Renormalization of Singular Potentials and Power Counting

Understand

We use a toy model to illustrate how to build effective theories for singular potentials.

• We consider a central attractive 1/r^2 potential perturbed by a 1/r^4 correction.
• The power-counting rule, an important ingredient of effective theory, is established by seeking the minimum set of short-range counterterms that renormalize the scattering amplitude.
• We show that leading-order counterterms are needed in all partial waves where the potential overcomes the centrifugal barrier, and that the additional counterterms at next-to-leading order are the ones expected on the basis of dimensional analysis.