The Uehling correction to the energy levels in a pionic atom
  We consider a correction to energy levels in a pionic atom induced by the Uehling potential, i.e., by a free electron vacuum-polarization loop. The calculation is performed for circular states (l=n-1). The result is obtained in a closed analytic form as a function of $ Z\alpha$ and the pion-to-electron mass ratio. Certain asymptotics of the result are also presented. 
S. G. Karshenboim, E. Yu. Korzinin, V. G. Ivanov