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Fragmentation of a jet system

In a system of many jets, each jet is fragmented independently. Since each jet by itself does not conserve the flavour, energy and momentum, as we have seen, then neither does a system of jets. At the end of the generation, special algorithms are therefore used to patch this up. The choice of approach has major consequences, e.g. for event shapes and $\alpha_{\mathrm{s}}$ determinations [Sjö84a].

Little attention is usually given to flavour conservation, and we only offer one scheme. When the fragmentation of all jets has been performed, independently of each other, the net initial flavour composition, i.e. number of $\u $ quarks minus number of $\overline{\mathrm{u}}$ quarks etc., is compared with the net final flavour composition. In case of an imbalance, the flavours of the hadron with lowest three-momentum are removed, and the imbalance is re-evaluated. If the remaining imbalance could be compensated by a suitable choice of new flavours for this hadron, flavours are so chosen, a new mass is found and the new energy can be evaluated, keeping the three-momentum of the original hadron. If the removal of flavours from the hadron with lowest momentum is not enough, flavours are removed from the one with next-lowest momentum, and so on until enough freedom is obtained, whereafter the necessary flavours are recombined at random to form the new hadrons. Occasionally one extra $\mathrm{q}_i \overline{\mathrm{q}}_i$ pair must be created, which is then done according to the customary probabilities.

Several different schemes for energy and momentum conservation have been devised. One [Hoy79] is to conserve transverse momentum locally within each jet, so that the final momentum vector of a jet is always parallel with that of the corresponding parton. Then longitudinal momenta may be rescaled separately for particles within each jet, such that the ratio of rescaled jet momentum to initial parton momentum is the same in all jets. Since the initial partons had net vanishing three-momentum, so do now the hadrons. The rescaling factors may be chosen such that also energy comes out right. Another common approach [Ali80] is to boost the event to the frame where the total hadronic momentum is vanishing. After that, energy conservation can be obtained by rescaling all particle three-momenta by a common factor.

The number of possible schemes is infinite. Two further options are available in the program. One is to shift all particle three-momenta by a common amount to give net vanishing momentum, and then rescale as before. Another is to shift all particle three-momenta, for each particle by an amount proportional to the longitudinal mass with respect to the imbalance direction, and with overall magnitude selected to give momentum conservation, and then rescale as before. In addition, there is a choice of whether to treat separate colour singlets (like $\mathrm{q}\overline{\mathrm{q}}'$ and $\mathrm{q}' \overline{\mathrm{q}}$ in a $\mathrm{q}\overline{\mathrm{q}}\mathrm{q}' \overline{\mathrm{q}}'$ event) separately or as one single big system.

A serious conceptual weakness of the IF framework is the issue of Lorentz invariance. The outcome of the fragmentation procedure depends on the coordinate frame chosen, a problem circumvented by requiring fragmentation always to be carried out in the c.m. frame. This is a consistent procedure for 2-jet events, but only a technical trick for multijets.

It should be noted, however, that a Lorentz covariant generalization of the independent fragmentation model exists, in which separate `gluon-type' and `quark-type' strings are used, the Montvay scheme [Mon79]. The `quark string' is characterized by the ordinary string constant $\kappa$, whereas a `gluon string' is taken to have a string constant $\kappa_{\mathrm{g}}$. If $\kappa_{\mathrm{g}} > 2 \kappa$ it is always energetically favourable to split a gluon string into two quark ones, and the ordinary Lund string model is recovered. Otherwise, for a 3-jet $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}$ event the three different string pieces are joined at a junction. The motion of this junction is given by the vector sum of string tensions acting on it. In particular, it is always possible to boost an event to a frame where this junction is at rest. In this frame, much of the standard naïve IF picture holds for the fragmentation of the three jets; additionally, a correct treatment would automatically give flavour, momentum and energy conservation. Unfortunately, the simplicity is lost when studying events with several gluon jets. In general, each event will contain a number of different junctions, resulting in a polypod shape with a number of quark and gluons strings sticking out from a skeleton of gluon strings. With the shift of emphasis from three-parton to multi-parton configurations, the simple option existing in JETSET 6.3 therefore is no longer included.

A second conceptual weakness of IF is the issue of collinear divergences. In a parton-shower picture, where a quark or gluon is expected to branch into several reasonably collimated partons, the independent fragmentation of one single parton or of a bunch of collinear ones gives quite different outcomes, e.g. with a much larger hadron multiplicity in the latter case. It is conceivable that a different set of fragmentation functions could be constructed in the shower case in order to circumvent this problem (local parton-hadron duality [Dok89] would correspond to having $f(z) = \delta(z-1)$).


next up previous contents
Next: Other Fragmentation Aspects Up: Independent Fragmentation Previous: Fragmentation of a single   Contents
Stephen_Mrenna 2012-10-24