5.4.1 Jet Overlapping

 In order to describe the non-perturbative features of hadronization processes, we usually assume two generalities ; one is the local parton-hadron duality and the other is the universality of $\alpha _s$ in low energy region. With these assumptions we can construct models for hadronizations resulting in various jet generating algorithms such as HERWIG from cluster model, JETSET or UCLA from string model, PYTHIA, PANDORA, ISAJET, SUSYGEN etc. with appropriate corrections. For these algorithms, one important point is that there exist differences between quark jet and gluon jet. These differences could be critical in analyzing many jet systems resulting from high energy, e⁺ e^- linear collider.  Another point to be considered in analyzing many jet systems is jet overlapping. For 2 jet system, there exists no problem since the configuration of 2 jets is linear. For 3 jets, the differences between quark jet and gluon jet turn out to be string effects with slight overlapping effects. However, for 4 jet system, the overlapping probability becomes significant. For example, let's estimate the probability to overlap by assuming jet shapes of cone structure with subtending solid angle $\frac{\pi}{16}$, which corresponds approximately to a cone with side angle $\frac{\pi}{6}$. For a fixed cone, the total solid angle for another cone to overlap becomes $\frac{9}{16}\pi$. Then the probability for 3 fixed cones to overlap with the remaining one is $\frac{27}{64}$, which is larger than $\frac{1}{3}$. This probability becomes $\frac{9}{16}$ for 5 jet system and increases to $\frac{45}{64}$ for 6 jets. Since $t\bar{t}$ processes in linear collider correspond to 6 jet system, it becomes problematic to analyze $t\bar{t}$ by just counting particle trajectories. The situation becomes worse if we want to get some information about particle polarizations. Moreover, the $t\bar{t}H$ processes result in 8 jets for which the overlap probability becomes $\frac{63}{64}$. In our simple estimation, jets cannot be separately observed in 8 jet system so that it is meaningless to follow particle trajectories to define jets.  Since we have to be prepared to account for many jets such as 10 or 20 jets in high energy linear collider, we need to construct new theoretical models which can be used to analyze jet overlapping. One possibility is the momentum space flux-tube model which will be explained in the following sections. We can classify flux-tubes and construct topological spaces and then it is possible to define probability amplitude to have quark pairs which can be used to predict particle multiplicities in jets. In this way, we can account for string effects in 3-jets by considering appropriate differences between quark and gluon jets.