Two-loop Bethe logarithms for non-Slevels

Abstract

Two-loop Bethe logarithms are calculated for excited P and D states in hydrogenlike systems, and estimates are presented for all states with higher angular momenta. These results complete our knowledge of the P and D energy levels in hydrogen at the order of α 8 me c 2 , where me is the electron mass and c is the speed of light, and scale as Z 6 , where Z is the nuclear charge number. Our analytic and numerical calculations are consistent with the complete absence of logarithmic terms of order (α/π) 2 (Zα) 6 ln[(Zα) −2 ] me c 2 for D states and all states with higher angular momenta. For higher excited P and D states, a number of poles from lower-lying levels have to subtracted in the numerical evaluation. We find that, surprisingly, the corrections of the "squared decayrate type" are the numerically dominant contributions in the order (α/π) 2 (Zα) 6 me c 2 for states with large angular momenta, and provide an estimate of the entire B60-coefficient for Rydberg states with high angular momentum quantum numbers. Our results reach the predictive limits of the quantum electrodynamic theory of the Lamb shift.

Proposal for the determination of nuclear masses by high-precision spectroscopy of Rydberg states

“…Calculations also exist for B 60 (see Ref. [37]), however only for states with ℓ ≤ 5. The three-loop term C 60 as well as the higher-order two-photon contributions not listed in Eq.…”

Two-loop effects and current status of the 4He+ Lamb shift

“…In the current section of this article, we are primarily concerned with recently obtained [8,16] results for the normalized (or "weighted") difference H (nS, Zα)−H (1S, Zα) of S states, whose importance for the determination of fundamental constants has been stressed in refs. In the current section of this article, we are primarily concerned with recently obtained [8,16] results for the normalized (or "weighted") difference H (nS, Zα)−H (1S, Zα) of S states, whose importance for the determination of fundamental constants has been stressed in refs.…”

“…25, which is currently under study and which will need to be evaluated before final conclusions can be drawn. 16) provide enough information to eliminate all theoretical uncertainty at the current level of accuracy. This uncertainty would comprise all higher order analytic terms, as it involves a comparison to a nonperturbative (in Zα) calculation.…”

“…1-8 and references therein) as well as within the nonperturbative (in Zα) numerical approach, as described in refs. We also review some very recent progress [16] regarding the B 60 coefficient for general excited hydrogenic states with nonvanishing angular momentum, with a special emphasis on states with P symmetry. 1.…”

Screened Radiative Corrections from Hyperfine-Split Dielectronic Resonances in Lithiumlike Scandium

“…For experimentally analyzing these quantum-electrodynamical (QED) effects, especially their convergence for multiple virtual-particle loops, lithiumlike ions play a crucial role as the transition energies between the fine structure levels 2s and 2p in their valence shell have particularly large relative QED corrections and are in the reach of precision measurements for many elements up to the highest nuclear charge Z [2 -10]. At high Z especially, these level energies presently offer the most precise access to single-electron radiative corrections at the forefront of experimental [9] and theoretical research [11,12], representing the corrections due to two loops of virtual particles. Nevertheless, few-body effects severely interfere in extracting the higher-order single-electron effects from such measurements.…”