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The recursive Neville algorithm allows one to calculate interpolating functions recursively. Upon a judicious choice of the abscissas used for the interpolation (and extrapolation), this algorithm leads to a method for convergence acceleration. For example, one can use the Neville algorithm in order to successively eliminate inverse powers of the upper limit of the summation from the partial sums of a given, slowly convergent input series. Here, we show that, for a particular choice of the abscissas used for the extrapolation, one can replace the recursive Neville scheme by a simple onestep transformation, while also obtaining access to subleading terms for the transformed series after convergence acceleration. The matrixbased, uniﬁed formulas allow one to estimate the rate of convergence of the partial sums of the input series to their limit. In particular, Bethe logarithms for hydrogen are calculated to 100 decimal digits. Generalizations of the method to series whose remainder terms can be expanded in terms of inverse factorial series, or series with halfinteger powers, are also discussed.more » « lessFree, publiclyaccessible full text available October 1, 2025

The calculation of higherorder binding corrections to bound systems is a fundamental problem of theoretical physics. For any nonrelativistic expansion, one needs the FoldyWouthuysen transformation, which disentangles the particle and the antiparticle degrees of freedom. This transformation is carried out here to eighth order in the momenta or to eighth order in the momentum operators, which is equivalent to the eighth order of the fine structure constant. Matrix elements of the eighthorder terms are evaluated for F5/2 and F7/2 states in hydrogenlike ions and compared with the DiracCoulomb energy levels.more » « lessFree, publiclyaccessible full text available July 1, 2025

We discuss numerical aspects of instantons in two and threedimensional${\varphi}^{4}$theories with an internal$O(N)$symmetry group, the socalled$N$vector model. By combining asymptotic transseries expansions for large arguments with convergence acceleration techniques, we obtain highprecision values for certain integrals of the instanton that naturally occur in loop corrections around instanton configurations. Knowledge of these numerical properties is necessary in order to evaluate corrections to the largeorder factorial growth of perturbation theory in${\varphi}^{4}$theories. The results contribute to the understanding of the mathematical structures underlying the instanton configurations.
Published by the American Physical Society 2024 Free, publiclyaccessible full text available August 1, 2025 
We present a complete reevaluation of the irreducible twoloop vacuumpolarization correction to the photon propagator in quantum electrodynamics, i.e. with an electronpositron pair in the fermion propagators. The integration is carried out by reducing the integrations to a limited set of master integrals, which are calculated using integrationbyparts identities. Dimensional regularization is used in$D=42\epsilon $dimensions, and onmass shell renormalization is employed. The oneloop effect is given to order$\epsilon $, to be combined with the$1/\epsilon $divergence of the twoloop amplitude. Master integrals are given. Final evaluations of twoloop energy shifts for$1S$,$2S$, and$2P$states are done analytically, and results are presented, with an emphasis on muonic hydrogen. For relativistic DiracCoulomb reference states, higherorder coefficients are obtained for the$Z\alpha $expansion. We compare the results obtained to the existing literature.
Published by the American Physical Society 2024 Free, publiclyaccessible full text available May 1, 2025 
We revisit the derivation of multipole contributions to the atomwall interaction previously presented in Łach et al. [G. Łach, M. DeKieviet, and U. D. Jentschura, Phys. Rev. A 81, 052507 (2010)]. A careful reconsideration of the angular momentum decomposition of the second, third, and fourthrank tensors composed of the derivatives of the electricfield modes leads to a modification for the results for the quadrupole, octupole, and hexadecupole contributions to the atomwall interaction. Asymptotic results are given for the asymptotic longrange forms of the multipole terms, in both the shortrange and longrange limits. Calculations are carried out for hydrogen and positronium in contact with αquartz; a reanalysis of analytic models of the dielectric function of αquartz is performed. Analytic results are provided for the multipole polarizabilities of hydrogen and positronium. The quadrupole correction is shown to be numerically significant for atomsurface interactions. The expansion into multipoles is shown to constitute a divergent, asymptotic series. Connections to van der Waals corrected densityfunctional theory and applications to physisorption are describedmore » « lessFree, publiclyaccessible full text available January 1, 2025

The socalled protonradius puzzle (the current discrepancy of proton radii determined from spectroscopic measurements in ordinary versus muonic hydrogen) could be addressed via an accurate measurement of the Rydberg constant because the proton radius and the Rydberg constant values are linked through highprecision optical spectroscopy. We argue that, with manageable additional experimental effort, it might be possible to improve circular Rydberg state spectroscopy, potentially leading to an important contribution to the clarification of the puzzle. Our proposal involves circular and nearcircular Rydberg states of hydrogen with a principal quantum number around n = 18, whose classical velocity on a Bohr orbit is slower than that of the fastest macroscopic manmade object, the Parker Solar Probe. We obtain improved estimates for the quality factor of pertinent transitions and illustrate a few recent improvements in instrumentation which facilitate pertinent experiments.more » « lessFree, publiclyaccessible full text available December 1, 2024

Abstract The nonperturbative LandauKhalatnikovFradkin (LKF) transformations describe how Green functions in quantum field theory transform under a change in the photon field’s linear covariant gauge parameter (denoted
ξ ). The transformations are framed most simply in coordinate space where they are multiplicative. They imply that information on gaugedependent contributions from higher order diagrams in the perturbative series is contained in lower order contributions, which is useful in multiloop calculations. We study the LKF transformations for the propagator and the vertex in both scalar and spinor QED, in some particular dimensions. A novelty of our work is to derive momentumspace integral representations of these transformations; our expressions are also applicable to the longitudinal and transverse parts of the vertex. Applying these transformations to the treelevel Green functions, we show that the oneloop terms obtained from the LKF transformation agree with the gauge dependent parts obtained from perturbation theory. Our results will be presented in more comprehensive form elsewhere.Free, publiclyaccessible full text available December 1, 2024 
We analyze the leading and higherorder quantum electrodynamic corrections to the energy levels for a single electron bound in a Penning trap, including the Bethe logarithm correction due to virtual excitations of the reference quantum cyclotron state. The effective coupling parameter αc in the Penning trap is identified as the square root of the ratio of the cyclotron frequency, converted to an energy via multiplication by the Planck constant, to the electron rest mass energy. We find a large, stateindependent, logarithmic oneloop selfenergy correction of order α α4c mc2 lnðα−2 c Þ, where m is the electron rest mass and c is the speed of light. Furthermore, we find a stateindependent “trapped” Bethe logarithm. We also obtain a statedependent higherorder logarithmic selfenergy correction of order α α6c mc2 lnðα−2 c Þ. In the highenergy part of the boundstate self energy, we need to consider terms with up to six magnetic interaction vertices inside the virtual photon loop.more » « less

We revisit the derivation of the apparatusdependent correction to the energy levels of quantum cyclotron states, as previously outlined [Boulware et al., Phys. Rev. D 32, 729 (1985)]. We evaluate the leading corrections to the axial, magnetron, cyclotron, and spinprojectiondependent energy levels due to the altered photon field quantization in the vicinity of a conducting wall. Our work significantly extends previous considerations. Quantum cyclotron states are used for the determination of the electron g factor in Penning traps. Our calculations show that the numerically largest apparatusdependent corrections can be expected for the axial and magnetron frequencies, where they can be as large as 10−8 in relative units. For the cyclotron frequency, one can expect corrections on the order of 10−12 , which can affect the determination of the anomalous magnetic moment of the electron.more » « less

Abstract The proton radius puzzle is known as the discrepancy of the proton radius, obtained from muonic hydrogen spectroscopy (obtained as being roughly equal to 0.84 fm), and the proton radius obtained from (ordinary) hydrogen spectroscopy where a number of measurements involving highly excited states have traditionally favored a value of about 0.88 fm. Recently, a number of measurements of hydrogen transitions by the Munich (Garching) groups (notably, several hyperfineresolved sublevels of the 2 S –4 P ) and by the group at the University of Toronto (2 S –2 P 1/2 ) have led to transition frequency data consistent with the smaller proton radius of about 0.84 fm. A recent measurement of the 2 S –8 D transition by a group at Colorado State University leads to a proton radius of about 0.86 fm, in between the two aforementioned results. The current situation points to a possible, purely experimental, resolution of the proton radius puzzle. However, a closer look at the situation reveals that the situation may be somewhat less clear, raising the question of whether or not the proton radius puzzle has been conclusively solved, and opening up interesting experimental possiblities at TRIUMF/ARIEL.more » « less