Corrections of Order Α(zα) to Hyperfine Splitting and Lamb Shift

Corrections to hyperfine splitting and Lamb shift of order α(Zα) induced by the diagrams with radiative photon insertions in the electron line are calculated in the Fried-Yennie gauge. These contributions are as large as −7.725(3)α(Zα)/(πn)(mr/m) m and −0.6711(7)α(Zα)/(πn)EF for the Lamb shift and hyperfine splitting, respectively. Phenomenological implications of these results are discussed with special emphasis on the accuracy of the theoretical predictions for the Lamb shift and experimental determination of the Rydberg constant. New more precise value of the Rydberg constant is obtained on the basis of the improved theory and experimental data. A steady and rapid progress in the spectroscopic measurements in recent years led to a dramatic increase of accuracy in the measurements of the Rydberg constant [1, 2], ground state 1S Lamb shift in hydrogen and deuterium [2, 3, 4], classic 2S−2P Lamb shift in hydrogen [5, 6, 8], and of the muonium hyperfine splitting in the ground state [9, 10] (see Table 1). These spectacular experimental achievements constitute a serious challenge to the theory and intensive theoretical efforts are necessary to match this experimental accuracy. Theoretical work on the high order corrections to hyperfine splitting (HFS) and Lamb shift concentrated recently on calculation of nonrecoil contributions of order α(Zα). Their magnitude may run up to several kilohertz for HFS in the ground state of muonium, to several tens of kilohertz for n = 2 Lamb shift in hydrogen and may be as large as hundreds of kilohertz for the ground state Lamb shift in hydrogen. Contributions of such order of magnitude are clearly crucial for comparison of the current and pending experimental results with the theory. As was shown in [11] for hyperfine splitting and in [12] for the Lamb shift there are six gauge invariant sets of diagrams (see Fig.1), which produce corrections of order α(Zα). All these diagrams may be obtained from the skeleton diagram, which contains two external photons attached to the electron line, with the help of different radiative insertions. All contributions induced by the diagrams in Figs.1a − 1e, containing closed electron loops, were obtained recently in papers [11, 13, 14, 15] for the case of hyperfine splitting and in papers [12, 16, 17, 18, 19] for the case of the Lamb shift. These theoretical results are now firmly established since all these corrections were calculated independently by two different groups and the results of these calculations are in excellent agreement. We report below on the results of our calculation of the contributions of order α(Zα) to HFS and Lamb shift induced by the last gauge invariant set of diagrams in Fig.1f . This set includes nineteen topologically different diagrams [20] presented in Fig.2. The simplest way to describe these graphs is to realize that they were obtained from the three graphs for the two-loop electron self-energy by insertion of two external photons in all possible ways. Really, graphs 2a−2c are obtained from the two-loop reducible electron selfenergy diagram, graphs 2d−2k are the result of all possible insertions of two external photons in the rainbow self-energy diagram, and diagrams 2l−2s are connected with the overlapping two-loop self-energy graph. We have already