Lamb Shift In Hydrogen Research Articles

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We derive an analytical expression for the contribution of the order mα2(Zα)6 to the hydrogen Lamb shift which comes from the diagrams for radiative corrections to the Wichmann-Kroll potential. We use modern methods of multiloop calculations, based on IBP reduction, DRA method and differential equations.

Two-loop corrections to Lamb shift and hyperfine splitting in hydrogen via multi-loop methods

We revisit the contributions of order α2(Zα)5m and α2(Zα)EF, respectively, to the Lamb shift and to the hyperfine splitting from mixed self-energy-vacuum-polarization diagrams, involving fermionic loop. We use modern multi-loop calculation techniques based on IBP reduction and differential equations. We construct the ϵ-regular basis [1] and explicitly demonstrate that it is compatible with the renormalization. We obtain analytic results in terms of one-fold integral involving elliptic function and dilogarithm. As a by-product, we obtain the analogous contribution for the limiting cases of heavy and light fermionic loop.

Multipolar quantum electrodynamics of localized charge-current distributions: Spectral theory and renormalization

We formulate a nonrelativistic quantum field theory to model interactions between quantized electromagnetic fields and localized charge-current distributions. The electronic degrees of freedom are encoded in microscopic polarization and magnetization field operators whose moments are identified with the multipole moments of the charge-current distribution. The multipolar Hamiltonian is obtained from the minimal coupling Hamiltonian through a unitary transformation, often referred to as the Power-Zienau-Woolley transformation; we renormalize this Hamiltonian using perturbation theory, the result of which is used to compute the leading-order radiative corrections to the electronic energy levels due to interactions between the electrons and quantum vacuum fluctuations in the electromagnetic field. Our renormalized energy shift constitutes a generalization of the Lamb shift in atomic hydrogen, valid for general localized assemblies of atoms and molecules, possibly with net charge but absent a free current. By expanding the fields in a series of multipole moments, our results can be used to study contributions to this energy shift coming from specific multipole moments of arbitrary order.

The Proton Structure in and out of Muonic Hydrogen

Laser spectroscopy of muonic atoms has been recently used to probe properties of light nuclei with unprecedented precision. We introduce nuclear effects in hydrogen-like atoms, nucleon structure quantities (form factors, structure functions, polarizabilities), and their effects in the Lamb shift and hyperfine splitting (HFS) of muonic hydrogen (μH). Updated theory predictions for the Lamb shift and HFS in μH are presented. We review the challenges of the ongoing effort to measure the ground-state HFS in μH and its impact on our understanding of the nucleon spin structure. To narrow down this search, we present a novel theory prediction obtained by scaling the measured HFS in hydrogen while leveraging radiative corrections. We also summarize recent developments in the spectroscopy of simple atomic and molecular systems and emphasize how they allow for precise determinations of fundamental constants, bound-state QED tests, and New Physics searches.

Fresh Extraction of the Proton Charge Radius from Electron Scattering

We present a novel method for extracting the proton radius from elastic electron-proton (ep) scattering data. The approach is based on interpolation via continued fractions augmented by statistical sampling and avoids any assumptions on the form of function used for the representation of data and subsequent extrapolation onto Q^{2}≃0. Applying the method to extant modern ep datasets, we find that all results are mutually consistent and, combining them, we arrive at r_{p}=0.847(8) fm. This result compares favorably with values obtained from contemporary measurements of the Lamb shift in muonic hydrogen, transitions in electronic hydrogen, and muonic deuterium spectroscopy.

Hadronic vacuum-polarization contribution to various QED observables

Due to precision tests of quantum electrodynamics (QED), determination of accurate values of fundamental constants, and constraints on new physics, it is important in a consistent way to evaluate a number of QED observables such as the Lamb shift in hydrogen-like atomic systems. Even in a pure leptonic case, those QED variables are in fact not pure QED ones since hadronic effects are involved through intermediate states while accounting for higher-order effects. One of them is hadronic vacuum polarization (hVP). Complex evaluations often involve a number of QED quantities, for which treatment of hVP is not consistent. The highest accuracy for a calculation of the hVP term is required for the anomalous magnetic moment of a muon. However, a standard data-driven treatment of hVP, based on a dispersion integration of experimental data on electron-positron annihilation to hadrons and some other phenomena, leads to a contradiction with the experimental value of a_mu . This experimental value can be considered as an indirect determination of the hVP contribution to a_mu and the scatter of theory and experiment allows one to obtain a conservative estimation of the related hVP contribution. In this paper, we derive exact and approximate relations between the leading-order (LO) hVP contributions to various observables. Using those relations, we obtain for them a consistent set of the results, based on the scatter of a_mu values. While calculating the LO hVP term, we have to remember that next-to-LO (NLO) hVP corrections are often comparable with the uncertainty of the LO term. Special attention is payed to hVP contribution to simple atoms. In particular, we discuss the NLO contribution to the Lamb shift in ordinary and muonic hydrogen and other two-body atoms for Zle 10. We also consider the NLO contribution of the muonic vacuum polarization to the Lamb shift in hydrogen-like atoms. With the a_mu puzzle unresolved, one may still require present-days values of the hVP contributions to various observable for comparison to experiment etc. the presence of contradicting values and a lack of consistency means an additional uncertainty for a_mu and for key contributions to it, including the LO hVP one. We present here an estimation of such a propagated uncertainty in hVP contributions to different QED observables and recommend a consistent set of the related LO hVP contributions.Graphic

Low-energy doubly virtual Compton scattering from dilepton electroproduction on a nucleon

We propose a new way to experimentally determine the subleading low-energy structure constant of doubly-virtual Compton scattering on a proton. Such empirical determination will reduce the theoretical model error in estimates of the hadronic correction to the muonic hydrogen Lamb shift. We demonstrate that the di-lepton forward-backward asymmetry in the $e^- p \to e^- p \, e^- e^+$ process, which can be accessed at electron scattering facilities, yields a large sensitivity to this so far unknown low-energy constant.

Tensor meson contribution to the Lamb shift and hyperfine splitting in muonic hydrogen

We calculate the tensor meson exchange contribution to the interaction operator of muon and proton, which is determined by the tensor meson coupling with two photon state. For the construction of transition form factor T → γγ we use the monopole parametrization over photon four-momenta and experimental data on the decay width ΓTγγ. It is shown that tensor meson f2(1270) exchange gives essential contribution to the Lamb shift in muonic hydrogen ΔΕLs (2P – 2S) which should be taken into account for a comparison with precise experimental data.

Higher-order logarithmic corrections and the two-loop self-energy of a 1s electron in hydrogen

The leading computational QED uncertainty of the Lamb shift in the $1s$ state in the hydrogen atom and helium ion is due to two-loop pure self-energy contributions. An exact evaluation in $Z\ensuremath{\alpha}$ has not yet been performed for $Z=1,2$, while a calculation based on the $Z\ensuremath{\alpha}$ expansion has a limited accuracy. To improve it, we calculate a leading logarithmic contribution in order ${\ensuremath{\alpha}}^{2}{(Z\ensuremath{\alpha})}^{8}m$ and estimate the nonleading subterms in orders ${\ensuremath{\alpha}}^{2}{(Z\ensuremath{\alpha})}^{7}m$ and ${\ensuremath{\alpha}}^{2}{(Z\ensuremath{\alpha})}^{8}m$. To further improve the accuracy, we take advantage of a combined evaluation of the numerical results of the exact calculations in $Z\ensuremath{\alpha}$ performed at $Z=10--30$, and the results of the $Z\ensuremath{\alpha}$ expansion in order to reach the most accurate result, which can be done by fitting the data. The quality of the fit, including its reliability and accuracy, strongly depends on a theoretical input, which is the main target of this paper. Strictly speaking, we develop not a fit, but a theoretical approximation for the pure self-energy contribution, which may serve as a base for a robust fit. It is also important to estimate a deviation of the truncated function from the true one. For this purpose, we calculate the leading logarithmic term and estimate the subleading ones in order ${\ensuremath{\alpha}}^{2}{(Z\ensuremath{\alpha})}^{9}m$. That allows us to find a reliable result which is more accurate than both fits over the numerical data and the results from the $Z\ensuremath{\alpha}$ expansion, which have been previously considered in literature. Performing the fit procedures, we focus our attention on the theoretical input which plays an essential role in the reliability and accuracy of the fit. Our results for the two-loop pure self-energy contribution to the $1s$ Lamb shift are expressed in a standard notation as ${B}_{60}=\ensuremath{-}72(7),\phantom{\rule{0.28em}{0ex}}{G}_{60}(Z=1)=\ensuremath{-}80(6)$, and ${G}_{60}(Z=2)=\ensuremath{-}83(5)$. That significantly improves the theoretical accuracy of the $1s$ Lamb shift in hydrogen, deuterium, and in the helium ions.

Unraveling the proton puzzle

Atomic Physics The discrepancy between the proton size deduced from the Lamb shift in muonic hydrogen and the average, textbook value based on regular (electronic) hydrogen has puzzled physicists for nearly a decade. One possible resolution could be that electrons interact with protons in a different way than muons do, which would require “new physics.” Bezginov et al. measured the Lamb shift in electronic hydrogen, which allowed for a direct comparison to the Lamb shift measured in muonic hydrogen. The two results agreed, but the discrepancy with the averaged value remains. Science , this issue p. [1007][1] [1]: /lookup/doi/10.1126/science.aau7807

A measurement of the atomic hydrogen Lamb shift and the proton charge radius.

The surprising discrepancy between results from different methods for measuring the proton charge radius is referred to as the proton radius puzzle. In particular, measurements using electrons seem to lead to a different radius compared with those using muons. Here, a direct measurement of the n = 2 Lamb shift of atomic hydrogen is presented. Our measurement determines the proton radius to be r p = 0.833 femtometers, with an uncertainty of ±0.010 femtometers. This electron-based measurement of r p agrees with that obtained from the analogous muon-based Lamb shift measurement but is not consistent with the larger radius that was obtained from the averaging of previous electron-based measurements.

The Contribution of the Sigma-Meson to the Lamb Shift of Muonic Hydrogen

The scalar meson exchange contribution to the potential of the muon-proton interaction in muonic hydrogen induced by the scalar meson coupling to two photon state is calculated. An estimate of transition form factor $$S \to \gamma \gamma $$ is given on the basis of quark model and experimental data on the decay widths $${{\Gamma }_{{S\gamma \gamma }}}.$$ It is shown that scalar meson exchange contribution to the Lamb shift in muonic hydrogen $$\Delta {{E}^{{Ls}}}(2P - 2S)$$ is large and important for a comparison with precise experimental data.

The Lamb shift of the 1s state in hydrogen: Two-loop and three-loop contributions

We consider the 1s Lamb shift in hydrogen and helium ions, a quantity, required for an accurate determination of the Rydberg constant and the proton charge radius by means of hydrogen spectroscopy, as well as for precision tests of the bound-state QED. The dominant QED contribution to the uncertainty originates from α8m external-field contributions (i.e., the contributions at the non-recoil limit). We discuss the two- and three-loop cases and in particular, we revisit calculations of the coefficients B61,B60,C50 in standard notation.We have found a missing logarithmic contribution of order α2(Zα)6m. We have also obtained leading pure self-energy logarithmic contributions of order α2(Zα)8m and α2(Zα)9m and estimated the subleading terms of order α2(Zα)7m, α2(Zα)8m, and α2(Zα)9m. The determination of those higher-order contributions enabled us to improve the overall accuracy of the evaluation of the two-loop self-energy of the electron.We investigated the asymptotic behavior of the integrand related to the next-to-leading three-loop term (order α3(Zα)5m, coefficient C50 in standard notation) and applied it to approximate integration over the loop momentum. Our result for contributions to the 1s Lamb shift for the total three loop next-to-leading term is (−3.3±10.5)(α3/π3)(Zα)5m.Altogether, we have completed the evaluation of the logarithmic contributions to the 1s Lamb shift of order α8m and reduced the overall α8m uncertainty by approximately a factor of three for H, D, and He+ as compared with the most recent CODATA compilation.

Effects of light-by-light scattering in the Lamb shift and hyperfine structure of muonic hydrogen

We calculate the meson exchange contribution to the interaction operator of muon and proton, which is determined by the meson coupling with two photon state. For the construction of transition form factor Meson → γγ we use the existing parametrizations based on experimental data including the monopole parametrization over photon four-momenta. For an estimate of the form factor value atzero photon four-momenta squared we use experimental data on the decay width ΓMeson→γγ. It is shown that scalar, pseudoscalar, axial vector and tensor mesons exchanges give significant contribution to the Lamb shift (LS) and hyperfine splitting (HFS) in muonic hydrogen which should be taken into account for a comparison with precise experimental data.

The sigma-meson exchange contribution to the muonic hydrogen Lamb shift

The sigma(ξ)meson exchange contribution to the potential of the muon-proton interactionin muonichydrogen inducedbythe ξ-meson coupling to two photons is estimated. The transition form factor ξ → γγ is deduced from the quark model and experimental data on the decay widths Γσγγ. It is shown that scalar meson exchange contribution to the Lamb shift in muonic hydrogen, △ELs(2P−2S ),is rather large and relevant for a comparison with coming precise experimental data.

Hadronic contribution of light by light scattering in the energy spectrum of muonic hydrogen

The contribution from scalar meson (S) exchange to the potential of the muon-proton interaction in muonic hydrogen induced by the scalar meson coupling to the two photon state is calculated. An estimate of transition form factor S → γγ is given on the basis of the quark model and experimental data on the decay widths $${\Gamma _{S\gamma \gamma }}$$. It is shown that scalar meson exchange contribution to the Lamb shift in muonic hydrogen ∆ELs(2P – 2S) is large and important for a comparison with precise experimental data.

ISR Experiment at A1-Collaboration

The discrepancy between the proton charge radius extracted from the muonic hydrogen Lamb shift measurement and the best present value obtained from the elastic scattering experiments, remains unexplained and represents a burning problem of today’s nuclear physics. In a pursuit of reconciling the puzzle an experiment is underway at MAMI, which exploits the radiative tail of the elastic peak to study the properties of electromagnetic processes and to extract the proton charge form factor $ \left( {\mathop G\nolimits_E^p } \right) $ at extremely small Q2. This paper reports on the latest results of the first such measurement performed at the three-spectrometer facility of the A1-Collaboration, which led to a precise validation of radiative corrections far away from elastic line and provided measurements of $ \mathop G\nolimits_E^p $ for 0.001 ≤ Q2 ≤ 0.017 (GeV/c)2.

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

AbstractTheoretical 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 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.

Internal Structure of Wormholes—Geometric Images of Charged Particles in General Relativity

Using an exact solution of the Einstein-Maxwell equations for a free electric field and dustlike matter, we study the internal structure of the wormhole-type space with two nonclosing static throats opening to two parallel vacuum spaces or to a single space. This geometry is considered as a particleantiparticle pair with two electric charges of opposite signs, ±e, and the rest mass m0 (the total mass of the particle’s gravitational internal world) specified on the throats. These fundamental constants emerge while solving the Cauchy problem in the form of first integrals. Using the energy conservation law, we consider an irremovable rotation of the internal structure, and the projection of the corresponding angular momentum onto the rotation axis is identified with the z-projection of the charged particle’s spin. The Gaussian 2D curvature radius of the throat is identified with the particle’s radius; the z-projection of the magnetic moment and the gyromagnetic ratio have been found. The plausibility of this gravitational model is confirmed by the fact that the proton and electron spins, s = 1/2, turn out to be independent of the particle radius and of the relativistic mass of the rotating throat; also, there is a coincidence with great accuracy between the proton radius value calculated in this model, 0.8412 × 10−13 cm, and the one obtained in measurements of the Lamb shift in muonic hydrogen, 0.8409× 10−13 cm. The electron turns out to be also a structured particle, with a radius of 3.8617 × 10−11 cm.