Defect Theory of Positronium and Nontrivial QED Relations

Abstract

An effective theory of the excited states of positronium is derived and some of its consequences are explored. At large physical separation, the binding of the electron and positron is assumed to be described completely by QED, whereas all short-ranged phenomena, including those within and beyond QED, can be accounted for with energy-dependent quantum defects. This theory has at least two practical applications. First, it provides an accurate and economical, yet largely QED-independent, means to fit the positronium spectrum in order to predict and compare the outcome of experiments. Second, matching the spectrum in this effective theory to that predicted by QED reveals nontrivial relationships that exist \emph{within} bound-state QED; some higher order contributions to the spectrum may be obtained from lower order contributions. These relations are verified up to order $m\alpha^{6}$, and predictions are made for the order $m\alpha^{7}$ and $m\alpha^{8}$ corrections. This theory and its extensions to other hydrogenic systems may provide a useful complement to bound-state QED.