Recoil corrections to the energy levels of hydrogenic atoms

TL;DR

This paper calculates precise recoil corrections of order $(Z \alpha)^6$ for hydrogenic atoms, applicable to various systems, and introduces methods for computing complex integrals relevant to higher-order corrections.

Contribution

It provides an exact calculation of recoil corrections for arbitrary mass ratios and develops analytic methods for complex two-loop integrals in bound state QED.

Findings

Recoil corrections of order $(Z \alpha)^6$ are now precisely computed.

Analytic expressions for two-loop master integrals are derived.

The methods can be extended to three-loop integrals for higher-order corrections.

Abstract

We have completed the calculation of pure-recoil corrections of order $(Z \alpha)^6$ to Coulombic bound states of two spin-1/2 fermions without approximation in the particle masses. Our result applies to systems of arbitrary mass ratio such as muonium and positronium, and also hydrogen and muonic hydrogen (with the neglect of proton structure effects). We have shown how the two-loop master integrals that occur in the relativistic region can be computed in analytic form, and suggest that the same method can be applied to the three-loop integrals that would be present in a calculation of order $(Z \alpha)^7$ corrections.