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

Abstract.

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 generalized rapidity range $\lambda$, which has been used as a measure on the partonic states. We identify $\lambda$ 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 ${\mathcal{P}}$-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.