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

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 A term-by-term comparison of all available sources, including new, updated, and so far unpublished calculations, reveals reliable values and uncertainties of the QED and nuclear structure-dependent contributions to the Lamb shift and the hyperfine splitting.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).