The Lamb shift of the 1s state in hydrogen: Two-loop and three-loop contributions

We consider the is 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 alpha(8)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 B-61, B-60, C-50 in standard notation. We have found a missing logarithmic contribution of order alpha(2)(Z alpha)(6)m. We have also obtained leading pure self-energy logarithmic contributions of order alpha(2)(Z alpha)(8)m and alpha(2)(Z alpha)(9)m and estimated the subleading terms of order alpha(2)(Z alpha)(7)m, alpha(2)(Z alpha)(8)m, and alpha(2)(Z alpha)(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 alpha(3)(Z alpha)(5)m, coefficient C-50 in standard notation) and applied it to approximate integration over the loop momentum. Our result for contributions to the is Lamb shift for the total three loop next-to-leading term is (-3.3 +/- 10.5)(alpha(3)/pi(3))(Z alpha)(5)m. Altogether, we have completed the evaluation of the logarithmic contributions to the is Lamb shift of order alpha(8)m and reduced the overall alpha(8)m uncertainty by approximately a factor of three for H, D, and He+ as compared with the most recent CODATA compilation. (C) 2019 The Authors. Published by Elsevier B.V.