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The Lamb shift in muonic hydrogen and the proton radius puzzle Proton radius puzzle Randolf Pohl · for the CREMA collaboration
Received: date / Accepted: date
Abstract Keywords Muonic atoms · Proton radius · Laser spectroscopy
1 Introduction
Muonic hydrogen is (µp) is the exotic hydrogen atom made from a proton and a negative muon µ− . Due to the large muon mass mµ ≈ 200 me , the Bohr orbits are about 200 times smaller in muonic hydrogen, compared to regular hydrogen. This results in a 2003 ≈ 107 times larger overlap of the muon’s wave function with the proton, leading to a dramatically increased sensitivity of muonic hydogen energy levels to the finite size of the proton. Fig. 1 shows the n = 2 energy levels of muonic hydrogen. The charge radius contribution to the Lamb shift (2S-2P energy splitting) in muonic hydrogen is as large as 1.8% of the total 2S Lamb shift. This makes muonic hydrogen the ideal, clean, atomic system to study the proton rms charge radius. The magnetic structure of the proton is encoded in the Zemach radius which gives a 0.8% contribution to the 2S hyperfine splitting in µp. We have recently measured two 2S-2P transitions in muonic hydrogen[1, 2], (see Fig. 1). From the Lamb shift we obtain a proton rms charge radius that is ten times more accurate, but differs by 7 standard deviations from the CODATA-2010 world average [3], deduced from hydrogen spectroscopy [4] and elastic electron proton scattering [5, 6] This is what’s now known as the “Proton Radius Puzzle” [7]. The measured 2S Lamb shift in muonic hydrogen is 0.3 meV larger than the expected value, based on the CODATA-2010 value of the proton charge radius. The measured resonances in µp [1, 2] are 75 GHz away from the expected position, which corresponds to 4 natural line widths, or more than 100 experimental error bars of the muonic hydrogen experiment. Randolf Pohl Max-Planck-Institut f¨ ur Quantenoptik Hans-Kopfermann-Str. 1 DE-85748 Garching, Germany Tel.: +49/89/32905-281 E-mail:
[email protected]
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Randolf Pohl, for the CREMA collaboration 2P fine structure 2P3/2 2P1/2
F=2 F=1 F=1 F=0
206 meV 50 THz 6 µm Lamb shift
225 meV 55 THz 5.5 µm
F=1
2S1/2 fin. size: 3.7 meV
2S hyperfine splitting F=0
Fig. 1 Energy levels in muonic hydrogen. The Lamb shift is sensitive to the rms charge radius of the proton whereas the 2S hyperfine splitting depends on its Zemach radius. The 2P wave function has a negligible overlap with the nucleus, so the 2P fine and hyperfine splittings can be calculated with very high accuracy.
Many attempts have been made to resolve the puzzle [7], but no solution has been found yet. The theory in muonic hydrogen has been carefully checked by various authors, but no large missing or wrong QED term has been found. Fig. 2 illustrates the sizes of the various contributions. Note that the missing 0.3 meV corresponds to the 5th largest term. This makes it higly unlikely that such a term has been overlooked. The theory of µp Lamb shift and hyperfine splitting has been summarized in Ref. [8].
2 New results from elastic electron proton scattering
The CODATA-2010 least squares adjustment [3] includes two values of of the proton rms charge radius rp from elastic electron scattering: Sick reanalyzed all the world data available in 2003 and obtained rp = 0.895(18) fm [5]. The new measureement from the Mainz MAMI A1 collaboration yielded a value of rp = 0.879(8) fm [6]. Some new results from elastic electron-proton scattering became available after the closing date of the CODATA 2010 adjustment. These include a new measurement of the proton form factor ratio µp GE /GM at low Q2 [9], which resulted in a value of rp = 0.875(10) fm, in agreement with the previous determinations of rp in electronic systems. This analysis uses the world data, but not the new Mainz results. Sick obtained a new value of rp = 0.886(8) fm [10] from both the world data and the new data from Mainz. The key improvement here was to constrain the behavious of the charge density at large radii to obtain a smaller uncertainty.
The Lamb shift in muonic hydrogen and the proton radius puzzle
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1-loop eVP proton size 2-loop eVP µSE and µVP discrepancy 1-loop eVP in 2 Coul. recoil 2-photon exchange hadronic VP proton SE 3-loop eVP light-by-light 10
-3
10
-2
10
-1
1
10
2
10 meV
Fig. 2 The Lamb shift in µp is dominated by the 1-loop electron vacuum polarization of 205 meV. The finite proton charge radius is the 2nd largest term. The discrepancy of 0.31 meV corresponds to the the 5th largest term, if attributed to a missing theory contribution, which seems very unlikely. The 2-photon-exchange contribution has recently gained cnsiderable interest (see text). For a detailed theroy summary, see Ref. [8].
All of the above analyses use the method of Rosenbluth separation to deduce the form factors from the measured data. These form factors are fitted by phenomenologically motivated functions (splines, polynomials, continued fractions, sum of Gaussians, etc.) and the fit is extrapolated to Q2 =0. Here, the slope of the electric form factor, GE (Q2 ), is directly related to the rms charge radius rp by
hrp2 i = −6
dGE (Q2 ) 2 dQ2 Q =0
(1)
In contrast, the the analysis of scattering data, both in the space- and the timelike region, by means of dispersion relations has traditionally resulted in values of rp around 0.84 fm, in agreement with the muonic hydrogen value. Belushkin et al. obtained rp ∈ 0.822..0.852 fm, depending on the model approach, using the data available in 2005 [11]. Recently, this group reanalyzed the new Mainz data within their framework and obtained rp = 0.84(1) fm [12], again in agreement with muonic hydrogen. Adamuscin et al. independently obtained rp = 0.849(7) fm [13] The dispersion analysis has been criticised for the seemingly inferior quality of the fit, indicated by a larger χ2 per degree of freedom. Such a worse χ2 /dof can of course originate from shortcomings of the model. It could, however, also point towards a lack of understanding of the behaviour of the electric form factor GE (Q2 ), e.g. around Q2 = 4m2π [14], or towards yet undiscovered systematic effects in the data. In any case, a detailed understanding is needed why the two approaches give systematically different results, even when the same data is used.
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Randolf Pohl, for the CREMA collaboration
Table 1 Some recent values of the proton rms charge radius. source
rp [fm]
Ref.
± 0.018 ± 0.008 ± 0.008 ± 0.010
[5] [6] [10] [9]
0.822..0.852 0.849 ± 0.007 0.84 ± 0.01
[11] [13] [12]
spectroscopy H and D, world data
0.876 ± 0.008
[3] Tab. XXXVIII
muonic hydrogen spectroscopy 2010 muonic hydrogen spectroscopy 2010+2013
0.8418 ± 0.0007 0.8409 ± 0.0004
[1] [2]
e-p e-p e-p e-p
scattering scattering scattering scattering
pre-2003 world data MAMI-A1 2010 world data + new Mainz world data + JLab µp GE /GM
dispersion analysis world data dispersion analysis world data dispersion analysis Mainz data
0.895 0.879 0.886 0.875
3 Two-photon exchange
The two-photon exchange (TPE) contribution, also called “proton polarizability” contribution, in muonic hydrogen has recently gained a lot of interest as it seems to be the only term that might have the potential to shift the muonic hydrogen result towards the electronic value of rp . Several authors have evaluated the polarizability term and found a numerical value around -10 meV, i.e. a factor of 30 too small to account for the discrepancy [15–20]. For an excellent review of the situation, see Ref. [20]. It has been noted, however, that loopholes exist [21, 22] in the traditional evaluation of the polarizability contribution using forward virtual Compton scattering data: A subtraction term required to make a dispersion relation converge is not well contrained neither from theory nor from experiment. In principle, one can construct a subtraction term that resolves the proton radius puzzle [23, 7]. Chiral perturbation theory, however, suggests that such a subtraction term is very unnatural [17, 19, 20]. Further studies are clearly indicated. The final answer on the muonic proton polarizability may only come from the recently approved muonproton scattering experiment at PSI [24].
4 New experiments
Several experiments are underway that will shed new light on the puzzle. New electron scattering experiments will study the proton, deuteron and helium nuclei with improved sensitivity and at lower momentum transfer [25, 26]. The MUSE experiment will for the first time measure elastic muon-proton scattering at low Q2 . It will be able to directly detect a possible difference between electron-proton and muon-proton interaction, as well as determine the two-photon exchange effects for muons [24]. Precision spectroscopy of hydrogen and hydrogen-like systems has also gained a lot of momentum. We have recently measured the 1S-2S transition in hydrogen to 4 parts in 1015 [27]. This very accurate measurement creates a strong link between the proton charge radius rp and the Rydberg constant R∞ (correlation coefficient 0.984 [3]). Hence any determination of the Rydberg constant is equivalent to a new determination of the proton charge radius.
The Lamb shift in muonic hydrogen and the proton radius puzzle
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Recently, a new value for the 1S-3S energy difference in H has been reported [28]. More measurements of this quantity are underway [29]. The 2S-n` (n > 2) transitions in H and D are sensitive to the Rydberg constant and can thus be used to obtain a new value of rp . The 2S − 6S, D transition is being remeasured [30, 31]. In parallel, we aim at a new measurement of the 2S-4P 1-photon-transition in H [32], which should suffer from different systematic effects than the 2-photon transitions that determine R∞ today [4]. The classical Lamb shift (2S-2P energy difference) in H has last been measured twenty years ago. A new measurement is underway [33] that will allow the determination of rp from hydrogen, and does not depend on the exact value of the Rydberg constant. We have also recently remeasured the isotope shift of the 1S-2S transition in hydrogen and deuterium [34]. This connects the charge radii of the proton and the deuteron (correlation coefficient 0.999 [3]) using state-of-the-art QED theory [35]. Therefore, a new measurement of the deuteron charge radius, e.g. from elastic electron scattering [26] or muonic deuterium, will give new insights into the proton radius puzzle. Laser spectroscopy of muonic deuterium and helium ions [36] has recently been performed by the CREMA collaboration at PSI. These results will hopefully help to solve the proton radius puzzle.
5 Outlook
The “proton radius puzzle”, i.e. the discrepancy between the measured Lamb shift in muonic hydrogen and its expected value, calculated using the world average of the proton rms charge radius, has created intense activity in the fields of atomic, nuclear and particle physics, as well as physics beyond the Standard Model. This has already resulted in deeper insights into the physics of muonic atoms, nucleon and nuclear stucture and polarizability, and alactron scattering. It has also seeded a wealth of new experiments that will almost certainly help solve the puzzle. Nobody knows today how the final solution may look like. Still, we can conclude today that we have already learned a lot from this exciting discrepancy. Acknowledgements The author would like to thank his many colleagues for the exciting discussions about the radius puzzle. I acknowledge support from the European Research Council under Stg. 279765.
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