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The effective Lagrangian and power counting rules for nonrelativistic gauge theories are derived via a systematic expansion in the large c limit. It is shown that the 1/c expansion leads to an effective field theory which incorporates a multipole expansion. Within this theory there is no need for heuristic arguments to determine the scalings of operators. After eliminating c from the lowest order Lagrangian the states of the theory become independent of c and the scaling of an operator is given simply by its overall coefficient. We show how this power counting works in the calculation of the Lamb shift within the effective field theory formalism.
Grinstein et al. (1998) studied this question.