassidy, D. B. Experimental progress in positronium laser physics. Eur. Phys. J. D 72, 53 (2018).

    ADS  Google Scholar 

  116. Adkins, G. S., Kim, M., Parsons, C. & Fell, R. N. Three-photon-annihilation contributions to positronium energies at order mα7. Phys. Rev. Lett. 115, 233401 (2015).

    ADS  Google Scholar 

  117. Crivelli, P. & Wichmann, G. in CPT and Lorentz Symmetry (ed. Kostelecký, V. A.) 5–8 (World Scientific, 2017).

  118. Herrmann, M. et al. Feasibility of coherent XUV spectroscopy on the 1S–2S transition in singly ionized helium. Phys. Rev. A 79, 052505 (2009).

    ADS  Google Scholar 

  119. Altmann, R. K., Galtier, S., Dreissen, L. S. & Eikema, K. S. E. High-precision Ramsey-comb spectroscopy at deep ultraviolet wavelengths. Phys. Rev. Lett. 117, 173201 (2016).

    ADS  Google Scholar 

  120. Krauth, J. J. et al. in Proc. International Conference on Precision Physics and Fundamental Physical Constants — PoS(FFK2019) Vol. 353, 049 (Sissa Medialab, 2019).

  121. Karshenboim, S. G., Ozawa, A., Shelyuto, V. A., Szafron, R. & Ivanov, V. G. The Lamb shift of the 1s state in hydrogen: two-loop and three-loop contributions. Phys. Lett. B 795, 432–437 (2019).

    ADS  Google Scholar 

  122. Pachucki, K., Patkóš, V. & Yerokhin, V. A. Testing fundamental interactions on the helium atom. Phys. Rev. A 95, 062510 (2017).

    ADS  Google Scholar 

  123. Rengelink, R. J. et al. Precision spectroscopy of helium in a magic wavelength optical dipole trap. Nat. Phys. 14, 1132–1137 (2018).

    Google Scholar 

  124. Zheng, X. et al. Measurement of the frequency of the 23S−23P transition of 4He. Phys. Rev. Lett. 119, 263002 (2017).

    ADS  Google Scholar 

  125. Diepold, M. et al. Theory of the Lamb shift and fine structure in muonic 4He ions and the muonic 3He–4He isotope shift. Ann. Phys. 396, 220–244 (2018).

    ADS  Google Scholar 

  126. Karr, J.-P., Hilico, L., Koelemeij, J. C. J. & Korobov, V. I. Hydrogen molecular ions for improved determination of fundamental constants. Phys. Rev. A 94, 050501 (2016).

    ADS  Google Scholar 

  127. Puchalski, M., Komasa, J., Czachorowski, P. & Pachucki, K. Complete α6m corrections to the ground state of H2. Phys. Rev. Lett. 117, 263002 (2016).

    ADS  Google Scholar 

  128. Hilico, L., Billy, N., Grémaud, B. & Delande, D. Polarizabilities, light shifts and two-photon transition probabilities between J = 0 states of the H\({}_{2}^{+}\) and D\({}_{2}^{+}\) molecular ions. J. Phys. B 34, 491–507 (2001).

    ADS  Google Scholar 

  129. Tran, V. Q., Karr, J.-P., Douillet, A., Koelemeij, J. C. J. & Hilico, L. Two-photon spectroscopy of trapped HD+ ions in the Lamb–Dicke regime. Phys. Rev. A 88, 033421 (2013).

    ADS  Google Scholar 

  130. Alighanbari, S., Giri, G. S., Constantin, F. L., Korobov, V. I. & Schiller, S. Precise test of quantum electrodynamics and determination of fundamental constants with HD+ ions. Nature 581, 152–158 (2020).

    ADS  Google Scholar 

  131. Patra, S. et al. Proton–electron mass ratio from laser spectroscopy of HD+ at the part-per-trillion level. Science 369, 1238–1241 (2020).

    ADS  Google Scholar 

  132. Korobov, V. I., Hilico, L. & Karr, J.-P. Fundamental transitions and ionization energies of the hydrogen molecular ions with few ppt uncertainty. Phys. Rev. Lett. 118, 233001 (2017).

    ADS  Google Scholar 

  133. Tao, L.-G. et al. Toward a determination of the proton-electron mass ratio from the Lamb-dip measurement of HD. Phys. Rev. Lett. 120, 153001 (2018).

    ADS  Google Scholar 

  134. Cozijn, F. M. J., Dupré, P., Salumbides, E. J., Eikema, K. S. E. & Ubachs, W. Sub-Doppler frequency metrology in HD for tests of fundamental physics. Phys. Rev. Lett. 120, 153002 (2018).

    ADS  Google Scholar 

  135. Hölsch, N. et al. Benchmarking theory with an improved measurement of the ionization and dissociation energies of H2. Phys. Rev. Lett. 122, 103002 (2019).

    ADS  Google Scholar 

  136. Puchalski, M., Komasa, J., Czachorowski, P. & Pachucki, K. Nonadiabatic QED correction to the dissociation energy of the hydrogen molecule. Phys. Rev. Lett. 122, 103003 (2019).

    ADS  Google Scholar 

  137. Jentschura, U. D., Mohr, P. J., Tan, J. N. & Wundt, B. J. Fundamental constants and tests of theory in Rydberg states of hydrogenlike ions. Phys. Rev. Lett. 100, 160404 (2008).

    ADS  Google Scholar 

  138. Ramos, A., Moore, K. & Raithel, G. Measuring the Rydberg constant using circular Rydberg atoms in an intensity-modulated optical lattice. Phys. Rev. A 96, 032513 (2017).

    ADS  Google Scholar 

  139. Schmidt, S. et al. The next generation of laser spectroscopy experiments using light muonic atoms. J. Phys. Conf. Ser. 1138, 012010 (2018).

    Google Scholar 

  140. Lehmann, P., Taylor, R. & Wilson, R. Electron–proton scattering at low momentum transfers. Phys. Rev. 126, 1183–1188 (1962).

    ADS  Google Scholar 

  141. Dudelzak, B., Sauvage, G. & Lehmann, P. Measurements of the form factors of the proton at momentum transfers q22 fermi−2. Nuovo Cim. 28, 18–24 (1963).

    ADS  Google Scholar 

  142. Frèrejacque, D., Benaksas, D. & Drickey, D. Proton form factors from observation of recoil protons. Phys. Rev. 141, 1308–1312 (1966).

    ADS  Google Scholar 

  143. Borkowski, F., Peuser, P., Simon, G., Walther, V. & Wendling, R. Electromagnetic form factors of the proton at low four-momentum transfer. Nucl. Phys. A 222, 269–275 (1974).

    ADS  Google Scholar 

  144. Borkowski, F., Simon, G., Walther, V. & Wendling, R. Electromagnetic form factors of the proton at low four-momentum transfer (II). Nucl. Phys. B 93, 461–478 (1975).

    ADS  Google Scholar 

  145. McCord, M. et al. Preliminary results of a new determination of the rms charge radius of the proton. Nucl. Instrum. Methods Phys. Res. B 56–57, 496–499 (1991).

    ADS  Google Scholar 

  146. Eschrich, I. et al. Measurement of the Σ− charge radius by Σ−-electron elastic scattering. Phys. Lett. B 522, 233–239 (2001).

    ADS  Google Scholar 

  147. Hand, L. N., Miller, D. G. & Wilson, R. Electric and magnetic form factors of the nucleon. Rev. Mod. Phys. 35, 335–349 (1963).

    ADS  Google Scholar 

  148. Rosenfelder, R. Coulomb corrections to elastic electron–proton scattering and the proton charge radius. Phys. Lett. B 479, 381–386 (2000).

    ADS  Google Scholar 

  149. Blunden, P. G. & Sick, I. Proton radii and two-photon exchange. Phys. Rev. C 72, 057601 (2005).

    ADS  Google Scholar 

  150. Borisyuk, D. Proton charge and magnetic rms radii from the elastic ep scattering data. Nucl. Phys. A 843, 59–67 (2010).

    ADS  Google Scholar 

  151. Zhan, X. et al. High-precision measurement of the proton elastic form factor ratio μpGE/GM at low Q2. Phys. Lett. B 705, 59–64 (2011).

    ADS  Google Scholar 

  152. Sick, I. Problems with proton radii. Prog. Part. Nucl. Phys. 67, 473–478 (2012).

    ADS  Google Scholar 

  153. Graczyk, K. M. & Juszczak, C. Proton radius from Bayesian inference. Phys. Rev. C 90, 054334 (2014).

    ADS  Google Scholar 

  154. Arrington, J. & Sick, I. Evaluation of the proton charge radius from electron–proton scattering. J. Phys. Chem. Ref. Data 44, 031204 (2015).

    ADS  Google Scholar 

  155. Horbatsch, M., Hessels, E. A. & Pineda, A. Proton radius from electron–proton scattering and chiral perturbation theory. Phys. Rev. C 95, 035203 (2017).

    ADS  Google Scholar 

  156. Sick, I. Proton charge radius from electron scattering. Atoms 6, 2 (2018).

    ADS  Google Scholar 

  157. Newton, G., Andrews, D. A. & Unsworth, P. J. A precision determination of the Lamb shift in hydrogen. Phil. Trans. R. Soc. Lond. A 290, 373–404 (1979).

    ADS  Google Scholar 

  158. Bourzeix, S. et al. High resolution spectroscopy of the hydrogen atom: determination of the 1S Lamb shift. Phys. Rev. Lett. 76, 384–387 (1996).

    ADS  Google Scholar 

  159. Hagley, E. W. & Pipkin, F. M. Separated oscillatory field measurement of hydrogen 2S1/2–2P3/2 fine structure interval. Phys. Rev. Lett. 72, 1172–1175 (1994).

    ADS  Google Scholar 

  160. Berkeland, D. J., Hinds, E. A. & Boshier, M. G. Precise optical measurement of Lamb shifts in atomic hydrogen. Phys. Rev. Lett. 75, 2470–2473 (1995).

    ADS  Google Scholar 

  161. Weitz, M. et al. Precision measurement of the 1S ground-state Lamb shift in atomic hydrogen and deuterium by frequency comparison. Phys. Rev. A 52, 2664–2681 (1995).

    ADS  Google Scholar 

  162. de Beauvoir, B. et al. Absolute frequency measurement of the 2S–8S/D transitions in hydrogen and deuterium: new determination of the Rydberg constant. Phys. Rev. Lett. 78, 440–443 (1997).

    ADS  Google Scholar 

  163. Schwob, C. et al. Optical frequency measurement of the 2S–12D transitions in hydrogen and deuterium: Rydberg constant and Lamb shift determinations. Phys. Rev. Lett. 82, 4960–4963 (1999).

    ADS  Google Scholar 

  164. Drickey, D. J. & Hand, L. N. Precise neutron and proton form factors at low momentum transfers. Phys. Rev. Lett. 9, 521–524 (1962).

    ADS  Google Scholar 

  165. Meyer, V. et al. Measurement of the 1s–2s energy interval in muonium. Phys. Rev. Lett. 84, 1136–1139 (2000).

    ADS  Google Scholar 

  166. Jungmann, K.-P. in The Hydrogen Atom (eds Karshenboim, S. G. et al.) 81–102 (Springer, 2001).

  167. Fee, M. S. et al. Measurement of the positronium 13S1−23S1 interval by continuous-wave two-photon excitation. Phys. Rev. Lett. 70, 1397–1400 (1993).

    ADS  Google Scholar 

  168. Bumiller, F., Croissiaux, M., Dally, E. & Hofstadter, R. Electromagnetic form factors of the proton. Phys. Rev. 124, 1623–1631 (1961).

    ADS  Google Scholar 

  169. Chen, K. W. et al. Measurement of proton electromagnetic form factors at high momentum transfers. Phys. Rev. 141, 1267–1285 (1966).

    ADS  Google Scholar 

  170. Bartel, W. et al. Small-angle electron-proton elastic scattering cross sections for momentum transfers between 10 and 105 f−2. Phys. Rev. Lett. 17, 608–611 (1966).

    ADS  Google Scholar 

  171. Janssens, T., Hofstadter, R., Hughes, E. B. & Yearian, M. R. Proton form factors from elastic electron–proton scattering. Phys. Rev. 142, 922–931 (1966).

    ADS  Google Scholar 

  172. Berger, C., Burkert, V., Knop, G., Langenbeck, B. & Rith, K. Electromagnetic form factors of the proton at squared four-momentum transfers between 10 and 50 fm−2. Phys. Lett. B 35, 87–89 (1971).

    ADS  Google Scholar 

  173. Price, L. E. et al. Backward-angle electron–proton elastic scattering and proton electromagnetic form factors. Phys. Rev. D 4, 45–53 (1971).

    ADS  Google Scholar 

  174. Bartel, W. et al. Measurement of proton and neutron electromagnetic form factors at squared four-momentum transfers up to 3 (GeV/c)2. Nucl. Phys. B 58, 429–475 (1973).

    ADS  Google Scholar 

  175. Walker, R. C. et al. Measurements of the proton elastic form factors for 1 ≤ Q2 ≤ 3 (GeV/c)2 at SLAC. Phys. Rev. D 49, 5671–5689 (1994).

    ADS  Google Scholar 

  176. Andivahis, L. et al. Measurements of the electric and magnetic form factors of the proton from Q2 = 1.75 to 8.83 (GeV/c)2. Phys. Rev. D 50, 5491–5517 (1994).

    ADS  Google Scholar 

  177. Christy, M. E. et al. Measurements of electron-proton elastic cross sections for 0.4 < Q2 < 5.5(GeV/c)2. Phys. Rev. C 70, 015206 (2004).

    ADS  Google Scholar 

  178. Qattan, I. A. et al. Precision Rosenbluth measurement of the proton elastic form factors. Phys. Rev. Lett. 94, 142301 (2005).

    ADS  Google Scholar 

  179. Milbrath, B. D. et al. Comparison of polarization observables in electron scattering from the proton and deuteron. Phys. Rev. Lett. 80, 452–455 (1998).

    ADS  Google Scholar 

  180. Dieterich, S. et al. Polarization transfer in the 4He(e →, e′p →)3H reaction. Phys. Lett. B 500, 47–52 (2001).

    ADS  Google Scholar 

  181. Gayou, O. et al. Measurements of the elastic electromagnetic form factor ratio μpGEp/GMp via polarization transfer. Phys. Rev. C 64, 038202 (2001).

    ADS  Google Scholar 

  182. Pospischil, T. et al. Measurement of GEp/GMp via polarization transfer at Q2 = 0.4GeV/c2. Eur. Phys. J. A 12, 125–127 (2001).

    ADS  Google Scholar 

  183. Strauch, S. et al. Polarization transfer in the \({}^{4}{\rm{H}}{\rm{e}}{\overrightarrow{e},{e}^{{\prime} }\overrightarrow{p}}^{3}{\rm{H}}\) reaction up to Q2 = 2.6(GeV/c)2. Phys. Rev. Lett. 91, 052301 (2003).

    ADS  Google Scholar 

  184. Punjabi, V. et al. Proton elastic form factor ratios to Q2 = 3.5GeV2 by polarization transfer. Phys. Rev. C 71, 055202 (2005).

    ADS  Google Scholar 

  185. MacLachlan, G. et al. The ratio of proton electromagnetic form factors via recoil polarimetry at Q2 = 1.13(GeV/c)2. Nucl. Phys. A 764, 261–273 (2006).

    ADS  Google Scholar 

  186. Jones, M. K. et al. Proton GE/GM from beam-target asymmetry. Phys. Rev. C 74, 035201 (2006).

    ADS  Google Scholar 

  187. Crawford, C. B. et al. Measurement of the proton’s electric to magnetic form factor ratio from \({}^{1}\overrightarrow{{\rm{H}}}(\overrightarrow{e},{e}^{{\prime} }p)\). Phys. Rev. Lett. 98, 052301 (2007).

    ADS  Google Scholar 

  188. Paolone, M. et al. Polarization transfer in the \({}^{4}{\rm{He}}{(\overrightarrow{e},{e}^{{\prime} }\overrightarrow{p})}^{3}{\rm{H}}\) reaction at Q2 = 0.8 and 1.3 (GeV/c)2. Phys. Rev. Lett. 105, 072001 (2010).

    ADS  Google Scholar 

  189. Meziane, M. et al. Search for effects beyond the Born approximation in polarization transfer observables in \(\overrightarrow{e}p\) elastic scattering. Phys. Rev. Lett. 106, 132501 (2011).

    ADS  Google Scholar 

  190. Ron, G. et al. Low-Q2 measurements of the proton form factor ratio μpGE/GM. Phys. Rev. C 84, 055204 (2011).

    ADS  Google Scholar 

  191. Puckett, A. J. R. et al. Final analysis of proton form factor ratio data at Q2 = 4.0, 4.8, and 5.6 GeV2. Phys. Rev. C 85, 045203 (2012).

    ADS  Google Scholar 

  192. Puckett, A. J. R. et al. Polarization transfer observables in elastic electron–proton scattering at Q2 = 2.5, 5.2, 6.8, and 8.5 GeV2. Phys. Rev. C 96, 055203 (2017).

    ADS  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de France, Paris, France

    Jean-Philippe Karr

  2. Université d’Evry-Val d’Essonne, Université Paris-Saclay, Evry, France

    Jean-Philippe Karr

  3. Laboratoire de Physique des 2 Infinis Irène Joliot-Curie (IJCLab), Université Paris-Saclay, CNRS/IN2P3, Orsay, France

    Dominique Marchand & Eric Voutier

Authors
  1. Jean-Philippe Karr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dominique Marchand
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eric Voutier
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors contributed equally to all aspects of the Review.

Corresponding author

Correspondence to Jean-Philippe Karr.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information

Nature Reviews Physics thanks A. Gasparian, T. Udem and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Related links

CODATA internationally recommended 2018 values of the fundamental physical constants: https://physics.nist.gov/cuu/Constants/

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karr, JP., Marchand, D. & Voutier, E. The proton size. Nat Rev Phys 2, 601–614 (2020). https://doi.org/10.1038/s42254-020-0229-x

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42254-020-0229-x

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing