The diagonalisation of the Lund fragmentation model I

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. 1 bo@thep.lu.se 2 fredrik@thep.lu.se

The Lund model at nonzero impact parameter

“…2(b) we depict the same process. 5 We calculate first the area of the sector defined by the breaking up points (σ i , τ i ) and (σ f , τ f ). The coordinates of the tip of the sector are:…”

Transverse momentum correlations of quarks in recursive jet models

“…To keep the two possibilities open, we introduced the parameter c in Eqs. (12)(13). With c = 0 the rectangles are included, with c = 1 they are excluded.…”

The Lund fragmentation process for a multi-gluon string according to the area law

“…The model is based upon a few general assumptions: (i) the final state particles stem from the breakup of a string-like force field spanned between the coloured constituents, (ii) there is causality and Lorentz invariance and (iii) the production of the particles can be described in terms of a stochastic process which obeys a saturation assumption. We have, in a recent paper [5], re-derived the major result for the (1+1)-dimensional model, which is applicable for events with a quark (q, a colour-3) and an antiquark (q, a colour-3) at the endpoints of the string but with no interior gluonic (g, colour-8) excitations. The result is that the (non-normalised) probability for the production of an n-particle final state of hadrons with energy momenta {p j } and masses {m j } is given by the Lund Area Law:…”

“…In [5] we have shown that it is possible to "diagonalise" the model, i.e. to express the result in Eq.…”

“…Another reason is to get a handle on the general structure of fragmentation, in particular to be able to treat also the multi-gluon fragmentation states by the analytical methods introduced in [5]. This is of particular interest for the transition region, i.e.…”