The deuteron is too small, too
The radius of the proton has remained a point of debate ever since the spectroscopy of muonic hydrogen indicated a large discrepancy from the previously accepted value. Pohl et al. add an important clue for solving this so-called proton radius puzzle. They determined the charge radius of the deuteron, a nucleus consisting of a proton and a neutron, from the transition frequencies in muonic deuterium. Mirroring the proton radius puzzle, the radius of the deuteron was several standard deviations smaller than the value inferred from previous spectroscopic measurements of electronic deuterium. This independent discrepancy points to experimental or theoretical error or even to physics beyond the standard model.
Science, this issue p. 669
Abstract
The deuteron is the simplest compound nucleus, composed of one proton and one neutron. Deuteron properties such as the root-mean-square charge radius rd and the polarizability serve as important benchmarks for understanding the nuclear forces and structure. Muonic deuterium μd is the exotic atom formed by a deuteron and a negative muon μ–. We measured three 2S-2P transitions in μd and obtain rd = 
fm, which is 2.7 times more accurate but 7.5σ smaller than the CODATA-2010 value rd = 
fm. The μd value is also 3.5σ smaller than the rd value from electronic deuterium spectroscopy. The smaller rd, when combined with the electronic isotope shift, yields a “small” proton radius rp, similar to the one from muonic hydrogen, amplifying the proton radius puzzle.
Precision spectroscopy of atomic energy levels can be used to determine properties of the nucleus (1). Deuterium (D), for example, is a heavier isotope of hydrogen (H), with a nucleus, the deuteron (d), composed of one proton and one neutron (2). D was discovered through a tiny shift in the Balmer spectral lines of D-enriched hydrogen (3). This shift is caused mainly by the mass difference between the proton and the deuteron. Today, the nuclear masses are accurately known from cyclotron frequency measurements in a Penning trap (1), and the measured isotope shift of the 1S-2S transition in H and D (4) determines the (squared) deuteron-proton charge radius difference (5)
(1)This is because the wave function of atomic S states is maximal at the origin, where the nucleus resides, and the wave function overlap with the extended nuclear charge distribution reduces the atomic binding energy. Equation 1 links measurements of transition frequencies in H and D. These, together with elastic electron scattering on protons (6) and deuterons (7), determine the Rydberg constant R∞, rp and rd in the CODATA adjustment of the fundamental physical constants (1).
Muonic atoms are a special class of “exotic” atoms that offer access to nuclear properties with much higher accuracy. In a muonic atom, the nucleus is orbited by one negative muon μ–, instead of the usual electrons e–. The muon’s larger mass 
results in a muonic Bohr radius that is smaller than the corresponding electronic Bohr radius by the ratio of reduced masses 
. Here 
is the mass of the lepton (muon μ– or electron e−), and 
is the mass of the nucleus. As the Bohr radius shrinks proportionally to 
, the overlap of the muon’s wave function with the nuclear charge distribution increases as 
. For μd, 
, and the wave function overlap is 
larger in μd than in D. A measurement of the Lamb shift (2P-2S energy difference) in μd is therefore extremely sensitive to the deuteron charge radius rd.
Our recent measurements of the Lamb shift in muonic hydrogen 
have resulted in a value of the proton charge radius 
= 0.84087(39) fm, which is 10 times more accurate, but 4%, or 
, smaller (8, 9) than the CODATA-2010 value (1), which is the most recently published CODATA compilation. This so-called “proton radius puzzle” has questioned the correctness of various experiments or quantum electrodynamics (QED) calculations, the value of the Rydberg constant, our understanding of the proton structure, or the standard model of particle physics (10, 11).
Here we present measurements of the three 2S-2P transitions in 
highlighted in Fig. 1, yielding a precise value of 
. The principle of the experiment is to form 
atoms in the metastable 2S state (12) and to measure the 2S-2P transitions by pulsed laser spectroscopy. Comparison with theory (13) reveals 
. The muonic deuterium data presented here were acquired in the same measurement period as the muonic hydrogen data in (8, 9). Independent and reliable calculations of QED (14–17) and nuclear structure effects (18–22) in 
, which are required to interpret the experiment, have recently become available and are summarized in (13).
The order of the 2P3/2 sublevels is changed by the nuclear quadrupole moment (13). The three measured transitions are indicated.
Measurement of the spectral lines of muonic deuterium
The experiment has been described before (8, 9). In brief, a 5 × 12 mm2 beam of low-energy negative muons 
(3-keV kinetic energy, average rate 600/s) is stopped in a 20-cm-long target filled with 1 hPa of D2 gas at 20°C. A pulsed laser system (23, 24) is triggered on the detection of a single arriving muon and provides pulses with an energy of ~0.25 mJ, tunable around a wavelength of 6 μm, and calibrated against water vapor absorption lines known within a few megahertz (25). A multipass mirror cavity (26) ensures good laser illumination of the muon stop volume. Large-area avalanche photo diodes (27, 28) detect the 2-keV Kα x-rays from the radiative 2P→1S transition that follows the laser-induced 2S→2P excitation of μd. The laser frequency is changed every few hours, and the resonances displayed in Fig. 2 are obtained by plotting the number of 2-keV x-rays (normalized to the number of stopped muons) detected in time coincidence with the laser pulse, as a function of laser frequency. On the peak of the resonance, we recorded up to 10 laser-induced x-rays (“events”) per hour with all data reduction cuts (9) applied. The background level of about 2 events per hour originates mainly from misidentified muon decay electrons. About a third of the recorded events are without laser light, providing the expected background level shown as horizontal bands in Fig. 2. The resonances are fitted with a flat background plus a Lorentzian line shape model that takes into account varying laser pulse energies and saturation effects.
The resonances are labeled #1 (A), and #2 and #3 (B). The signal (y axis) is “normalized number of events” as described in (8). Predicted resonance positions are shown based on Eqs. 7 and 8: The CODATA-2010 deuteron radius (pink, Eq. 10) would correspond to ~104 GHz lower resonance positions, which is a difference of 7.5σ. The “expected” deuteron radius Eq. 13, (“
+ iso,” brown) obtained by combining the proton radius from muonic hydrogen (9) and the electronic isotope shift (“iso”), Eq. 1, is consistent with the observed resonance positions within ~2.6σ. The top and bottom panel’s data were recorded in 1 week and 2 days, respectively. As an example, the three highest points around the peak of resonance #1 contain a total of 260 events, recorded in 21 hours.
The three resonances shown in Fig. 2 are the μd transitions 
, 
, and 
, abbreviated as #1, #2, and #3, respectively. Their positions and uncertainties are
(2)
(3)
(4)The systematic uncertainties of 0.35 GHz arise from laser frequency fluctuations (8) and Zeeman shifts from a conceivable small admixture of circular polarized light and the 5 T magnetic field of the muon beam line. Line-pulling effects from off-resonant excitation of neighboring levels are negligible (29).
Deuteron charge radius
For the fit of line #1, the Lorentzian width was fixed to the natural radiative line width of 
GHz (8, 9), as the freely fitted value 
GHz is 
too small. Both fits agreed on the line center within 0.33 GHz, and the uncertainty quoted in Eq. 2 is the larger one from the fit with fixed width. The difference 
GHz from the fit is in good agreement (
) with the theoretical value of 88.045 GHz (13). The amplitude of line #3 is larger than zero only with a significance of 
, but it serves to identify line #2 unambiguously. The alternative—namely, that the left peak in Fig. 2 (bottom) is in fact line #3—is disfavored with 
significance thanks to the absence of a peak with twice the amplitude ~90 GHz left of line #2.
Combining the three measured frequencies and using the theoretical 2P fine structure and 2P3/2 hyperfine splittings (13), we determine the 2P-2S Lamb shift (LS) and 2S hyperfine splitting (HFS) in μd
(5)
(6)with total experimental uncertainties of 
and 
meV, respectively. The measured 2S HFS is in excellent agreement with the theoretical value, 
meV (13).
The Lamb shift in μd is extraordinarily sensitive (13) to the root mean square (RMS) deuteron charge radius
(7)
where
(8)is the deuteron polarizability contribution (13) from two-photon exchange (TPE), recently calculated with good accuracy (18–22). The charge radius effect in Eq. 7 contributes as much as 14% to the 2P-2S Lamb shift, which explains the excellent sensitivity of our measurement to rd. We obtain rd from equating Eqs. 5 and 7, and using Eq. 8, which yields
(9)where the theory uncertainty is almost exclusively from 
(Eq. 8). This radius is in 
disagreement with the CODATA value (1), which is the best estimate of the deuteron radius obtained from precision spectroscopy of H and D and electron scattering on protons and deuterons,
(10)(see Fig. 3). We are hence faced with the fact that precision determinations of the Lamb shift in both 
and 
, from a total of five measured resonances, each show a 
discrepancy to the predictions based on fundamental physical constants from the self-consistent CODATA world average (1), carefully checked QED calculations (13, 30), and physics within the standard model.
Our value Eq. 9 (“
,” red) has a 7.5σ discrepancy with the CODATA-2010 value (1), but is within 2.6σ of the smaller “expected” value Eq. 13 (“
+ iso,” brown, see text), obtained by combining the proton radius from muonic hydrogen (9) and the electronic isotope shift, Eq. 1. The value from laser spectroscopy of electronic deuterium (“D spectroscopy,” blue, Eq. 11) is obtained as detailed in (31) and is 3.5σ larger than the 
value. The world average from elastic e-d scattering (7) (“e-d scatt,” green) is also shown.
The CODATA deuteron radius rd is tightly linked to the CODATA proton radius rp, by virtue of Eq. 1. However, as detailed in (31), we have deduced a deuteron charge radius considering spectroscopy data in regular deuterium alone–i.e., without relying on the value of the proton radius. This yields a value of
(11)in excellent agreement with the CODATA value, but 
larger than the value obtained here from muonic deuterium (see Fig. 3, blue point, “D spectroscopy”).
This distinct 
discrepancy between the atomic physics determinations of rd from D and 
is almost as severe as the 
atomic physics discrepancy between the rp values from H spectroscopy [see (1), table XXXVIII, adjustment 8] and 
(9) (see Fig. 4). These two discrepancies are independent, as explained in (31).
The difference between the deuteron radii from the spectroscopy of electronic and muonic deuterium is only 0.017 fm, or 0.8%. Thus, even though the deuteron charge radius 
fm, extracted from elastic electron-deuteron scattering (7), is accurate to 0.5%, it is unfortunately not accurate enough to distinguish between the values from 
and CODATA.
Proton and deuteron radius puzzle
Many attempts to explain the proton radius discrepancy exist (10, 11). Our muonic deuterium result provides fresh insight, as the so-called “proton radius puzzle” is in fact not limited to the proton; there is a distinct deuteron radius puzzle. Using rd (CODATA) in Eq. 7 yields a Lamb shift that is 
meV smaller than the measured value, Eq. 5, and hence resonance frequencies that are 
104 GHz smaller than observed (Fig. 2). The 
is even somewhat larger than the proton radius discrepancy 
meV between the LS we observed in 
and the one calculated with the CODATA value of rp (9).
The ratio of discrepancies in 
and 
, 
is in agreement with the ratio of the wave-function overlap from the reduced mass ratio, 
. Such a scaling is expected for several beyond–standard model (BSM) physics scenarios (10, 11, 32–34), where a new force between muons and protons is responsible for the change in the observed LS and can at the same time explain the long-standing 
discrepancy in the muon g-2 value (35). In these models, the coupling of such a new force to neutrons must be negligible to fulfill other experimental constraints. The same scaling is also expected for explanations based on an unexpectedly large TPE contribution to proton polarizability (36), or an effect of a “sea of leptons” inside the proton (37, 38).
Before resorting to BSM solutions, however, one should investigate what it would take to “solve” the two discrepancies within SM physics. As noted before (8–11), and explained in more detail in (31), the reconciliation of electronic and muonic spectroscopy data still requires rather drastic measures.
For one, the CODATA Rydberg constant could be wrong by 
—for example, because of a yet- undiscovered, common systematic effect in the most precise measurements of transitions from the 2S to the 8S, 8D, and 12D states in H and D (39). Such a change of 
would shift the proton radius from H to the smaller 
value (8–10, 31). It would also bring the deuteron radius from D to within 
of the 
value (see below).
Alternatively, the QED theory of the Lamb shift in electronic H and D could be missing a large contribution of ~110 kHz, which corresponds to about 44 times the claimed theory uncertainty (
) of ~2.5 kHz. Such a missing QED contribution would bring the charge radii from H and D spectroscopy into agreement with their muonic counterparts, without changing the Rydberg constant (31).
Third, a systematic shift of all spectroscopic muonic measurements by 
(corresponding to 80 GHz in 
and 104 GHz in 
), or a missing theory term in the Lamb shift of muonic atoms that accounts for the missing 
, could be the source of the discrepancies. This theory error would correspond to 
in 
, and 
in 
, where the uncertainty of the TPE contribution is about 10 times larger (13). The claimed uncertainty of the pure QED (i.e., non-TPE) contributions in 
(
) is about 440 (220) times smaller than the 
(13).
Neither a shift of 
by 
from the CODATA value, nor a change of the LS in H and D by 
kHz, will, however, appreciably affect Eq. 1 (5). Hence, we can proceed and draw conclusions from the fact that the muonic isotope shift
(12)is compatible within 
with the “electronic” isotope shift, Eq. 1, but five times less accurate. The absolute values of 
and 
from the muonic 2S-2P measurements are thus roughly consistent with the size difference from the electronic 1S-2S measurement (4, 5), Eq. 1.
The dominant source of uncertainty in Eq. 12 is the calculated TPE contribution (Eq. 8), whose effect on the uncertainty of 
from 
, Eq. 9, is about six times larger than the experimental uncertainty. Hence, we are tempted to ascribe the remaining 
discrepancy between the electronic and muonic isotope shift to the TPE contribution to the LS in 
.
We can thus use the muonic proton radius from 
(9), 
fm, and the electronic isotope shift, Eq. 1, to obtain a precise value of the deuteron charge radius in an indirect way. The resulting value
(13)was given in (9) and is indicated as “
+ iso” in Figs. 2 and 3. It is the most accurate value of the deuteron RMS charge radius and is independent of the TPE contribution in 
.
Using this “expected” deuteron radius from Eq. 13 in the theory expression for the LS in 
, Eq. 7, yields an experimental value for the TPE contribution to the LS in 
(14)from the measured LS in Eq. 5. It is 
larger than the calculated value, Eq. 8, but three times more accurate, making it a benchmark for ab initio calculations of the deuteron (2, 19, 20, 22) or analysis of virtual Compton scattering data (21).
In a similar manner, we determine the experimental value of the polarizability—i.e., the inelastic part of the TPE contribution to the 2S-HFS—using our measured HFS, Eq. 6, Eq. 42 of (13), and the Zemach radius of the deuteron 
fm from (40). We obtain
(15)where the experimental uncertainty is by far the dominant one. This agrees with the theoretical value 
meV, which has been calculated only recently (16).
Finally, we note that the reasoning that leads to Eq. 13 can of course be inverted. Using the measured muonic deuteron charge radius, Eq. 9, and the electronic isotope shift, Eq. 1, we obtain a new value for the proton radius
(16)confirming the “small” proton charge radius from muonic hydrogen (8, 9), further amplifying the “proton radius puzzle” (10, 11) (see Fig. 4).
Ultimately, only new experiments can shed more light on the proton and deuteron radius discrepancies. A lot of activity exists in elastic electron scattering (41, 42), with the hope for refined values of 
and 
. Muon scattering on the proton will be able to check the BSM hypothesis (43). Moreover, several atomic physics measurements are underway to verify and improve the Rydberg constant and the proton and deuteron radius from regular (electronic) hydrogen and deuterium (44–46).
Shown are the values deduced from muonic hydrogen (9) (“
,” brown) and from muonic deuterium (Eq. 9, “
+ iso,” red), which both differ by >7σ from the CODATA-2010 value (1). Many other determinations exist, and we highlight the values from spectroscopy of H (but not D), from CODATA-2010 [(1), table XXXVIII, adjustment 8] (“H spectroscopy”, blue); elastic electron-proton scattering (dark green) from (6) and (47); and electron scattering data analyses based on dispersion relations (light green), both less recent (48) and more recent (49) than the 
value (8, 9). Many more values exist (50).
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- Acknowledgments: We thank E. Borie for the calculations that made this measurement possible; I. Sick for insightful discussions; L.M. Simons and B. Leoni for setting up the cyclotron trap; R. Rosenfelder and C. Hoffman for support; 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; Th. Udem for insightful discussions; the MEG-collaboration for a share of valuable 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 and thank K. Kirch for his continuous support. We also thank the PSI accelerator division, the Hallendienst, the workshops at PSI, MPQ, and Fribourg, and other support groups for their valuable help. We acknowledge support from the European Research Council (ERC StG. 279765), the Max Planck Society and the Max Planck Foundation, the Swiss National Science Foundation (project 200020-100632, 200021L_138175, 200020_159755, 200021_165854) and the Swiss Academy of Engineering Sciences, the BQR de l’UFR de physique fondamentale et appliquée de l’Université Pierre et Marie Curie- Paris 6, the program PAI Germaine de Staël no. 07819NH du ministère des affaires étrangères France, the Fundação para a Ciência e a Tecnologia (Portugal) and FEDER (project PTDC/FIS/102110/2008 and grants SFRH/BPD/46611/2008, SFRH/BPD/74775/2010, and SFRH/BPD/76842/2011), Deutsche Forschungsgemeinschaft (DFG) GR 3172/9-1 within the D-A-CH framework, and Ministry of Science and Technology, Taiwan, no. 100-2112-M-007-006-MY3. P.I. acknowledges support by the “ExtreMe Matter Institute, Helmholtz Alliance HA216/EMMI.” Reasonable requests for sharing the data should be addressed to R.P. All authors contributed substantially to this work.
