${\alpha }^2\mathbf{(}Z\alpha {\mathbf{)}}^{4}m$ 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 ${\alpha }^2(Z\alpha {)}^{4}m$. 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.

Figure 1

Leading-order diagrams, which include the electronic ($e$) light-by-light scattering block. The horizontal double line is for the reduced Green function of a muon ($\mu$) at the Coulomb field.

Figure 2

The eVP potentials are required for a nonrelativistic calculation of the eVP two-loop contribution. They include the Uehling potential and a reducible and irreducible part of the Källen-Sabry potential [38].

Figure 3

Effective potentials for the NRPT calculation of the relativistic recoil one-loop eVP contribution.

Figure 4

Diagrams for the relativistic recoil one-loop eVP correction.

Figure 5

Diagrams of the nonrelativistic perturbation theory for the calculation of the relativistic recoil two-loop eVP contributions.

Figure 6

Diagrams for various classes of specific corrections, contributing at order ${\alpha }^2(Z\alpha {)}^{4}m$. $h$ stands for hadronic vacuum polarization.

Figure 7

The leading hadronic VP contribution to the Lamb shift.