Potential NRQED: The Positronium Case

We discuss in detail potential NRQED (pNRQED), a previously proposed effective field theory for ultrasoft photons. The pNRQED lagrangian for the equal mass case is presented and it is shown that it correctly reproduces the positronium spectrum at order $m\alpha^5$. The pNRQED lagrangian for the unequal mass case is also presented at the same order. Dimensional regularization is used throughout.