Nonrelativistic contributions of order α5mμc2 to the Lamb shift in muonic hydrogen and deuterium, and in the muonic helium ion

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 α5mμ. 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 α5mμ contributions for muonic hydrogen. The results are also adjusted for muonic deuterium atom and helium ion.