Relativistic recoil effects for energy levels in a muonic atom within a Grotch-type approach: An application to the one-loop electronic vacuum polarization

We continue our account of relativistic recoil effects in muonic atoms and present explicitly analytic results at first order in electron-vacuum-polarization effects. The results are obtained within a Grotch-type approach based on an effective Dirac equation. Some expressions are cumbersome and we investigate their asymptotic behavior. Previously relativistic two-body effects due to the one-loop electron vacuum polarization were studied by several groups. Our results found here are consistent with the previous result derived within a Breit-type approach (including ours) and disagree with a recent attempt to apply a Grotch-type approach.