Bound states and power counting in effective field theories

The problem of bound states in effective field theories is studied. A rescaled version of nonrelativistic effective field theory is formulated which makes the velocity power counting of operators manifest. Results obtained using the rescaled theory are compared with known results from NRQCD. The same ideas are then applied to study Yukawa bound states in $1+1$ and $3+1$ dimensions, and to analyze when the Yukawa potential can be replaced by a δ-function potential. The implications of these results for the study of nucleon-nucleon scattering in chiral perturbation theory are discussed.