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.     

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。     

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.     

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.     

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.     

The size of the proton

The root-mean-square (rms) charge radius r _p of the proton has so far been known only with a surprisingly low precision of about 1% from both electron scattering and precision spectroscopy of hydrogen. We have recently determined r _p by means of laser spectroscopy of the Lamb shift in the exotic “muonic hydrogen” atom. Here, the muon, which is the 200 times heavier cousin of the electron, orbits the proton with a 200 times smaller Bohr radius. This enhances the sensitivity to the proton’s finite size tremendously. Our new value r _p?=?0.84184 (67) fm is ten times more precise than the generally accepted CODATA-value, but it differs by 5 standard deviations from it. A lively discussion about possible solutions to the “proton size puzzle” has started. Our measurement, together with precise measurements of the 1S–2S transition in regular hydrogen and deuterium, also yields improved values of the Rydberg constant, R _?∞??=?10,973,731.568160 (16) m^???1.     

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].     

Nuclear-structure corrections to the energy spectra of ordinary and muonic deuterium

One-loop nuclear-structure-induced corrections of order (Zα)⁵ to the Lamb shift and to the hyperfine structure of deuterium are calculated. The contribution of deuteron-structure effects to the (ep)-(ed) and (μp)-(μd) isotopic shifts for the 1S–2S splitting is obtained with the aid of modern experimental data on the electromagnetic form factors for the deuteron. A comparison with the analogous contributions to the Lamb shift for ordinary and muonic hydrogen shows that the relative contribution of corrections associated with the nuclear structure increases as we go over from the hydrogen to the deuterium atom owing to the growth of the nuclear size.     

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.     

A nonlinear electrodynamical shift of spectral lines in the hydrogen atom and hydrogen-like ions

The shift of spectral lines in the hydrogen atom and hydrogen-like ions is calculated using Born-Infeld nonlinear electrodynamics. The formulas obtained show that the position of the s levels in hydrogen-like ions and muonic atoms should be most strongly affected by the nonlinearity of electrodynamics of vacuum.     

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.     

Deceleration of muonic hydrogen atoms in solid hydrogens

Results of recent calculations of cross-sections for muonic hydrogen atom scattering in solid hydrogen isotope targets are presented. The coherent parts of these cross-sections, namely, the elastic Bragg scattering and phonon coherent scattering, are calculated accurately for the first time. A fine structure of Bragg peaks is obtained in the case of Bravais fcc structure of hydrogen targets frozen rapidly at 3 K. The one-phonon coherent cross-section is estimated using the Debye approximation. The calculated differential cross-sections are used for Monte Carlo simulations of muonic atom diffusion and slowing down in solid hydrogens. Also is calculated the energy-dependent rate of resonant ddμ molecule formation in 3 K solid deuterium quantum crystal, using the Debye model and Van Hove's formalism of the response function. This rate is very different from that obtained for the 3 K gas model. The influence of dμ atom slowing down on the average ddμ formation rate is considered. It is shown that very slow dμ deceleration below 10 meV is important for explanation of experimental results. This revised version was published online in August 2006 with corrections to the Cover Date.     

Hyperfine ground-state structure of muonic hydrogen

Corrections of orders α⁵ and α⁶ to the superfine ground-state structure of the muonic hydrogen atom were calculated. The calculations took into account the effects of the structure of the nucleus on one-and two-loop Feynman amplitudes with the help of the electromagnetic form factors of the proton and the modification of the superfine part of the Breit potential caused by the electronic polarization of the vacuum. The total splitting of the 1S state is 182.725 meV; this value can be used as a reliable estimate in conducting a corresponding experiment with an accuracy of 30 ppm.     

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 polarizability contribution: Muonic hydrogen Lamb shift and elastic scattering

The uncertainty in the contribution to the Lamb shift in muonic hydrogen, ΔE^subt$\mathrm{\Delta }{E}^{\mathit{subt}}$ arising from proton polarizability effects in the two-photon exchange diagram at large virtual photon momenta is shown large enough to account for the proton radius puzzle. This is because ΔE^subt$\mathrm{\Delta }{E}^{\mathit{subt}}$ is determined by an integrand that falls very slowly with very large virtual photon momenta. We evaluate the necessary integral using a set of chosen form factors and also a dimensional regularization procedure which makes explicit the need for a low energy constant. The consequences of our two-photon exchange interaction for low-energy elastic lepton–proton scattering are evaluated and could be observable in a planned low energy lepton–proton scattering experiment planned to run at PSI.     

Infrared spectroscopic studies of the rare gas atom perturbed S1(0) rovibron band of solid parahydrogen

We report high resolution infrared absorption studies of rare gas (Rg) atom doped solid parahydrogen in the hydrogen S₁(0) region around 4486 cm⁻¹. At low Rg atom concentrations (∼0.1%), satellite transitions appear in the S₁(0) region due to rovibrational excitation of parahydrogen molecules with one nearest-neighbor Rg atom. The Ne satellite feature differs qualitatively from the Ar, Kr, and Xe satellite features for reasons described within. The frequency of the S₁(0) satellite features linearly shift to lower energy as the polarizability of the Rg atom increases while the absorption coefficients increase with the square of the Rg atom polarizability. Rotational calculations are performed for H₂ with a nearest-neighbor Rg atom assuming a rigid hexagonal close-packed lattice structure. The calculated fine structure of the S₁(0) satellite features agree qualitatively with lifting of the 2J+1 degeneracy of the v = 1, J = 2 upper state caused by the anisotropy in the Rg-H₂ intermolecular potential. The discrepancy between the calculated and measured Rg atom S₁(0) satellite features may signal partial delocalization of the J = 2 roton onto neighboring parahydrogen molecules.     

Scattering of muonic hydrogen atoms

Our measurement compares the energy dependence of the scattering cross-sections of muonic deuterium and tritium on hydrogen molecules for collisions in the energy range 0.1–45 eV. A time-of-flight method was used to measure the scattering cross-section as a function of the muonic atom beam energy and shows clearly the Ramsauer–Townsend effect. The results are compared with theoretical calculations by using Monte Carlo simulations. The molecular pdμ and ptμ formation creates background processes. We measure the formation rates in solid hydrogen by detecting the 5.5 MeV (pdμ) and 19.8 MeV (ptμ) γ-rays emitted during the subsequent nuclear fusion processes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Charge exchange and Coulomb de-excitation of muonic atoms in low energy collisions

The muon transfer and Coulomb de-excitation rates at the collisions of (pμ) _n , (dμ) _n and (tμ) _n muonic atoms in excited states n = 3, 4, 5 with hydrogen isotopes p, d, t are calculated for all possible combinations of hydrogen isotopes. The advanced adiabatic approach (AAA) [1–3] is adapted and used to the specific case of muonic atom collisions. This revised version was published online in August 2006 with corrections to the Cover Date.     

The hydrogen hyperfine structure and inelastic electron proton scattering experiments

It is shown that present experiments on inelastic electron proton scattering can be used to set bounds to part of the proton polarizability contribution to the hyperfine structure of the ground state in the hydrogen atom.     

Proton-polarizability effect in the Lamb shift for the hydrogen atom

The proton-polarizability contribution to the Lamb shift for the ordinary and muonic hydrogen atoms is calculated on the basis of the isobar model and experimental data on structure functions for deep-inelastic lepton-nucleon scattering. The contributions of Born terms, vector mesons, and nucleon resonances are taken into account in constructing cross sections for the absorption of transversely and longitudinally polarized virtual photons, σ _T,L.