Effective Hamiltonian approach to the bound state: Positronium hyperfine structure

Abstract

An effective Hamiltonian approach is developed for the perturbative calculation of bound state energies in quantum electrodynamics. In this approach a recoil correction of order ${\alpha }^2{E}_F$ to the hyperfine structure of hydrogenic systems with the arbitrary mass ratio is obtained. In the case of the positronium it amounts to $\Delta E=$ $\frac{1}{3}$ $m{\alpha }^6\left[1.130\right(5)\mathrm{}-$ $\frac{1}{2}$ $\ln \alpha ]$.