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.Comment: Submitted to Phys.Rev.A; cross refs are added in v.2&

Vacuum polarization in muonic atoms: the Lamb shift at low and medium Z

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 Delta n = 0 and Delta l \neq 0. Here we consider the Lamb shift in the leading nonrelativistic approximation, i.e., within an approach based on a certain Schrodinger equation. That is valid for low and medium $Z$ as long as (Z alpha)^2 >> 1. The result is a function of a few parameters, including kappa = Z alpha m_ mu/m_e, n and l. We present various asymptotics and in particular we study a region of validity of asymptotics with large and small kappa. Special attention is paid to circular states, which are considered in a limit of n >> 1

Relativistic recoil effects in a muonic atom within a Grotch-type approach: General approach

Recently we calculated relativistic recoil corrections to the energy levels of the low lying states in muonic hydrogen induced by electron vacuum polarization effects. The results were obtained by Breit-type and Grotch-type calculations. The former were described in our previous papers in detail, and here we present the latter. The Grotch equation was originally developed for pure Coulomb systems and allowed to express the relativistic recoil correction in order $(Z\alpha)^4m^2/M$ in terms of the relativistic non-recoil contribution $(Z\alpha)^4m$. Certain attempts to adjust the method to electronic vacuum polarization took place in the past, however, the consideration was incomplete and the results were incorrect. Here we present a Groth-type approach to the problem and in a series of papers consider relativistic recoil effects in order $\alpha(Z\alpha)^4m^2/M$ and $\alpha^2(Z\alpha)^4m^2/M$. That is the first paper of the series and it presents a general approach, while two other papers present results of calculations of the $\alpha(Z\alpha)^4m^2/M$ and $\alpha^2(Z\alpha)^4m^2/M$ contributions in detail. In contrast to our previous calculation, we address now a variety of states in muonic atoms with a certain range of the nuclear charge $Z$.Comment: Submitted to Phys.Rev.A; cross refs are added in v.2&

Relativistic recoil effects to energy levels in a muonic atom: a Grotch-type calculation of the second-order vacuum-polarization contributions

Adjusting a previously developed Grotch-type approach to a perturbative calculation of the electronic vacuum-polarization effects in muonic atoms, we find here the two-loop vacuum polarization relativistic recoil correction of order $\alpha^2(Z\alpha)^4m^2/M$ in light muonic atoms. The result is in perfect agreement with the one previously obtained within the Breit-type approach. We also discuss here simple approximations of the irreducible part of the two-loop vacuum-polarization dispersion density, which was applied to test our calculations and can be useful for other evaluations with an uncertainty better than 1%.Comment: Submitted to Phys.Rev.A; cross refs are added in v.2&

The α2(Zα)4m\alpha^2(Z\alpha)^4m contributions to the Lamb shift and the fine structure in light muonic atoms

Corrections to energy levels in light muonic atoms are investigated in order $\alpha^2(Z\alpha)^4m$. We pay attention to corrections which are specific for muonic atoms and include the electron vacuum polarization loop. In particular, we calculate relativistic and relativistic-recoil two-loop electron vacuum polarization contributions. The results are obtained for the levels with $n=1,2$ and in particular for the Lamb shift ($2p_{1/2}-2s_{1/2}$) and fine-structure intervals ($2p_{3/2}-2p_{1/2}$) in muonic hydrogen, deuterium, and muonic helium ions.Comment: Accepted by Phys.Rev.D; cross refs are added in v.

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.Comment: Submitted to Phys. Rev. A; in v.2 results for muonic helium are correcte

Subtractions and the effective Salpeter term for the Lamb shift in muonic atoms with the nuclear spin

While taking into account the nuclear-structure contributions to the Lamb shift, one has to make various subtractions for the two-photon exchange contributions. Such subtractions should be consistent with the structureless part of theory. We study here the subtractions for a two-body atomic systems which consist of a pointlike lepton (an electron or a muon) and a nucleus with spin 0, 1/2, and 1, and find the recoil contribution in order (Zα)5 due to the subtractions for I = 0, 1. (The related contribution to the energy levels for I = 1∕2 of order (Zα)5m is called the Salpeter term.