Theoretical Accuracy of S-State Lamb Shifts

Prom the theoretical point of view the accuracy of calculations is limited by the magnitude of the yet uncalculated contributions to the Lamb shift. Corrections to the P levels are known now with a higher accuracy than the corrections to the S levels, and do not limit the results of the comparison between theory and experiment. 

Corrections of order a Za) m are the largest uncalculated contributions to the energy levels for 5-states. These corrections are pure numbers and are estimated by comparing them with corrections of order a Za) m. They could be as large as 1 kHz for the 15-state and about 0.1 kHz for the 25-state. 

Only the leading logarithm squared contribution to the recoil correction of order Za) m/M) is known now [6, 7]. Numerically the contribution in (4.24) is below 1 kHz. Due to linear dependence of the recoil correction on the electron-nucleus mass ratio, the respective contribution to the hydrogen-deuterium isotope shift (see Subsect. 12.1.7 below) is phenomenologically 

Numerical results for recoil corrections obtained without expansion over Za [8, 9] indicate that the contribution of the single logarithmic and nonlog-arithmic recoil corrections of order [Zaf [m/M) is about 0.64(1) kHz for the IS level in hydrogen, and exceeds the total logarithm squared contribution. It would be interesting to calculate respective coefficients perturbatively. 

All other unknown theoretical contributions to the Lamb shift are much smaller, and 6 kHz for the IS-state and 0.8 kHz for the 2S-state are reasonable estimates of the total theoretical uncertainty of the expression for the Lamb shift. Theoretical uncertainties for the higher S levels may be obtained from the lA-state uncertainty ignoring its state-dependence and scaling it with the principal quantum number n. 

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