A Non-perturbative Estimate of Vacuum Polarization in QED
  We present a new estimate of the fine structure constant and the $\beta$-function of QED at an arbitrary scale. Using the non-perturbative but convergent series expression of the one loop effective action of QED that has been available recently we make a non-perturbative estimate of the running coupling and the $\beta$-function, and prove the renormalization group invariance of the effective action. The contrast between our result and the perturbative result is remarkable. 
Y. M. Cho and D. G. Pak