The Lund Area Law describes the probability for the production of a set of colourless hadrons from an initial set of partons, in the Lund string fragmentation model. It was derived from classical probability concepts but has later been interpreted as the result of gauge invariance in terms of the Wilson gauge loop integrals. In this paper we will present a general method to implement the Area Law for a multi-gluon string state. In this case the world surface of the massless relativistic string is a geometrically bent (1 + 1)-dimensional surface embedded in the (1 + 3)-dimensional Minkowski space. The partonic states are in general given by a perturbative QCD cascade and are consequently defined only down to a cutoff in the energy momentum fluctuations. We will show that our method defines the states down to the hadronic mass scale inside an analytically calculable scenario.We will then show that there is a differential version of our process which is closely related to the generalised rapidity range λ, which has been used as a measure on the partonic states. We identify λ as the area spanned between the directrix curve (the curve given by the parton energy momentum vectors laid out in colour order, which determines the string surface) and the average curve (to be called the P-curve) of the stochastic X-curves (curves obtained when the hadronic energy-momentum vectors are laid out in rank order). Finally we show that from the X-curve corresponding to a particular stochastic fragmentation situation it is possible to reproduce the directrix curve (up to one starting vector and a set of sign choices, one for each hadron). This relationship provides an analytical formulation of the notion of Parton-Hadron Duality. The whole effort is made in order to get a new handle to treat the transition region between where we expect perturbative QCD to work and where the hadronic features become noticeable.

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

A brief introduction to the String Fragmentation Model and its consequences is presented. We discuss the fragmentation of a general multi-gluon string and show that it can be formulated as the production of a set of "plaquettes" between the hadronic curve and the directrix. We also discuss certain interesting scaling properties of the partonic states obtained from standard parton shower algorithms like those in Pythia and Ariadne, which are communicated to the final state hadrons obtained through our hadronisation procedure.

We demonstrate that the multiple gluon emission phase space in the dipole cascade model has a strong linear correlation with the number of gluons emitted. The number of gluons per unit available phase space at a certain resolution scale is found to be remarkably independent of the cms energy and global event properties like thrust, and even changes in the ordering variable or resolution scale. We show that the distribution of sizes of gluon-gluon dipoles in a parton cascade has stability properties which are sufficient to account for those of the phase space variable. We observe that certain more abstract entities, defined in the context of hadronisation and related to the gluon emission phase space, share those properties of colour dipoles and name them Generalised Dipoles. We also present an analytical model to qualitatively describe our findings. 1 sandipan@thep.lu.se 2 fredrik@thep.lu.se