Hyperfine structure of the ground state muonic He-3 atom

On the basis of the perturbation theory in the fine structure constant $\alpha$ and the ratio of the electron to muon masses we calculate one-loop vacuum polarization and electron vertex corrections and the nuclear structure corrections to the hyperfine splitting of the ground state of muonic helium atom $(\mu e{}_2^{3}He)$. We obtain total result for the ground state hyperfine splitting $\mathrm{\Delta }{\nu }^{hfs}=4166.471$ MHz which improves the previous calculation of Lakdawala and Mohr due to the account of new corrections of orders ${\alpha }^5$ and ${\alpha }^6$. The remaining difference between our theoretical result and experimental value of the hyperfine splitting lies in the range of theoretical and experimental errors and requires the subsequent investigation of higher order corrections.