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

摘要

Adjusting a previously developed Grotch-type approach to a perturbative calculation of the electronic vacuumpolarization effects in muonic atoms, we find here the two-loop vacuum-polarization relativistic recoil correction of order α~2(Zα)~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 are applied to test our calculations and could be useful for other evaluations with an uncertainty better than 1%.