The relativistic recoil contributions to the Uehling corrections are revisited. We consider a controversy in recent calculations, which are 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 to retardation and two-photon-exchange effects were missed. We have evaluated such effects and obtained a consistent result.We present a correct expression for the Grotch-type approach which produces a correct gaugeinvariant 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.

The complete contribution of diagrams with the light-by-light scattering to the Lamb shift is found for muonic hydrogen, deuterium and helium ion. The results are obtained in the static muon approximation and a part of the paper is devoted to the verification of this approximation and analysis of its uncertainty.

Contributions to the energy levels in light muonic atoms and, in particular, to the Lamb shift fall into a few well-distinguished classes. The related diagrams are calculated using different approaches. In particular, there is a specific type of nonrelativistic (NR) contribution. Here, we consider such corrections to the Lamb shift of order α 5 m µ . These contributions are due to free vacuum-polarization loops as well as to various effects of light-by-light scattering. The closed loop in the related diagrams is an electronic one, which allows an NR consideration of the muon. Both types of contributions have been known for some time, however, the results obtained to date are only partial results. We complete a calculation of the α 5 m µ contributions for muonic hydrogen. The results are also adjusted for muonic deuterium atom and helium ion.

Abstract. In muonic atoms the Uehling potential (an effect of a free electronic vacuum polarization loop) is responsible for the leading contribution to the Lamb shift causing the splitting of states with ∆n = 0 and ∆l = 0. Here we consider the Lamb shift in the leading nonrelativistic approximation, i.e., within an approach based on a certain Schrödinger equation. That is valid for low and medium Z as long as (Zα) 2 ≪ 1. The result is a function of a few parameters, including κ = Zαmµ/me, n and l. We present various asymptotics and in particular we study a region of validity of asymptotics with large and small κ. Special attention is paid to circular states, which are considered in a limit of n ≫ 1.PACS. 36.10.Gv Mesonic atoms and molecules, hyperonic atoms and molecules -31.30.Jv Relativistic and quantum electrodynamic effects in atoms and molecules