ar X iv : h ep - p h / 05 01 20 3 v 3 2 8 Fe b 20 05 Soft and hard processes in QCD

QCD equations for the generating functions are applied to separate soft and hard jets in e + e −-processes of multiparticle production. The dependence of average multiplicities and higher moments of multiplicity distributions of particles created in a " newly born " soft subjets on the share of energy devoted to them is calculated in fixed coupling gluody-namics. This dependence is the same as for the total multiplicity up to a constant factor if soft jets are defined as those carrying out a fixed share of initial energy at all energies. The constant factor depends on this share in a non-trivial way. Other definitions are also proposed. The relation between these quantities for soft and hard processes is discussed. In multiparticle production, it is quite common procedure to separate all processes into the soft and hard ones. Even though the intuitive approach is appealing, the criteria of the separation differ. It is shown below that QCD equations for the generating functions can be applied to this problem. It is demonstrated how the average multiplicities of soft and hard processes depend on the parameter which is used to distinguish them. The same method can be applied to any moment of the multiplicity distributions as is explicitely shown for the second moment (dispersion). The QCD equations for the generating functions (functionals) are known since long ago (e.g., see the book [1]). This is the system of two integro-differential equations which describe the quark and gluon jets evolution. They are quite useful for prediction and description of many properties of high energy jets (for the reviews see, e.g., [2, 3, 4]). It has been found that the main qualitative features of the process can be safely predicted by considering the single equation for gluon jets evolution. In that way one neglects quarks and treats the gluodynamics in place of the chromodynamics. Moreover, its solution may be further simplified if one disregards the running property of the QCD coupling strength and considers it as a fixed one (see papers [5]). To avoid some technicalities, we adopt this approach in what follows and treat the multiplicity distributions of gluon jets. Both quark and gluon jets with running coupling strength will be considered in the QCD context elsewhere. When an initial gluon splits into two gluons (subjets), its energy E is additively shared among them, and the multiplicity of the whole process is a sum of …