Lamb shift in muonic hydrogen—I. Verification and update of theoretical predictions

In view of the recently observed discrepancy of theory and experiment for muonic hydrogen [R. Pohl et al., Nature 466 (2010) 213], we reexamine the theory on which the quantum electrodynamic (QED) predictions are based. In particular, we update the theory of the 2P–2S Lamb shift, by calculating the self-energy of the bound muon in the full Coulomb + vacuum polarization (Uehling) potential. We also investigate the relativistic two-body corrections to the vacuum polarization shift, and we analyze the influence of the shape of the nuclear charge distribution on the proton radius determination. The uncertainty associated with the third Zemach moment 〈r³〉₂ in the determination of the proton radius from the measurement is estimated. An updated theoretical prediction for the 2S–2P transition is given.     

Theory of the 2S–2P Lamb shift and 2S hyperfine splitting in muonic hydrogen

The 7σ$7\sigma$ discrepancy between the proton rms charge radius from muonic hydrogen and the CODATA-2010 value from hydrogen spectroscopy and electron-scattering has caused considerable discussions. Here, we review the theory of the 2S–2P Lamb shift and 2S hyperfine splitting in muonic hydrogen combining the published contributions and theoretical approaches. The prediction of these quantities is necessary for the determination of both proton charge and Zemach radii from the two 2S–2P transition frequencies measured in muonic hydrogen; see Pohl et al. (2010) [9] and Antognini et al. (2013) [71].     

应用光核反应计算μ原子中的核极化效应

近年来,瑞士保罗谢勒研究所的CREMA合作组通过测量$\mu$氢原子兰姆位移显著提升了质子半径的测量精确度。然而这一新实验结果与已知质子半径标准值(CODATA)相差5.6个标准差,被称为质子半径之谜,受到众多物理学家的关注。受此启发,CREMA合作组在不同的$\mu$原子中展开了一系列兰姆位移光谱的测量实验。他们计划从这些$\mu$原子的测量中得到轻核(包括$^{2,3}{\rm{H}}$,$^{3,4}{\rm{He}}$)的电荷半径。除了对光谱测量精度的要求外,轻核半径的实验精度当前仍被一项理论输入量限制:核极化效应对$\mu$原子光谱的修正。核极化效应体现了$\mu$子与原子核进行双光子交换中对核的虚激发,进而对$\mu$原子能谱产生高阶修正。因此,这一效应与光核反应以及康普顿散射直接相关。核极化效应对兰姆位移的修正可通过计算光核吸收截面以及虚光子康普顿振幅的求和规则而得到。本工作运用第一性原理的核结构计算方法,研究了$\mu$原子中的核极化效应。通过结合现代核力模型与超球简谐基展开多体方法,计算了一系列与核极化相关的光核反应及康普顿散射求和规则。这一理论研究为从$\mu$原子光谱测量中对核半径的精确提取提供了关键性的理论输入。     

The Lamb shift in muonic hydrogen and the proton radius puzzle

The recent measurement of the Lamb shift in muonic hydrogen resulted in a tenfold improved value of the rms charge radius of the proton. The value is, however, 7 standard deviations discrepant from the world average of this quantitiy which is obtained from elastic electron-proton scattering and precision spectroscopy of hydrogen and deuterium. New input from both theory and experiment is needed to resolve this so-called “proton radius puzzle”.     

What’s New with the Neutron and Proton

The existence and importance of the proton radius puzzle, observed via a Lamb shift measurement in muonic atoms, is discussed. Possible resolutions of the puzzle are discussed. Then the broader question of the meaning of the proton radius is addressed and examples of correctly defined charge densities are presented.     

Mössbauer study of Fe in GaAs following 57Mn + implantation

The metastable 2S state in muonic hydrogen is particularily interesting because a measurement of the Lamb shift could reduce the uncertainty in the proton charge radius by an order of magnitude. The most important prerequisite for such a measurement is a sufficiently large population and lifetime of the 2S state. We have determined the long-lived and short-lived 2S population, the deexcitation mechanisms and lifetimes, and the cascade time in μp.     

Proton polarizability and lamb shift in the muonic hydrogen atom

The corrections that the structure of the proton and its polarizability induce in the Lamb shift in the muonic and the conventional hydrogen atom are calculated on the basis of up-to-date experimental data on the structure functions in deep-inelastic scattering. Numerically, the contribution from proton polarizability to the 2P-2S shift in the muonic hydrogen atom is 4.4 GHz.     

Lamb shift in muonic hydrogen—II. Analysis of the discrepancy of theory and experiment

Currently, both the g factor measurement of the muon as well as the Lamb shift 2S–2P measurement in muonic hydrogen are in disagreement with theory. Here, we investigate possible theoretical explanations, including proton structure effects and small modifications of the vacuum polarization potential. In particular, we investigate a conceivable small modification of the spectral function of vacuum polarization in between the electron and muon energy scales due to a virtual millicharged particle and due to an unstable vector boson originating from a hidden sector of an extended standard model. We find that a virtual millicharged particle which could explain the muonic Lamb shift discrepancy alters theoretical predictions for the muon anomalous magnetic moment by many standard deviations and therefore is in conflict with experiment. Also, we find no parameterizations of an unstable virtual vector boson which could simultaneously explain both “muonic” discrepancies without significantly altering theoretical predictions for electronic hydrogen, where theory and experiment currently are in excellent agreement. A process-dependent correction involving electron screening is evaluated to have the right sign and order-of-magnitude to explain the observed effect in muonic hydrogen. Additional experimental evidence from light muonic atoms and ions is needed in order to reach further clarification.     

Proton polarisability contribution to the Lamb shift in muonic hydrogen at fourth order in chiral perturbation theory

We calculate the amplitude T₁ for forward doubly virtual Compton scattering in heavy-baryon chiral perturbation theory, to fourth order in the chiral expansion and with the leading contribution of the $ \gamma$ N $ \Delta$ form factor. This provides a model-independent expression for the amplitude in the low-momentum region, which is the dominant one for its contribution to the Lamb shift. It allows us to significantly reduce the theoretical uncertainty in the proton polarisability contributions to the Lamb shift in muonic hydrogen. We also stress the importance of consistency between the definitions of the Born and structure parts of the amplitude. Our result leaves no room for any effect large enough to explain the discrepancy between proton charge radii as determined from muonic and normal hydrogen.     

Constraints on proton structure from precision atomic-physics measurements

Ground-state hyperfine splittings in hydrogen and muonium are very well measured. Their difference, after correcting for magnetic moment and reduced mass effects, is due solely to proton structure-the large QED contributions for a pointlike nucleus essentially cancel. The rescaled hyperfine difference depends on the Zemach radius, a fundamental measure of the proton, computed as an integral over a product of electric and magnetic proton form factors. The determination of the Zemach radius, (1.019+/-0.016) fm, from atomic physics tightly constrains fits to accelerator measurements of proton form factors. Conversely, we can use muonium data to extract an experimental value for QED corrections to hydrogenic hyperfine data. There is a significant discrepancy between measurement and theory, in the same direction as a corresponding discrepancy in positronium.     

Proton size anomaly

A measurement of the Lamb shift in muonic hydrogen yields a charge radius of the proton that is smaller than the CODATA value by about 5 standard deviations. We explore the possibility that new scalar, pseudoscalar, vector, and tensor flavor-conserving nonuniversal interactions may be responsible for the discrepancy. We consider exotic particles that, among leptons, couple preferentially to muons and mediate an attractive nucleon-muon interaction. We find that the many constraints from low energy data disfavor new spin-0, spin-1, and spin-2 particles as an explanation.     

Correlation between muonic levels and nuclear structure in muonic atoms

A method that deals with the nucleons and the muon unitedly is employed to investigate the muonic lead, with which the correlation between the muon and nucleus can be studied distinctly. A “kink” appears in the muonic isotope shift at a neutron magic number where the nuclear shell structure plays a key role. This behavior may have very important implications for the experimentally probing the shell structure of the nuclei far away from the β-stable line. We investigate the variations of the nuclear structure due to the interaction with the muon in the muonic atom and find that the nuclear structure remains basically unaltered. Therefore, the muon is a clean and reliable probe for studying the nuclear structure. In addition, a correction that the muon-induced slight change in the proton density distribution in turn shifts the muonic levels is investigated. This correction to muonic level is as important as the Lamb shift and high order vacuum polarization correction, but is larger than anomalous magnetic moment and electron shielding correction.     

Effects of vacuum polarization and of proton polarizability in the Lamb shift of muonic hydrogen

The contributions to the Lamb shift in muonic hydrogen from hadronic vacuum polarization and from the correction associated with electron vacuum polarization and with the proton polarizability are calculated by using present-day experimental data on the cross section for e ⁺ e ⁻ annihilation into hadrons and on structure functions for deep-inelastic ep scattering. The numerical value of the total contribution to the (2P-2S) shift in muonic hydrogen is found to be 10.95 μeV. __________ Translated from Yadernaya Fizika, Vol. 64, No. 7, 2001, pp. 1358–1363. Original Russian Text Copyright ? 2001 by Martynenko, Faustov.     

Lamb shift in light muonic atoms — Revisited

In connection with recent and proposed experiments, and new theoretical results, my previous calculations of the Lamb shift in muonic hydrogen will be reviewed and compared with other work. In addition, numerical results for muonic deuterium and helium will be presented. Some previously neglected (but very small) effects are included.     

Hyperfine structure effects in the collisions of the 2s metastable muonic hydrogen with hydrogen atoms

Cross sections for elastic and spin-flip scattering of metastable 2s muonic hydrogen on hydrogen atoms are calculated for the symmetric systems pμ+p, dμ+d and tμ+t. The calculations are performed in the two-level approximation of the adiabatic method with inclusion of the Lamb shift and electron screening. Due to the large polarizability of the 2s state, the obtained cross sections significantly exceed their counterparts for the ground-state muonic hydrogen.     

Theory of the Lamb Shift in Hydrogen and Light Hydrogen‐Like Ions

Theoretical calculations of the Lamb shift provide the basis required for the determination of the Rydberg constant from spectroscopic measurements in hydrogen. The recent high‐precision determination of the proton charge radius drastically reduces the uncertainty in the hydrogen Lamb shift originating from the proton size. As a result, the dominant theoretical uncertainty now comes from the two‐ and three‐loop QED effects, which calls for further advances in their calculations. The present status of theoretical calculations of the Lamb shift in hydrogen and light hydrogen‐like ions with the nuclear charge number up to $Z=5$ is reviewed. Theoretical errors due to various effects are critically examined and estimated.     

The Mu-MASS (muonium laser spectroscopy) experiment

We present a new experiment, Mu-MASS, aiming for a 1000-fold improvement in the determination of the 1S-2S transition frequency of Muonium (M), the positive-muon/electron bound state. This substantial improvement beyond the current state-of-the-art relies on the novel cryogenic M converters and confinement techniques we developed, on the new excitation and detection schemes which we implemented for positronium spectroscopy and the tremendous advances in generation of UV radiation. This experiment is planned to be performed at the Paul Scherrer Institute (PSI). Interesting anomalies in the muon sector have accumulated: notably the famous anomalous muon magnetic moment (g-2) and the muonic hydrogen Lamb shift measurement which prompted the so-called proton charge radius puzzle. These tantalizing results triggered vibrant activity on both experimental and theoretical sides. Different explanations have been put forward including exciting solutions invoking New Physics beyond the Standard Model. Mu-MASS could contribute to clarifying the origin of these anomalies by providing robust and reliable values of fundamental constants such as the muon mass and a value of the Rydberg constant independent of finite size effects.     

Theory of the muonic hydrogen-muonic deuterium isotope shift

Corrections of the α³, α⁴, and α⁵ orders are calculated for the Lamb shift of the 1S and 2S energy levels of muonic hydrogen μp and muonic deuterium μd. The nuclear structure effects are taken into account in terms of the charge radii of the proton r _p and deuteron r _d for one-photon interaction, as well as in terms of the electromagnetic form factors of the proton and deuteron for the case of one-loop amplitudes. The μd-μp isotope shift for the 1S-2S splitting is found to be equal to 101003.3495 meV, which can be treated as a reliable estimate when conducting the corresponding experiment with an accuracy of 10^?6. The fine-structure intervals E(1S)-8E(2S) in muonic hydrogen and muonic deuteron are calculated.     

Measurement of isotope shift and hyperfine splitting of190, 191, 193, 197Pb isotopes by collinear laser spectroscopy

We report here on the measurement of isotope shift and hyperfine splitting of^{190, 191, 193, 197}Pb for the 723 nm atomic optical transition. Detailed analysis of the optical data has been done by combining them with the available muonic and electronicx-ray isotope shift data. The magnetic dipole moments and the electric quadrupole moments of the odd isotopes have been extracted from the hyperfine coupling constants of the atomic states involved in the optical transition used.     

Update on nuclear structure effects in light muonic atoms

We present calculations of the nuclear structure corrections to the Lamb shift in light muonic atoms, using state-of-the-art nuclear potentials. We outline updated results on finite nucleon size contributions.