Eeective Field Theory and Matching in Non-relativistic Gauge Theories

The eeective Lagrangian and power counting rules for non-relativistic gauge theories are derived via a systematic expansion in the large c limit. It is shown that the 1=c expansion leads to an eeective eld theory which incorporates a multi-pole 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 coeecient. We show how this power counting works in the calculation of the Lamb shift within the eeective eld theory formalism.