The Single Photon Annihilation Contributions to the Positronium Hyperfine Splitting to Order $m_eα^6$

The single photon annihilation contributions for the positronium ground state hyperfine splitting are calculated analytically to order $m_e\alpha^6$ using NRQED. Based on intuitive physical arguments the same result can also be determined by a trivial calculation using results from existing literature. Our result completes the hyperfine splitting calculation to order $m_e\alpha^6$. We compare the theoretical prediction with the most recent experimental measurement.