Renormalization Group Flow of the Aharonov-Bohm Scattering Amplitude

TL;DR

This paper investigates the renormalization group flow of the Aharonov-Bohm scattering amplitude, revealing symmetry breaking, new length scales, and phase transitions in the presence of quantized magnetic flux.

Contribution

It introduces a renormalization procedure for the scattering amplitude, connecting cylindrical Schrödinger and Calogero problems, and analyzes the resulting phase transitions.

Findings

Renormalization introduces new length scales and breaks conformal symmetry.

The renormalized cross-section remains non-zero for quantized flux, similar to a delta potential.

A BKT-like phase transition occurs at specific coupling values.

Abstract

The Aharonov-Bohm elastic scattering with incident particles described by plane waves is revisited by using the phase-shifts method. The formal equivalence between the cylindrical Schr\"odinger equation and the one-dimensional Calogero problem allows us to show that up to two scattering phase-shifts modes in the cylindrical waves expansion must be renormalized. The renormalization procedure introduces new length scales giving rise to spontaneous breaking of the conformal symmetry. The new renormalized cross-section has an amazing property of being non-vanishing even for a quantized magnetic flux, coinciding with the case of Dirac delta function potential. The knowledge of the exact beta function permits us to describe the renormalization group flows within the two-parametric family of renormalized Aharonov-Bohm scattering amplitudes. Our analysis demonstrates that for quantized magnetic fluxes a BKT-like phase transition at the coupling space occurs.