Second-order corrections to the wave function at origin in muonic hydrogen and pionium

TL;DR

This paper calculates second-order non-relativistic corrections to the wave function at origin in muonic and exotic atoms, focusing on vacuum polarization effects relevant for precise atomic measurements.

Contribution

It provides new quantitative results for wave function corrections in muonic hydrogen, deuterium, and pionium, incorporating electronic vacuum polarization effects.

Findings

Corrections are significant for hyperfine splitting and Lamb shift calculations.

Results improve the accuracy of theoretical predictions for exotic atom properties.

Applications to pionium lifetime and finite nuclear size effects are demonstrated.

Abstract

Non-relativisitic second-order corrections to the wave function at origin in muonic and exotic atoms are considered. The corrections are due to the electronic vacuum polarization. Such corrections are of interest due to various effective approaches, which take into account QED and hadronic effects. The wave function at origin plays a key role in the calculation of the pionium lifetime, various finite nuclear size effects and the hyperfine splitting. The results are obtained for the $1s$ and $2s$ states in pionic and muonic hydrogen and deuterium and in pionium, a bound system of $\pi^+$ and $\pi^-$. Applications to the hyperfine structure and the Lamb shift in muonic hydrogen are also considered.