Higher order corrections to bound state energy levels in QED: an effective field theory approach

In T.Zhang, L.Xiao and R.Koniuk, Can. J. Phys. {\bf 70}, 670 (1992), a new method was developed to calculate energy levels in QED nonrelativistic bound states, up to order m \alpha^5. Whether or not this method could be used beyond this order was left as an open question. In this paper, we answer this question with the help of a nonrelativistic effective field theory, NRQED. We find that, aside from infrared terms which had to be discarded without justification in Zhang et al, the order m \alpha^4 and m \alpha^5 calculations give the same results in both methods. It is shown, however, that this is due to the special physical origin of these contributions and that the method of Zhang et al would not give the right answer at order m \alpha^6 or higher. The separation of scale provided by NRQED is essential to the derivation of this result.