Theory of the Lamb shift and Fine Structure in muonic $\mathrm{^4He}$ ions and the muonic $\mathrm{^3He-^4He}$ Isotope Shift

TL;DR

This paper summarizes the theoretical calculations of the Lamb shift and fine structure in muonic helium ions, crucial for extracting nuclear charge radii and isotope shifts, with a focus on reducing uncertainties and understanding model dependencies.

Contribution

It provides a comprehensive and updated theoretical analysis of the Lamb shift and fine structure in muonic helium ions, including model dependence and uncertainty assessments, supporting precise nuclear radius measurements.

Findings

Theoretical contributions are compiled for charge radius extraction.

Model dependence on electric form factor is analyzed.

Theory uncertainty is smaller than existing measurement discrepancies.

Abstract

We provide an up to date summary of the theory contributions to the 2S-2P Lamb shift and the fine structure of the 2P state in the muonic helium ion $(\mathrm{\mu^4He})^+$. This summary serves as the basis for the extraction of the alpha particle charge radius from the muonic helium Lamb shift measurements at the Paul Scherrer Institute, Switzerland. Individual theory contributions needed for a charge radius extraction are compared and compiled into a consistent summary. The influence of the alpha particle charge distribution on the elastic two-photon exchange is studied to take into account possible model-dependencies of the energy levels on the electric form factor of the nucleus. We also discuss the theory uncertainty which enters the extraction of the $\mathrm{^3He-^4He}$ isotope shift from the muonic measurements. The theory uncertainty of the extraction is much smaller than a present discrepancy between previous isotope shift measurements. This work completes our series of $n=2$ theory compilations in light muonic atoms which we have performed already for muonic hydrogen, deuterium, and helium-3 ions.