The size of the proton

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The proton is the primary building block of the visible Universe, but many of its properties—such as its charge radius and its anomalous magnetic moment—are not well understood. The root-mean-square charge radius, rp, has been determined with an accuracy of 2 per cent (at best) by electron–proton scattering experiments1, 2. The present most accurate value of rp (with an uncertainty of 1 per cent) is given by the CODATA compilation of physical constants3. This value is based mainly on precision spectroscopy of atomic hydrogen4, 5, 6, 7 and calculations of bound-state quantum electrodynamics (QED; refs 8, 9). The accuracy of rp as deduced from electron–proton scattering limits the testing of bound-state QED in atomic hydrogen as well as the determination of the Rydberg constant (currently the most accurately measured fundamental physical constant3). An attractive means to improve the accuracy in the measurement of rp is provided by muonic hydrogen (a proton orbited by a negative muon); its much smaller Bohr radius compared to ordinary atomic hydrogen causes enhancement of effects related to the finite size of the proton. In particular, the Lamb shift10 (the energy difference between the 2S1/2 and 2P1/2 states) is affected by as much as 2 per cent. Here we use pulsed laser spectroscopy to measure a muonic Lamb shift of 49,881.88(76) GHz. On the basis of present calculations11, 12, 13, 14, 15 of fine and hyperfine splittings and QED terms, we find rp = 0.84184(67)fm, which differs by 5.0 standard deviations from the CODATA value3 of 0.8768(69)fm. Our result implies that either the Rydberg constant has to be shifted by −110kHz/c (4.9 standard deviations), or the calculations of the QED effects in atomic hydrogen or muonic hydrogen atoms are insufficient.

At a glance


  1. Energy levels, cascade and experimental principle in muonic hydrogen.
    Figure 1: Energy levels, cascade and experimental principle in muonic hydrogen.

    a, About 99% of the muons proceed directly to the 1S ground state during the muonic cascade, emitting ‘prompt’ K-series X-rays (blue). 1% remain in the metastable 2S state (red). b, The μp(2S) atoms are illuminated by a laser pulse (green) at ‘delayed’ times. If the laser is on resonance, delayed Kα X-rays are observed (red). c, Vacuum polarization dominates the Lamb shift in μp. The proton's finite size effect on the 2S state is large. The green arrow indicates the observed laser transition at λ = 6μm.

  2. Muon beam.
    Figure 2: Muon beam.

    Muons (blue) entering the final stage of the muon beam line pass two stacks of ultra-thin carbon foils (S1, S2). The released electrons (red) are separated from the slower muons by E×B drift in an electric field E applied perpendicularly to the B = 5T magnetic field and are detected in plastic scintillators read out by photomultiplier tubes (PM1–3). The muon stop volume is evenly illuminated by the laser light using a multipass cavity.

  3. Laser system.
    Figure 3: Laser system.

    The c.w. light of the Ti:sapphire (Ti:Sa) ring laser (top right) is used to seed the pulsed Ti:sapphire oscillator (‘osc.’; middle). A detected muon triggers the Yb:YAG thin-disk lasers (top left). After second harmonic generation (SHG), this light pumps the pulsed Ti:Sa oscillator and amplifier (‘amp.’; middle) which emits 5 ns short pulses at the wavelength given by the c.w. Ti:Sa laser. These short pulses are shifted to the required λ6µm via three sequential Stokes shifts in the Raman cell (bottom). The c.w. Ti:Sa is permanently locked to a I2/Cs calibrated Fabry-Perot reference cavity (FP). Frequency calibration is always performed at λ = 6µm using H2O absorption. See Online Methods for details.

  4. Summed X-ray time spectra.
    Figure 4: Summed X-ray time spectra.

    Spectra were recorded on resonance (a) and off resonance (b). The laser light illuminates the muonic atoms in the laser time window t[0.887, 0.962]µs indicated in red. The ‘prompt’ X-rays are marked in blue (see text and Fig. 1). Inset, plots showing complete data; total number of events are shown.

  5. Resonance.
    Figure 5: Resonance.

    Filled blue circles, number of events in the laser time window normalized to the number of ‘prompt’ events as a function of the laser frequency. The fit (red) is a Lorentzian on top of a flat background, and gives a χ2/d.f. of 28.1/28. The predictions for the line position using the proton radius from CODATA3 or electron scattering1, 2 are indicated (yellow data points, top left). Our result is also shown (‘our value’). All error bars are the ±1 s.d. regions. One of the calibration measurements using water absorption is also shown (black filled circles, green line).


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Author information


  1. Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany

    • Randolf Pohl,
    • Aldo Antognini,
    • Theodor W. Hänsch &
    • Tobias Nebel
  2. Laboratoire Kastler Brossel, École Normale Supérieure, CNRS, and Université P. et M. Curie-Paris 6, 75252 Paris, Cedex 05, France

    • François Nez,
    • François Biraben,
    • Paul Indelicato,
    • Lucile Julien,
    • Eric-Olivier Le Bigot &
    • Catherine Schwob
  3. Departamento de Física, Universidade de Coimbra, 3004-516 Coimbra, Portugal

    • Fernando D. Amaro,
    • João M. R. Cardoso,
    • Daniel S. Covita,
    • Luis M. P. Fernandes,
    • José A. M. Lopes,
    • Cristina M. B. Monteiro &
    • Joaquim M. F. dos Santos
  4. I3N, Departamento de Física, Universidade de Aveiro, 3810-193 Aveiro, Portugal

    • Daniel S. Covita &
    • João F. C. A. Veloso
  5. Physics Department, Yale University, New Haven, Connecticut 06520-8121, USA

    • Andreas Dax &
    • Satish Dhawan
  6. Institut für Strahlwerkzeuge, Universität Stuttgart, 70569 Stuttgart, Germany

    • Adolf Giesen &
    • Thomas Graf
  7. Physics Department, National Tsing Hua University, Hsinchu 300, Taiwan

    • Cheng-Yang Kao &
    • Yi-Wei Liu
  8. Département de Physique, Université de Fribourg, 1700 Fribourg, Switzerland

    • Paul Knowles,
    • Livia Ludhova,
    • Françoise Mulhauser &
    • Lukas A. Schaller
  9. Department of Chemistry, Princeton University, Princeton, New Jersey 08544-1009, USA

    • Paul Rabinowitz
  10. Dausinger & Giesen GmbH, Rotebühlstr. 87, 70178 Stuttgart, Germany

    • Karsten Schuhmann
  11. Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland

    • David Taqqu
  12. Institut für Teilchenphysik, ETH Zürich, 8093 Zürich, Switzerland

    • Franz Kottmann
  13. Present addresses: Deutsches Zentrum für Luft- und Raumfahrt e.V. in der Helmholtz-Gemeinschaft, 70569 Stuttgart, Germany (A.G.); International Atomic Energy Agency, A-1400 Vienna, Austria (F.M.).

    • Adolf Giesen &
    • Françoise Mulhauser


R.P., A.A., F.N., F.D.A., F.B., A.D., A.G., T.G., T.W.H., L.J., C.-Y.K., Y.-W.L., T.N., P.R., K.S., C.S. and F.K. designed, built and operated parts of the laser system. R.P., A.A., F.N., D.S.C., L.M.P.F., P.K., Y.-W.L., J.A.M.L., L.L., C.M.B.M., F.M., T.N., J.M.F.d.S., L.A.S., K.S., D.T., J.F.C.A.V. and F.K. planned, built and set up the various detectors of the experiment. R.P., A.A., D.S.C., F.M., D.T., J.F.C.A.V. and F.K. designed, built, set up and operated the muon beam line. R.P., A.A., F.N., J.M.R.C., D.S.C., A.D., S.D., L.M.P.F., C.-Y.K., P.K., Y.-W.L., F.M., T.N., J.M.F.d.S., K.S., D.T., J.F.C.A.V. and F.K. designed and implemented the electronics used in the experiment. R.P., A.A., J.M.R.C., P.I., P.K., E.-O.L.B. and T.N. set up the computing infrastructure, wrote software and realized the data acquisition system. R.P., A.A., F.N., F.D.A. F.B., J.M.R.C., D.S.C., A.D., L.M.P.F., P.I., L.J., C.-Y.K., P.K., E.-O.L.B., Y.-W.L., J.A.M.L., L.L., C.M.B.M., F.M., T.N., J.M.F.d.S., K.S., C.S., D.T., J.F.C.A.V. and F.K. took part in the months-long data-taking runs. E.-O.L.B., P.I. and F.K. did work on QED theory. R.P., A.A., F.N., F.B., P.I., L.J., P.K., L.L., T.N., D.T. and F.K. analysed the data and wrote the initial manuscript. The manuscript was then read, improved and finally approved by all authors.

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