Semi-Analytic Approach to Higher-Order Corrections in Simple Muonic   Bound Systems: Vacuum Polarization, Self-Energy and Radiative-Recoil
  The current discrepancy of theory and experiment observed recently in muonic hydrogen necessitates a reinvestigation of all corrections to contribute to the Lamb shift in muonic hydrogen muH, muonic deuterium muD, the muonic 3He ion, as well as in the muonic 4He ion. Here, we choose a semi-analytic approach and evaluate a number of higher-order corrections to vacuum polarization (VP) semi-analytically, while remaining integrals over the spectral density of VP are performed numerically. We obtain semi-analytic results for the second-order correction, and for the relativistic correction to VP. The self-energy correction to VP is calculated, including the perturbations of the Bethe logarithms by vacuum polarization. Subleading logarithmic terms in the radiative-recoil correction to the 2S-2P Lamb shift of order alpha (Zalpha)^5 mu^3 ln(Zalpha)/(m_mu m_N) are also obtained. All calculations are nonperturbative in the mass ratio of orbiting particle and nucleus. 
U. D. Jentschura and B. J. Wundt
physics.atom-ph hep-ph