Relativistic recoil corrections to the electron-vacuum-polarization contribution in light muonic atoms

The relativistic recoil contributions to the Uehling corrections are revisited. A controversy in recent calculations is considered, which is based on different approaches including Breit-type and Grotch-type calculations. It is found that calculations in those works were in fact done in different gauges and in some of those gauges contributions to retardation and two-photon-exchange effects were missed. Such effects are evaluated and a consistent result is obtained. A correct expression for the Grotch-type approach is presented, which produces a correct gauge-invariant result. A finite-nuclear-size correction for the Uehling term is also considered. The results are presented for muonic hydrogen and deuterium atoms and for muonic ³He and ⁴He ions.