Two-loop polarization contributions to radiative-recoil corrections to hyperfine splitting in muonium

We calculate radiative-recoil corrections of order ${\alpha }^2(Z\alpha {\left)\right(m/M)E}_F$ to hyperfine splitting in muonium generated by the diagrams with electron and muon polarization loops. These corrections are enhanced by the large logarithm of the electron-muon mass ratio. The leading logarithm cubed and logarithm squared contributions were obtained a long time ago. The single-logarithmic and nonlogarithmic contributions calculated here improve the theory of hyperfine splitting, and affect the value of the electron-muon mass ratio extracted from the experimental data on the muonium hyperfine splitting.