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

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