Effective Field Theory and Matching in Non-Relativistic Gauge Theories

The effective Lagrangian and power counting rules for non-relativistic gauge theories are derived via an expansion in $1/c$. It is shown that the $1/c$ expansion leads to an effective field theory which incorporates a multipole expansion. Failure to use this expansion in the low energy theory leads to a loss of predictive power as the matching procedure becomes intractable. It is emphasized that there is not a one-to-one correspondence between operators in the modified low energy theory and those in the heavy quark effective theory. We give an explicit example where the $1/c$ expansion leads to different results from previously presented methods. We also discuss the Lamb shift as a pedagogical example of the power counting scheme.