Fermion scattering on topological solitons in the nonlinear $O(3)$ $\sigma$-model

TL;DR

This paper investigates how Dirac fermions scatter off topological solitons in a (2+1)-dimensional nonlinear $O(3)$ sigma-model, providing analytical formulas and numerical analysis of scattering properties and symmetries.

Contribution

It offers new analytical expressions for fermion scattering and explores the symmetry and asymptotic behavior of scattering amplitudes in the nonlinear sigma-model.

Findings

Analytical formulas for scattering amplitudes and cross-sections.

Symmetry properties of the $S$-matrix elements.

Numerical analysis of $S$-matrix element dependence on fermion momentum.

Abstract

The scattering of Dirac fermions in the background fields of topological solitons of the $(2+1)$-dimensional nonlinear $O(3)$ $\sigma$-model is studied using both analytical and numerical methods. General formulae describing fermion scattering are obtained and the symmetry properties of the partial scattering amplitudes and elements of the $S$-matrix are determined. Within the framework of the Born approximation, the scattering amplitudes, differential cross-sections, and total cross-sections of fermion-soliton scattering are obtained in analytical forms, and their symmetry properties and asymptotic behavior are investigated. The dependences of the first several partial elements of the $S$-matrix on the momentum of the fermion are obtained using numerical methods, and some properties of these dependences are ascertained and discussed.