The size of the proton


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 −110 kHz/c (4.9 standard deviations), or the calculations of the QED effects in atomic hydrogen or muonic hydrogen atoms are insufficient.

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Figure 1: Energy levels, cascade and experimental principle in muonic hydrogen.
Figure 2: Muon beam.
Figure 3: Laser system.
Figure 4: Summed X-ray time spectra.
Figure 5: Resonance.


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We thank L. Simons and B. Leoni for setting up the cyclotron trap, H. Brückner, K. Linner, W. Simon, O. Huot and Z. Hochman for technical support, P. Maier-Komor, K. Nacke, M. Horisberger, A. Weber, L. Meier and J. Hehner for thin foils and windows, N. Schlumpf, U. Hartmann and M. Gaspar for electronics, S. Spielmann-Jaeggi and L. Carroll for optical measurements, Ch. Parthey and M. Herrmann for their help, the MEG-collaboration for a share of beam-time, and A. Voss, B. Weichelt and J. Fruechtenicht for the loan of a laser pump diode. We acknowledge the essential contributions of H. Hofer and V.W. Hughes in the initial stages of the experiment. We also thank the PSI accelerator division, the Hallendienst, the workshops at PSI, MPQ and Fribourg, and other support groups for their help. We acknowledge support from the Max Planck Society and the Max Planck Foundation, the Swiss National Science Foundation (project 200020-100632) and the Swiss Academy of Engineering Sciences, the BQR de l'UFR de physique fondamentale et appliquée de l'Université Paris 6, the program PAI Germaine de Staël no. 07819NH du ministère des affaires étrangères France, and the Fundação para a Ciência e a Tecnologia (Portugal) and FEDER (project PTDC/FIS/82006/2006 and grant SFRH/BPD/46611/2008). P.I. and E.-O.L.B. acknowledge support from the ‘ExtreMe Matter Institute, Helmholtz Alliance HA216/EMMI’.

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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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Correspondence to Randolf Pohl.

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Pohl, R., Antognini, A., Nez, F. et al. The size of the proton. Nature 466, 213–216 (2010).

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