Details on the $O(m_{e}\alpha^{6})$ 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 nonrelativistic effective theories. The current status of the theoretical description of the positronium ground state hyperfine splitting is reviewed.