Renormalization of the Inverse Square Potential

TL;DR

This paper applies field-theoretic renormalization to the quantum inverse square potential, revealing a strong-coupling regime with broken scale symmetry, a bound state, and an energy-dependent scattering matrix.

Contribution

It introduces a renormalized framework for analyzing the inverse square potential, including solutions for bound states and scattering using cutoff and dimensional regularization.

Findings

Existence of a strong-coupling regime with broken scale symmetry

Creation of a single bound state through dimensional transmutation

Energy-dependent s-wave scattering matrix element

Abstract

The quantum-mechanical D-dimensional inverse square potential is analyzed using field-theoretic renormalization techniques. A solution is presented for both the bound-state and scattering sectors of the theory using cutoff and dimensional regularization. In the renormalized version of the theory, there is a strong-coupling regime where quantum-mechanical breaking of scale symmetry takes place through dimensional transmutation, with the creation of a single bound state and of an energy-dependent s-wave scattering matrix element.