Nuclear recoil with electron vacuum polarization in muonic atoms

Using the heavy-particle QED approach, we derive the last missing α\,(Zα)⁵\,m²/M nuclear recoil correction in muonic atoms, which originates from the combination of the (Zα)⁵\,m²/M recoil with the electron vacuum polarization. Numerical results are presented for the $1S$, $2S$, and $2P$ states of $μ$H, $μ$D, μ³He⁺, and μ⁴He⁺. For the $2P-2S$ Lamb shift in muonic helium ions, our results are of opposite sign and about four times smaller in magnitude than the previous logarithmic estimates.