Bound-state QED Research Articles

Hydrogen energy levels from the anomalous energy-momentum QED trace

Energy levels of hydrogen are calculated as one-loop matrix elements of the QED energy-momentum tensor trace in the external field approximation. An explicit connection established between the one-loop trace diagrams and the standard Lamb shift one-loop diagrams. Our calculations provide an argument against inclusion of the anomalous trace contribution as a separate term in the decomposition of the QED quantum field Hamiltonian and serve as an illustration how the trace anomaly is realized in the bound state QED. Published by the American Physical Society 2024

High-Precision Determination of g Factors and Masses of ^{20}Ne^{9+} and ^{22}Ne^{9+}.

We present the measurements of individual bound electron g factors of ^{20}Ne^{9+} and ^{22}Ne^{9+} on the relative level of 0.1 parts per billion. The comparison with theory represents the most stringent test of bound-state QED in strong electric fields. A dedicated mass measurement results in m(^{20}Ne)=19.992 440 168 77(9) u, which improves the current literature value by a factor of 18, disagrees by 4 standard deviations, and represents the most precisely measured mass value in atomic mass units. Together, these measurements yield an electron mass on the relative level of 0.1ppb with m_{e}=5.485 799 090 99(59)×10^{-4} u as well as a factor of seven improved m(^{22}Ne)=21.991 385 098 2(26) u.

Ab initio calculations of the 2p3/2→2s transition in He-, Li-, and Be-like uranium

The bound-state QED approach is applied to calculations of the $2p_{3/2} \rightarrow 2s$ transition energies in He-, Li-, and Be-like uranium. For U$^{90+}$ and U$^{89+}$, standard perturbation theory for a single level is employed, while the calculations of U$^{88+}$ have required its counterpart for quasidegenerate levels. The utilized approach merges the rigorous QED treatment up to the second order of perturbation theory with the higher-order electron-correlation contributions evaluated within the Breit approximation. The higher-order screened QED effects are estimated by means of the model-QED operator. The nuclear recoil, nuclear polarization, and nuclear deformation effects are taken into account as well. Along with the transition energies, their pairwise differences are calculated. The comprehensive analysis of the uncertainties due to uncalculated effects is carried out, and the most accurate theoretical predictions, which are in perfect agreement with available experimental data, are obtained.

Relativistic Ritz approach to hydrogenlike atoms: Theoretical considerations

The Rydberg formula along with the Ritz quantum defect ansatz has been a standard theoretical tool used in atomic physics since before the advent of quantum mechanics, yet this approach has remained limited by its non-relativistic foundation. Here I present a long-distance relativistic effective theory describing hydrogen-like systems with arbitrary mass ratios, thereby extending the canonical Ritz-like approach. Fitting the relativistic theory to the hydrogen energy levels predicted by bound-state QED indicates that it is superior to the canonical, nonrelativistic approach. An analytic analysis reveals nonlinear consistency relations within the bound-state QED level predictions that relate higher-order corrections to those at lower order, providing guideposts for future perturbative calculations as well as insights into the asymptotic behavior of Bethe logarithms. Applications of the approach include fitting to atomic spectroscopic data, allowing for the determination the fine-structure constant from large spectral data sets and also to check for internal consistency of the data independently from bound-state QED.

Measurement of the transition frequency from 2S1/2, F = 0 to 2P1/2, F = 1 states in Muonium

Muons are puzzling physicists since their discovery when they were first thought to be the meson predicted by Yukawa to mediate the strong force. The recent result at Fermilab on the muon g-2 anomaly puts the muonic sector once more under the spotlight and calls for further measurements with this particle. Here, we present the results of the measurement of the 2S1/2, F = 0 → 2P1/2, F = 1 transition in Muonium. The measured value of 580.6(6.8) MHz is in agreement with the theoretical calculations. A value of the Lamb shift of 1045.5(6.8) MHz is extracted, compatible with previous experiments. We also determine the 2S hyperfine splitting in Muonium to be 559.6(7.2) MHz. The measured transition being isolated from the other hyperfine levels holds the promise to provide an improved determination of the Muonium Lamb shift at a level where bound state QED recoil corrections not accessible in hydrogen could be tested. This result would be sensitive to new physics in the muonic sector, e.g., to new bosons which might provide an explanation of the g-2 muon anomaly and allow to test Lorentz and CPT violation. We also present the observation of Muonium in the n = 3 excited state opening up the possibility of additional precise microwave measurements.

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.

Defect theory of positronium and nontrivial QED relations

An effective theory of the excited states of positronium is derived and some of its consequences are explored. At large physical separation, the binding of the electron and positron is assumed to be described completely by QED, whereas all short-ranged phenomena, including those within and beyond QED, can be accounted for with energy-dependent quantum defects. This theory has at least two practical applications. First, it provides an accurate and economical, yet largely QED-independent, means to fit the positronium spectrum in order to predict and compare the outcome of experiments. Second, matching the spectrum in this effective theory to that predicted by QED reveals nontrivial relationships that exist \emph{within} bound-state QED; some higher order contributions to the spectrum may be obtained from lower order contributions. These relations are verified up to order $m\alpha^{6}$, and predictions are made for the order $m\alpha^{7}$ and $m\alpha^{8}$ corrections. This theory and its extensions to other hydrogenic systems may provide a useful complement to bound-state QED.

One-Electron Energy Spectra of Heavy Highly Charged Quasimolecules: Finite-Basis-Set Approach

The generalized dual-kinetic-balance approach for axially symmetric systems is employed to solve the two-center Dirac problem. The spectra of one-electron homonuclear quasimolecules are calculated and compared with the previous calculations. The analysis of the monopole approximation with two different choices of the origin is performed. Special attention is paid to the lead and xenon dimers, Pb82+–Pb82+–e− and Xe54+–Xe54+–e−, where the energies of the ground and several excited σ-states are presented in the wide range of internuclear distances. The developed method provides the quasicomplete finite basis set and allows for the construction of perturbation theory, including within the bound-state QED.

Many-Electron QED with Redefined Vacuum Approach

The redefined vacuum approach, which is frequently employed in the many-body perturbation theory, proved to be a powerful tool for formula derivation. Here, we elaborate this approach within the bound-state QED perturbation theory. In addition to general formulation, we consider the particular example of a single particle (electron or vacancy) excitation with respect to the redefined vacuum. Starting with simple one-electron QED diagrams, we deduce first- and second-order many-electron contributions: screened self-energy, screened vacuum polarization, one-photon exchange, and two-photon exchange. The redefined vacuum approach provides a straightforward and streamlined derivation and facilitates its application to any electronic configuration. Moreover, based on the gauge invariance of the one-electron diagrams, we can identify various gauge-invariant subsets within derived many-electron QED contributions.

Redefined vacuum approach and gauge-invariant subsets in two-photon-exchange diagrams for a closed-shell system with a valence electron

The two-photon-exchange diagrams for atoms with single valence electrons are investigated. Calculation formulas are derived for an arbitrary state within the rigorous bound-state QED framework utilizing the redefined vacuum formalism. In contrast to other methods, the redefined vacuum approach enables the identification of eight gauge-invariant subsets and, thus, efficiently checks the consistency of the obtained results. The gauge invariance of found subsets is demonstrated both analytically (for an arbitrary state) as well as numerically for 2s, 2p1/2, and 2p3/2 valence electrons in Li-like ions. Identifying gauge-invariant subsets in the framework of the proposed approach opens a way to tackle more complex diagrams, e.g., three-photon exchange, where the fragmentation on simpler subsets is crucial for its successful calculation.

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.

The hyperfine puzzle of strong-field bound-state QED

The hyperfine splitting in heavy highly charged ions provide the means to test QED in extremely strong magnetic fields. In order to provide a meaningful test, the splitting has to be measured in H-like and Li-like ions to remove uncertainties from nuclear structure. This has been achieved at the experimental storage ring ESR but a discrepancy to the theoretical prediction of more than 7σ was observed. We report on these measurements as well as on NMR measurements that were performed to solve this issue.

Many-body study of the g factor in boronlike argon

Highly accurate measurements of the $g$-factor of boronlike Ar are currently implemented within the ARTEMIS experiment at GSI (Darmstadt, Germany) and within the ALPHATRAP experiment at the MPIK (Heidelberg, Germany). A comparison with the corresponding theoretical predictions will allow one to test the modern methods of bound-state QED. However, at least three different theoretical values of the \emph{g}-factor have been published up to date. The systematic study of the \emph{g} factor value of Ar$^{13+}$ in the ground $[(1s)^2(2s)^2 2p]^2P_{1/2}$ and the first excited $[(1s)^2(2s)^2 2p]^2P_{3/2}$ states is performed within the high order coupled cluster and configuration interaction theories up to the full configuration interaction treatment. Correlation contributions are discussed and results are compared with previous studies.

The positronium hyperfine structure: Progress towards a direct measurement of the 23S1 → 21S0 transition in vacuum

We present the current status for the direct measurement of the positronium hyperfine structure using the 23S1 → 21S0 transition. This experiment, currently being commissioned at the slow positron beam facility at ETH Zurich, will be the first measurement of this transition and the first positronium hyperfine splitting experiment conducted in vacuum altogether. This experiment will be free of systematic effects found in earlier experiments, namely the inhomogeneity in static magnetic fields and the extrapolation from dense gases to vacuum. The achievable precision is expected to be on the order of 10 ppm while the systematic uncertainty is estimated to be within a few ppm. This would allow to check recent bound state QED calculations and a 3σ discrepancy with earlier experiments.

High precision measurement of muonium hyperfine structure

MuSEUM (Muonium Spectroscopy Experiment Using Microwave) collaboration aims to measure the muonium hyperfine structure (MuHFS,vhfs) with a few ppb (parts per billion). MuHFS spectroscopy is a stringent test of the bound-state QED. From this measurement, the muon-proton magnetic moment ratio (μμ/μP) and the muon-electron mass ratio (mμ/me) can also be determined by applying a high magnetic field. In the previous MuHFS measurement carried out at Los Alamos Meson Physics Facility (LAMPF), the main uncertainty was caused by the lack of the statistics. MuSEUM collaboration uses the high intense pulsed muon beam at Japan Proton Accelerator Research Complex (J-PARC) to improve the statistical uncertainty. We are also developing our own experimental setups for the improvement of the systematic uncertainties. From June 2016, MuSEUM collaboration have carried out the MuHFS measurement in an extremely low magnetic field within 100 nT and now we are preparing for the measurement in a 1.7 T high magnetic field.

The nuclear magnetic moment of 208Bi and its relevance for a test of bound-state strong-field QED

The hyperfine structure splitting in the 6p3S3/24→6p27sP1/24 transition at 307 nm in atomic 208Bi was measured with collinear laser spectroscopy at ISOLDE, CERN. The hyperfine A and B factors of both states were determined with an order of magnitude improved accuracy. Based on these measurements, theoretical input for the hyperfine structure anomaly, and results from hyperfine measurements on hydrogen-like and lithium-like 209Bi80+,82+, the nuclear magnetic moment of 208Bi has been determined to μ(Bi208)=+4.570(10)μN. Using this value, the transition energy of the ground-state hyperfine splitting in hydrogen-like and lithium-like 208Bi80+,82+ and their specific difference of −67.491(5)(148) meV are predicted. This provides a means for an experimental confirmation of the cancellation of nuclear structure effects in the specific difference in order to exclude such contributions as the cause of the hyperfine puzzle, the recently reported 7-σ discrepancy between experiment and bound-state strong-field QED calculations of the specific difference in the hyperfine structure splitting of 209Bi80+,82+.

Testing fundamental interactions on the helium atom

We critically examine the current status of theoretical calculations of the energies, the fine structure, and the isotope shift of the lowest-lying states of helium, searching for unresolved discrepancies with experiments. Calculations are performed within the quantum electrodynamics expansion in powers of the fine structure constant $\alpha$ and the electron-to-nucleus mass ratio $m/M$. For energies, theoretical results are complete through orders $\alpha^6m$ and $\alpha^6m^2/M$, with the resulting accuracy ranging from $0.5$ to $2$~MHz for the $n=2$ states. The fine-structure splitting of the $2^3P$ state is predicted with a much better accuracy, 1.7~kHz, as a consequence of a calculation of the next-order $\alpha^7m$ effect. An excellent agreement of the theoretical predictions with the recent measurements of the fine structure provides one of the best tests of the bound-state QED in few-electron systems and determines the fine-structure constant $\alpha$ with an accuracy of 31~ppb. The isotope shift between $^3$He and $^4$He is treated theoretically with a sub-kHz accuracy, which allows for a high-precision determination of the differences of the nuclear charge radii $\delta r^2$. Several such determinations, however, yield results that are in a 4$\sigma$ disagreement with each other, what remains unexplained. Apart from this, we find no significant discrepancies between theory and experiment for the helium atom. In the future, a calculation of the next-order $\alpha^7m$ effect for energy levels will enable determinations of the nuclear charge radii from atomic transition frequencies with an accuracy better than 1\%. Combined with the complementary determinations from muonic atoms, this will provide a sensitive test of universality in electromagnetic interactions of leptons and contribute to the solution of the proton charge radius puzzle.

Frequency-comb referenced collinear laser spectroscopy of Be+ for nuclear structure investigations and many-body QED tests

Absolute transition frequencies of the $2s\,^2{\rm{S}}_{1/2}$ $\rightarrow$ $2p\,^2{\rm{P}}_{1/2,3/2}$ transitions in Be$^+$ were measured with a frequency comb in stable and short-lived isotopes at ISOLDE (CERN) using collinear laser spectroscopy. Quasi-simultaneous measurements in copropagating and counterpropagating geometry were performed to become independent from acceleration voltage determinations for Doppler-shift corrections of the fast ion beam. Isotope shifts and fine structure splittings were obtained from the absolute transition frequencies with accuracies better than 1\,MHz and led to a precise determination of the nuclear charge radii of $^{7,10-12}$Be relative to the stable isotope $^9$Be. Moreover, an accurate determination of the $2p$ fine structure splitting allowed a test of high-precision bound-state QED calculations in the three-electron system. Here, we describe the laser spectroscopic method in detail, including several tests that were carried out to determine or estimate systematic uncertainties. Final values from two experimental runs at ISOLDE are presented and the results are discussed.

Penning-trap experiments for spectroscopy of highly-charged ions at HITRAP

Highly charged ions offer the possibility to measure electronic fine structures and hyperfine structures with precisions of optical lasers. Microwave spectroscopy of transitions between Zeeman substates further yields magnetic moments (g-factors) of bound electrons, making tests of calculations in the framework of bound-state QED possible in the strong-field regime. We present the SPECTRAP and ARTEMIS experiments, which are currently being commissioned with highly charged ions in the framework of the HITRAP facility at GSI, Germany. We present the scientific outline, the experimental setups and first results with confined ions.

Transition energy measurements in hydrogenlike and heliumlike ions strongly supporting bound-state QED calculations

Transition energy measurements in hydrogenlike and heliumlike ions strongly supporting bound-state QED calculations