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 α⁸ m 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 α^{2(Zα) 6} m. We have also obtained leading pure self-energy logarithmic contributions of order α^{2(Zα) 8} m and α^{2(Zα) 9} m and estimated the subleading terms of order α^{2(Zα) 7} m, α^{2(Zα) 8} m, and α^{2(Zα) 9} m. 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 α^{3(Zα) 5} m, 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) 5} m.

Altogether, we have completed the evaluation of the logarithmic contributions to the 1s Lamb shift of order α⁸ m and reduced the overall α⁸ m uncertainty by approximately a factor of three for H, D, and He⁺ as compared with the most recent CODATA compilation.