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

physical review. a, atomic, molecular, and optical physics/physical review, a, atomic, molecular, and optical physics

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