Nuclear recoil with electron vacuum polarization in muonic atoms

Abstract

Using the heavy-particle QED approach, we derive the last missing $\alpha (Z\alpha {)}^5{m}^2/M$ nuclear recoil correction in muonic atoms, which originates from the combination of the $(Z\alpha {)}^5{m}^2/M$ recoil with the electron vacuum polarization. Numerical results are presented for the $1S$, $2S$, and $2P$ states of $\mu$H, $\mu$D, ${\mu }^3$He${}^{+}$, and ${\mu }^4$He${}^{+}$. 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.