# Renormalization group procedure for potential $-g/r^2$

## TL;DR

This paper introduces a renormalization group approach using Gaussian elimination to analyze the Schrödinger equation with a $-g/r^2$ potential, revealing complex behaviors including limit cycles, asymptotic freedom, triviality, and multiple fixed points.

## Contribution

It develops a novel RG procedure that directly yields equations for renormalized Hamiltonians, uncovering richer structures than previously known in the $-g/r^2$ potential problem.

## Key findings

- Identification of limit-cycle, asymptotic freedom, triviality, and fixed points.
- Discovery of multiple pairs of fixed points across different partial waves.
- Enhanced understanding of the potential's behavior at different scales.

## Abstract

Schr\"odinger equation with potential $-g/r^2$ exhibits a limit cycle, described in the literature in a broad range of contexts using various regularizations of the singularity at $r=0$. Instead, we use the renormalization group transformation based on Gaussian elimination, from the Hamiltonian eigenvalue problem, of high momentum modes above a finite, floating cutoff scale. The procedure identifies a richer structure than the one we found in the literature. Namely, it directly yields an equation that determines the renormalized Hamiltonians as functions of the floating cutoff: solutions to this equation exhibit, in addition to the limit-cycle, also the asymptotic-freedom, triviality, and fixed-point behaviors, the latter in vicinity of infinitely many separate pairs of fixed points in different partial waves for different values of $g$.