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 $\alpha^8m$ 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 $\alpha^2(Z\alpha)^6m$. We have also obtained leading pure self-energy logarithmic contributions of order $\alpha^2(Z\alpha)^8m$ and $\alpha^2(Z\alpha)^9m$ and estimated the subleading terms of order $\alpha^2(Z\alpha)^7m$, $\alpha^2(Z\alpha)^8m$, and $\alpha^2(Z\alpha)^9m$. 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 $\alpha^3(Z\alpha)^5m$, coefficient $C_{50}$ 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 $(-3.3\pm10.5)(\alpha^3/\pi^3)(Z\alpha)^5m$. Altogether, we have completed the evaluation of the logarithmic contributions to the $1s$ Lamb shift of order $\alpha^8m$ and reduced the overall $\alpha^8m$ uncertainty by approximately a factor of three for H, D, and He$^+$ as compared with the most recent CODATA compilation.