  For muonic hydrogen, positronium, and ordinary hydrogen, we show that the
existence of a relativistic two-body wave equation whose energy levels are
physically accurate to order $(Z\alpha )^4$ (where $Z\alpha$ is the binding
coupling constant) implies a previously unknown two-body Sommerfeld energy
formula which can be used to predict energy terms to order $(Z\alpha )^6$ using
simple algebra. Two such terms are verified to be physically correct by earlier
$(Z\alpha )^6$ calculations for positronium. For muonic hydrogen and ordinary
hydrogen, these terms are predictions.

High Energy Physics - Phenomenology