Vacuum polarization in muonic and antiprotonic atoms: the fine structure   at medium Z
  Effects of vacuum polarization modify the energy levels in atoms with an orbiting particle heavier than an electron. The dominant effect is due to the Uehling potential. In this paper we consider the relativistic corrections to the energy levels caused by the Uehling potential and in particular the fine structure in muonic and antiprotonic atoms. We derive general expressions and consider in detail specific regions of parameters which allow simple asymptotic expansion. We take into account the recoil effects and anomalous magnetic moment in the case of an antiproton as the orbiting particle. 
E. Yu. Korzinin, V. G. Ivanov, S. G. Karshenboim