We consider the most general loop integral that appears in non-relativistic
effective field theories with no light particles. The divergences of this
integral are in correspondence with simple poles in the space of complex sp
ace-time dimensions. Integrals related to the original integral by subtract
ion of one or more poles in dimensions other than D = 4 lead to non-minimal
subtraction schemes. Subtraction of all poles in correspondence with ultra
violet divergences of the loop integral leads naturally to a regularization
scheme which is precisely equivalent to cut-off regularization. We therefo
re recover cut-off regularization from dimensional regularization with a no
n-minimal subtraction scheme. We then discuss the power counting for non-re
lativistic effective field theories which arises in these alternative schem
es.