Vacuum polarization corrections to the n=2 and n=3 levels in muonic He4

Wayne W Repko,Duane A Dicus,Stanley F Radford

doi:10.1103/physrevd.101.053005

Abstract

The $n=2$ and $n=3$ levels for muonic Helium are calculated using a potential that includes all one-loop and recoil effects. Electronic vacuum polarization corrections are calculated using an extension of Kinoshita and Nio method. For $n=2$, the results are $2p_{1/2}-2s_{1/2}=1375.05\pm 1.4$ meV and $2p_{3/2}-2s_{1/2}=1521.65\pm 1.4$ meV, essentially in agreement with the latest summary of the current calculations. The $n=3$ results are summarized in tabular form and give $3p_{1/2}-3s_{1/2}=394.76\pm 0.43$ meV and $3d_{3/2}-3p_{3/2}=111.40$ meV.

Highlights

The tension between the determination of the proton size measured using muonic hydrogen energy levels [1,2] and electron scattering data [3] has focused renewed attention on accurate calculations of the level splitting in muonic atoms [4,5,6,7,8,9]
Part of the problem in the muonic hydrogen extraction of the proton radius is related to the appropriate modeling of its complicated electromagnetic structure consisting of both electric and magnetic form factors
The primary effect of including the eevacuum polarization corrections is the modification of the Coulomb potential

Summary

INTRODUCTION

The tension between the determination of the proton size measured using muonic hydrogen energy levels [1,2] and electron scattering data [3] has focused renewed attention on accurate calculations of the level splitting in muonic atoms [4,5,6,7,8,9]. QED corrections since the presence of an additional spin means that there are hyperfine and tensor interactions as well as mixing between energy levels While these corrections are well understood, they, in some sense, get in the way of figuring out the proper way to model the size parameters of the proton. The spin-independent terms should be verified by calculating the one-loop correction to the Breit-Pauli potential. The largest corrections to Eq (1) are QED electron vacuum polarization corrections to the Coulomb interaction and modifications due to the nuclear form factor. These will be treated in the two sections.

VACUUM POLARIZATION CORRECTIONS

NUCLEAR SIZE CORRECTIONS

ONE-LOOP CONTRIBUTIONS

RESULTS

CONCLUSIONS