Folding in the Skyrme Model

TL;DR

This paper investigates the singularity structures of Skyrme configurations, introduces a non-holomorphic ansatz to improve approximations, and analyzes baryon density features for various baryon numbers.

Contribution

It presents a new non-holomorphic ansatz for Skyrme configurations, improving approximation accuracy and revealing detailed singularity and baryon density structures.

Findings

Non-holomorphic ansatz yields closer energy solutions for baryon numbers 2, 3, 4.

Tiny negative baryon density observed for baryon number 3.

Singularity structures include folds, cusps, and swallowtails, with specific features for different baryon numbers.

Abstract

There are only three stable singularities of a differentiable map between three-dimensional manifolds, namely folds, cusps and swallowtails. A Skyrme configuration is a map from space to SU(2), and its singularities correspond to the points where the baryon density vanishes. In this paper we consider the singularity structure of Skyrme configurations. The Skyrme model can only be solved numerically. However, there are good analytic ansaetze. The simplest of these, the rational map ansatz, has a non-generic singularity structure. This leads us to introduce a non-holomorphic ansatz as a generalization. For baryon number two, three and four, the approximate solutions derived from this ansatz are closer in energy to the true solutions than any other ansatz solution. We find that there is a tiny amount of negative baryon density for baryon number three, but none for two or four. We comment briefly on the relationship to Bogomolny-Prasad-Sommerfield monopoles.