Two-loop corrections to Lamb shift and hyperfine splitting in hydrogen via multi-loop methods

We revisit the contributions of order $\alpha^2(Z\alpha)^5m$ and $\alpha^2(Z\alpha)E_F$, respectively, to the Lamb shift and to the hyperfine splitting from mixed self-energy-vacuum-polarization diagrams, involving fermionic loop. We use modern multi-loop calculation techniques based on IBP reduction and differential equations. We construct the $\epsilon$-regular basis [LeeOnishchenko2019] and explicitly demonstrate that it is compatible with the renormalization. We obtain analytic results in terms of one-fold integral involving elliptic function and dilogarithm. As a by-product, we obtain the analogous contribution for the limiting cases of heavy and light fermionic loop.