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Ground state hyperfine structure in muonic lithium ions

On the basis of perturbation theory in fine structure constant alpha and the ratio of electron to muon masses we calculate one-loop vacuum polarization, electron vertex corrections, nuclear structure and recoil corrections to hyperfine splitting of the ground state in muonic lithium ions $(μ e\ ^6_3Li)^+$ and $(μ e\ ^7_3Li)^+$. We obtain total results for the ground state small hyperfine splittings in $(μ e\ ^6_3Li)^+$ $Δν_1=21572.16$ MHz and $Δν_2=14152.56$ MHz and in $(μ e\ ^7_3Li)^+$ $Δν_1=21733.06$ MHz and $Δν_2=13994.35$ MHz which can be considered as a reliable estimate for a comparison with future experimental data.