Soliton quantization and internal symmetry

TL;DR

This paper applies collective coordinate quantization to a 2D soliton model with a global U(1) symmetry, analyzing charged excitations, their interactions with mesons, and deriving decay properties relevant to hadronic resonances.

Contribution

It introduces a method to sum all tree-level diagrams for meson interactions with quantized solitons, providing a new way to compute decay amplitudes and widths.

Findings

Identifies a pole in the one-point meson function indicating a physical state.

Derives an effective Yukawa coupling for charged solitons and mesons.

Calculates the semi-classical decay width of excited soliton states.

Abstract

We apply the method of collective coordinate quantization to a model of solitons in two spacetime dimensions with a global $U(1)$ symmetry. In particular we consider the dynamics of the charged states associated with rotational excitations of the soliton in the internal space and their interactions with the quanta of the background field (mesons). By solving a system of coupled saddle-point equations we effectively sum all tree-graphs contributing to the one-point Green's function of the meson field in the background of a rotating soliton. We find that the resulting one-point function evaluated between soliton states of definite $U(1)$ charge exhibits a pole on the meson mass shell and we extract the corresponding S-matrix element for the decay of an excited state via the emission of a single meson using the standard LSZ reduction formula. This S-matrix element has a natural interpretation in terms of an effective Lagrangian for the charged soliton states with an explicit Yukawa coupling to the meson field. We calculate the leading-order semi-classical decay width of the excited soliton states discuss the consequences of these results for the hadronic decay of the $\Delta$ resonance in the Skyrme model.