Renormalization of Singular Potentials and Power Counting

TL;DR

This paper demonstrates how to develop effective theories for singular potentials using a toy model, establishing power-counting rules and identifying necessary counterterms for renormalization across different partial waves.

Contribution

It introduces a systematic approach to determine the minimal set of counterterms for renormalizing singular potentials in effective theories.

Findings

Leading-order counterterms are needed where the potential dominates over the centrifugal barrier.

Next-to-leading order counterterms align with dimensional analysis expectations.

A clear power-counting rule is established for singular potentials.

Abstract

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.