Muonic hydrogen and the proton radius puzzle

A. Antognini and F. Kottmann on behalf of the CREMA collaboration
Institut für Teilchenphysik, ETH Zürich, 8093 Zürich and Paul Scherrer Institute, 5232 Villigen PSI


Figure 1: Part of the laser system used to measure the muonic hydrogen 2S-2P splitting.
Figure 2: Energy levels, cascade and experimental principle in muonic hydrogen.
(a) About 99% of the muons stopped in H2 proceed directly to the 1S ground state during the muonic cascade, emitting X-rays of the Lyman series at around 2 keV. 1% of the muons form long-lived metastable 2S states with a lifetime t2S = 1 μs at 1 hPa (1 mbar) hydrogen gas.
(b) The muonic hydrogen atoms are illuminated by a short laser pulse 1 μs after their formation. If the laser frequency is on resonance, 2 keV X-rays from 2P-1S transitions in time coincidence with the laser light are observed.
(c) Muonic hydrogen energy levels with details of the fine- and hyperfine-splittings of the n = 2 states. The proton finite size effect accounts for 2% of the total splitting which is dominated by vacuum polarization.
Figure 3: 2S - 2P resonance in muonic hydrogen. The number of 2 keV X-rays in coincidence with the laser pulse, normalized to the number of ’prompt’ events, is plotted as a function of the laser frequency. On resonance we have six events per hour. The predictions (orange points) for the line position assume the proton radius from the CODATA group [2] and from world average electron scattering data [3]. One of the frequency calibration measurements using water absorption is also shown (in green).


The hydrogen atom is unique since physical theories can be applied to it “without” approximations. Any discrepancy between theoretical prediction and experimental measurement which may be unveiled at any increase of theoretical and experimental accuracy thus holds the potential for new fundamental insights.

Nothing can hide in hydrogen, not even the proton at its center. In fact, measurements with hydrogen beams by Stern in 1933 revealed that the magnetic moment of the proton deviated from the prediction of the Dirac relativistic theory. This was the first indication that the proton - contrary to the electron - has a structure. In 1947 measurements of the 2S-2P (Lamb shift) and 1S-hyperfine splitting in hydrogen deviated from those predicted by the Dirac equation. This was the initiation for the development of quantum electrodynamics (QED). In the last four decades, the goal to measure hydrogen energy levels with greater accuracy has lead to advances in high resolution spectroscopy and metrology. This peaked with the invention of the frequency comb laser by Hänsch in the late 90ies. The high accuracy obtained with such techniques provided cornerstones to test bound-state QED, to determine the Rydberg constant and the proton radius (assuming the correctness of the theory), and to search for slow time variations of fundamental constants.

Hydrogen energy levels are slightly modified by the fact that in contrast to the electron the proton has a size. Hence, to precisely predict these energy levels an accurate knowledge of the root-mean-square charge radius of the proton is necessary. The historical method of determining the proton radius was based upon scattering electrons on protons, in effect by scattering an electron beam on a liquid hydrogen target. The uncertainty related to the knowledge of the proton radius extracted from electron-proton scattering limited the prediction accuracy of the hydrogen energy levels, and consequently it was limiting the comparison between theory and measurements. Therefore to advance the check (comparison between prediction and measurement) of bound-state QED describing the hydrogen energy levels it was necessary to have a more precise determination of the proton radius. This was one of the main motivations for our experiment: to measure the 2S-2P energy difference in muonic hydrogen (µp), an exotic atom composed by a negative muon and a proton. The single electron of a hydrogen atom is replaced by a negative muon which has a lifetime of only 2 microseconds and is 200 times heavier than the electron. According to the laws of quantum mechanics the muon wave functions in S-states overlap therefore more with the proton and the corresponding µp energy levels are sensitive to the proton size. By measuring the 2S-2P transition frequency in muonic hydrogen it is thus possible to extract with great accuracy the proton radius, assuming that the main QED contributions to the 2S-2P splitting are correctly predicted by theory.

Method and measurements

Our experiment is based upon laser spectroscopy of muonic hydrogen. The main components which had to be developed for this experiment are a low energy muon beam, an infrared laser system used to drive 2S-2P transitions in µp, and detectors for 2 keV X-rays emitted from 2P-1S transitions. More details are given in Ref. [1]. Muonic hydrogen is produced by stopping negative muons in hydrogen gas. Only at the Paul Scherrer Institut (PSI), Switzerland, is there a suffciently strong low energy muon beam suited for such an experiment. The µp atoms are produced at highly excited states (around n=14). Most of these de-excite quickly to the 1S-ground state, but ~1% populate the long-lived 2S-state (Fig. 2 (a)). A short laser pulse tunable to a wavelength around λ ≈ 6 µm (corresponding to the 2S-2P energy splitting) illuminates the muonic atom. 2S → 2P transitions are induced by the laser light (Fig. 2 (b)), immediately followed by 2P → 1S de-excitation via emission of a 2 keV X-ray (lifetime t2P = 8.5 ps). The transition from the 2S to the 2P state and the subsequent emission of X-ray only occurs if the laser frequency corresponds to the energy difference between 2S and 2P levels.

Fig. 3 shows the resulting resonance curve obtained by plotting the number of 2 keV X-rays at different laser wavelengths that occur in time-coincidence with the laser pulse. The centroid position is determined with a statistical uncertainty of 700 MHz. The linewidth is in good agreement with the theoretical prediction. The laser frequency is known over the whole region with 300 MHz accuracy and was determined with two independent methods. The systematics are completely dominated by the laser frequency calibration. Effects like Zeeman shift, Doppler shift, AC- and DC-Stark shifts and pressure shift are smaller than 50 MHz.

In summary, we have measured the muonic hydrogen transition at a frequency of 49881.88(76) GHz which corresponds to an energy of 206.2949(32) meV [1]. The position of this line strongly disagrees with predictions (shown by the orange points in Fig. 3) which have been computed assuming the proton radius extracted from hydrogen spectroscopy and theory, and the proton radius from electron-proton scattering experiments.

Interpretation of the measured transition

Comparison of the measured transition energy, ∆Eexp=206.2949(32)meV, with calculations of the corresponding µp 2S-2P energy difference, ∆Etheo=209.9779(49) − 5.2262 + 0.0347 meV (where rp is given in fm) results in a determination of the only parameter which is not well known: the root-mean-square proton charge radius . The resulting value rp=0.84184(36)exp(56)theo fm = 0.84184(67) fm is 10 times more precise but 5σ smaller than the previous CODATA value [2] which is essentially based on hydrogen spectroscopy. It is also 3σ smaller than the electron-proton world average scattering result [3].

Fig. 4 shows the several proton radius values from various experiments and calculations. It includes a very recent result from electron-proton scattering data obtained at Mainz [4] which also deviates by 5σ from the µp value. The origin of the discrepancies is not yet known. It may come from theory of the muonic hydrogen energy levels (used to deduce our new value), from problems in hydrogen spectroscopy experiments or hydrogen energy level theory (both used to deduce the “H” value by CODATA), from inconsistent definitions of the proton radius in the various systems, or it occurs from uncalculated or new effects.

Figure 4: Our new proton radius value from muonic hydrogen spectroscopy (μp) is in strong disagreement with the values extracted from hydrogen spectroscopy (H) [2], with the value from the world average electron scattering (e-p scatt.) data [3], and with the new electron scattering value from Mainz [4]. Values from “lattice QCD” calculations [5] and a “dispersion” fit of scattering data [6] are model dependent.



First of all we need to understand the origin of the observed discrepancy. It may be a computational mistake of the energy levels in muonic hydrogen or hydrogen, a fundamental error in bound-state QED, an unknown effect related to the proton or the muon, or a contribution which has been neglected. In addition it may be that the Rydberg constant, the most precise constant in physics, has to be slightly corrected.

As soon as the discrepancy is solved the new precise proton radius value will pave the way to check hydrogen energy level theory to an unprecedented level of accuracy. Hydrogen theory is intriguing, since bound-state QED is challenging both from mathematical and from fundamental (binding effects, relativistic two-body system) points of view. Bound-state QED in hydrogen has to deal with several expansion parameters which account for radiative, recoil, relativistic, binding, and nuclear structure effects. Calculation of the one-loop radiative contributions has taken more than five decades because of the complexity of the binding properties. Several groups are presently working on the two-loop contributions. Two different approaches to the binding effects have been developed: one perturbative and one all-order / non-perturbative. Our experiment has opened the way for checking these problematic terms and the various approaches.

Hydrogen may therefore be considered as a platform for developing tools for even stronger bound systems. In addition our measurement will improve on the determination of the Rydberg constant by a factor of 5 to a level of 1×10-12. The Rydberg constant is the best know physical constant and plays a very important role because it connects fundamental constants like the fine-structure constant a, the Planck constant and the electron mass. Also, the proton radius itself is an interesting benchmark for lattice quantum chromodynamics (QCD), aiming to model the proton from its constituents: the quarks and gluons.

Spectroscopy in hydrogen-like atoms continues to challenge our understanding of physics. We have also measured other 2S-2P transitions in muonic hydrogen and deuterium which are presently being analyzed, and a similar measurement of the Lamb shift in muonic helium ions is being prepared at PSI. A combined analysis of these experiments will help to clarify whether the problem arises in the muonic sector or if it is related to the proton, or the Rydberg constant, or if it originates from bound-state QED.


[1] R. Pohl, A. Antognini, F. Nez, et al., Nature 466, 213 (2010).
[2] CODATA-06: P. J. Mohr, B. N. Taylor and D. B. Newell, Rev. Mod. Phys. 80, 633 (2008).
[3] I. Sick, Phys. Lett. B 576, 62 (2003); P. G. Blunden and I. Sick, Phys. Rev. C 72, 057601 (2005).
[4] J. C. Bernauer et al., arXiv nucl-ex, 1007.5076 (2010).
[5] P. Wang et al., Phys. Rev. D 79, 094001 (2009).
[6] M. A. Belushkin, H. W. Hammer, and U. G. Meissner, Phys. Rev. C 75, 035202 (2007).


An animation of our experiment is found here.
Negative pions are injected into a magnetic trap where they decay into muons of MeV energy. The muon is slowed down and when its momentum is sufficiently low it escapes from the magnetic trap and goes through a toroidal assembly of coils which act as momentum filter. Then the muon enters a 5 Tesla solenoid containing the hydrogen target and a muon detector based on stacks of carbon foils. When a muon crosses these foils electrons are released. The successive E×B field separates electrons from the muons. The muon continues and reaches a second stack of carbon foil before it stops in hydrogen. The electrons released by the foils are detected and serve as a trigger for the laser system. The muon stopped in hydrogen forms muonic hydrogen. The laser system delivers a laser pulse which illuminates the muon stopping volume. The laser-induced 2 keV X-ray is detected with large-area avalanche photodiodes (LAAPD).


[Released: October 2010]