# Atomic Two-Body $(Z\alpha )^6$ Predictions

## TL;DR

This paper derives a new two-body Sommerfeld energy formula that predicts higher-order energy corrections for systems like muonic hydrogen and positronium, validated by existing calculations and providing new predictions.

## Contribution

It introduces a novel two-body Sommerfeld energy formula enabling predictions of $(Zeta)^6$ order energy terms for two-body systems.

## Key findings

- Validated formulas against positronium calculations
- Predicted energy terms for muonic hydrogen and hydrogen
- Established a link between relativistic two-body equations and energy predictions

## Abstract

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.