# Theory of the n=2 levels in muonic helium-3 ions

## TL;DR

This paper reviews the current theoretical understanding of the energy levels, including Lamb shift and hyperfine structure, in muonic helium-3 ions, preparing for upcoming experimental measurements to determine nuclear properties.

## Contribution

It provides a comprehensive comparison of existing calculations and updates, offering reliable values for QED and nuclear effects crucial for interpreting future experimental results.

## Key findings

- Reliable values for Lamb shift and hyperfine splitting
- Updated calculations including unpublished data
- Preparation for experimental determination of helion charge radius

## Abstract

The present knowledge of Lamb shift, fine-, and hyperfine structure of the 2S and 2P states in muonic helium-3 ions is reviewed in anticipation of the results of a first measurement of several $\mathrm{2S\rightarrow2P}$ transition frequencies in the muonic helium-3 ion, $\mathrm{\mu^3He^+}$. This ion is the bound state of a single negative muon $\mu^-$ and a bare helium-3 nucleus (helion), $\mathrm{^3He^{++}}$. A term-by-term comparison of all available sources, including new, updated, and so far unpublished calculations, reveals reliable values and uncertainties of the QED and nuclear structure-dependent contributions to the Lamb shift and the hyperfine splitting. These values are essential for the determination of the helion rms charge radius and the nuclear structure effects to the hyperfine splitting in $\mathrm{\mu^3He^+}$. With this review we continue our series of theory summaries in light muonic atoms; see Antognini et al., Ann. Phys. 331, 127 (2013), Krauth et al., Ann.Phys. 366, 168 (2016), and Diepold et al., ArXiv 1606.05231 (2016).