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

TL;DR

This paper calculates two-loop quantum electrodynamics corrections to the Lamb shift and hyperfine splitting in hydrogen using advanced multi-loop techniques, providing analytic results involving elliptic functions and dilogarithms.

Contribution

It introduces a modern multi-loop calculation approach with IBP reduction and differential equations to evaluate complex fermionic loop contributions in hydrogen energy levels.

Findings

Analytic expressions for two-loop corrections involving elliptic functions.

Explicit demonstration of renormalization compatibility with the epsilon-regular basis.

Results for limiting cases of heavy and light fermionic loops.

Abstract

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.