Details on the $\mathcal{O}(m_e{\alpha }^6)$ Positronium Hyperfine Splitting due to Single Photon Annihilation

A detailed presentation is given of the analytic calculation of the single-photon annihilation contributions to the positronium ground state hyperfine splitting to order $m_e{\alpha }^6$ in the framework of non-relativistic effective theories. The current status of the theoretical description of the positronium ground state hyperfine splitting is reviewed.