The Diagonalisation of the Lund Fragmentation Model I

TL;DR

This paper demonstrates that the Lund Fragmentation Model can be diagonalized by expressing its core law as a product of transition operators with a discrete spectrum, enabling analytical expressions for observables.

Contribution

It introduces a method to diagonalize the Lund Fragmentation Model using harmonic oscillator functions, providing a new analytical framework for the model's observables.

Findings

Transition operators have a discrete spectrum of eigenfunctions and eigenvalues.

Eigenfunctions are harmonic oscillator functions extended analytically.

All model observables can be expressed through analytical formulas.

Abstract

We will in this note show that it is possible to diagonalise the Lund Fragmentation Model. We show that the basic original result, the Lund Area law, can be factorised into a product of transition operators, each describing the production of a single particle and the two adjacent breakup points (vertex positions) of the string field. The transition operator has a discrete spectrum of (orthonormal) eigenfunctions, describing the vertex positions (which in a dual way corresponds to the momentum transfers between the produced particles) and discrete eigenvalues, which only depend upon the particle produced. The eigenfunctions turn out to be the well-known two- dimensional harmonic oscillator functions and the eigenvalues are the analytic continuations of these functions to time-like values (corresponding to the particle mass). In this way all observables in the model can be expressed in terms of analytical formulas. In this note only the 1+1-dimensional version of the model is treated but we end with remarks on the extensions to gluonic radiation, transverse momentum generation etc, to be performed in future papers.