Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics. II: Rotational Invariance

TL;DR

This paper investigates how dimensional regularization affects rotationally invariant, scale-invariant potentials in quantum mechanics, revealing symmetry breaking, bound states, and scattering behavior in critical coupling regimes.

Contribution

It provides a detailed analysis of rotationally invariant potentials using dimensional regularization, highlighting symmetry breaking and bound state formation in quantum mechanics.

Findings

Existence of a critical coupling for symmetry breaking

Appearance of a unique bound state in strong coupling

Logarithmic energy dependence in scattering processes

Abstract

A thorough analysis is presented of the class of central fields of force that exhibit: (i) dimensional transmutation and (ii) rotational invariance. Using dimensional regularization, the two-dimensional delta-function potential and the $D$-dimensional inverse square potential are studied. In particular, the following features are analyzed: the existence of a critical coupling, the boundary condition at the origin, the relationship between the bound-state and scattering sectors, and the similarities displayed by both potentials. It is found that, for rotationally symmetric scale-invariant potentials, there is a strong-coupling regime, for which quantum-mechanical breaking of symmetry takes place, with the appearance of a unique bound state as well as of a logarithmic energy dependence of the scattering with respect to the energy.