Nonrelativistic QED and next-to-leading hyperfine splitting in positronium

We compute the next-to-leading $O\left(m{\alpha }^5\right)$ contribution to the hyperfine splitting in positronium within the framework of nonrelativistic QED (NRQED). When applied to the ground state, our calculation reproduces the known result, providing a further test of NRQED techniques. In addition to providing a very simple method of calculation of this standard result, we also obtain new expressions for excited states of positronium with negligible additional effort. Our calculation requires the complete next-to-leading matching of the lowest-dimension NRQED four-Fermi couplings.