The renormalization and regularization of 1/r^n potentials*

Potentials derived from effective field theory that contain exchange particles usually contain interactions that fall off as 1/rⁿ at short distances (here r denotes the relative distance between two particles). I will discuss the renormalization of the 1/r⁶ interaction that is of relevance for the interactions of alkali atoms but plays also a central in the nuclear interaction as a contribution arising in two-pion exchange. Because of the singularity of these interactions (for n>1) a counterterm has to be added to this interaction to obtain predictive results.  I will put emphasis on how regularization impacts the results and will consider the "leading order" renormalization with one momentum-independent counterterm and "next-to-leading" order with an additional momentum-dependent counterterm. We will show that traditional RG invariance can not be achieved with two counterterns included non-pertubatively due to finetungs in the low-energy sector that occur as a result of cutoff variation. These results are relevant to the construction of new odering scheme in the NN sector that do not employ one-pion exchange at leading order.