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 ${}^3$He and ${}^4$He ions.

Figure 1

One-photon exchange diagram: for the pure photon exchange (a) and for the eVP contributions (b).

Figure 2

Two-photon exchange diagram: (a) for the pure photon exchange and (b) for the eVP contributions.

Figure 3

Poles for the unsubtracted two-photon exchange diagrams (see Fig. 2). The upper plot is for the ladder diagrams (right-side graphs in Fig. 2), while the lower one is for the cross diagrams (left-side graphs). For muonic hydrogen with $\kappa \sim 1$ the pole structure for free diagrams and for eVP contributions [see Figs. 2a and 2b, respectively] is similar. Here, $N$ stands for nuclear poles, $\gamma$ is for photonic poles, and $\mu$ is for muonic poles.