Small mass systems

A hadronic event is conventionally subdivided into sets of partons that form separate colour singlets. These sets are represented by strings, that e.g. stretch from a quark end via a number of intermediate gluons to an antiquark end. Three string mass regions may be distinguished for the hadronization.

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Normal string fragmentation. In the ideal situation, each string has a large invariant mass. Then the standard iterative fragmentation scheme above works well. In practice, this approach can be used for all strings above some cut-off mass of a few GeV.

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Cluster decay. If a string is produced with a small invariant mass, maybe only two-body final states are kinematically accessible. The traditional iterative Lund scheme is then not applicable. We call such a low-mass string a cluster, and consider it separately from above. The modelling is still intended to give a smooth match on to the standard string scheme in the high-cluster-mass limit [Nor98].

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Cluster collapse. This is the extreme case of the above situation, where the string mass is so small that the cluster cannot decay into two hadrons. It is then assumed to collapse directly into a single hadron, which inherits the flavour content of the string endpoints. The original continuum of string/cluster masses is replaced by a discrete set of hadron masses. Energy and momentum then cannot be conserved inside the cluster, but must be exchanged with the rest of the event [Nor98].

String systems below a threshold mass are handled by the cluster machinery. In it, an attempt is first made to produce two hadrons, by having the string break in the middle by the production of a new $\mathrm{q}\overline{\mathrm{q}}$ pair, with flavours and hadron spins selected according to the normal string rules. If the sum of the hadron masses is larger than the cluster mass, repeated attempts can be made to find allowed hadrons; the default is two tries. If an allowed set is found, the angular distribution of the decay products in the cluster rest framed is picked isotropically near the threshold, but then gradually more elongated along the string direction, to provide a smooth match to the string description at larger masses. This also includes a forward-backward asymmetry, so that each hadron is preferentially in the same hemisphere as the respective original quark it inherits.

If the attempts to find two hadrons fail, one single hadron is formed from the given flavour content. The basic strategy thereafter is to exchange some minimal amount of energy and momentum between the collapsing cluster and other string pieces in the neighbourhood. The momentum transfer can be in either direction, depending on whether the hadron is lighter or heavier than the cluster it comes from. When lighter, the excess momentum is split off and put as an extra `gluon' on the nearest string piece, where `nearest' is defined by a space-time history-based distance measure. When the hadron is heavier, momentum is instead borrowed from the endpoints of the nearest string piece.

The free parameters of the model can be tuned to data, especially to the significant asymmetries observed between the production of $\mathrm{D}$ and $\overline{\mathrm{D}}$ mesons in $\pi^- \mathrm{p}$ collisions, with hadrons that share some of the $\pi^-$ flavour content very much favoured at large $x_F$ in the $\pi^-$ fragmentation region [Ada93]. These spectra and asymmetries are closely related to the cluster collapse mechanism, and also to other effects of the colour topology of the event (`beam drag') [Nor98]. The most direct parameters are the choice of compensation scheme (MSTJ(16)), the number of attempts to find a kinematically valid two-body decay (MSTJ(16)) and the border between cluster and string descriptions (PARJ(32)). Also many other parameters enter the description, however, such as the effective charm mass (PMAS(4,1)), the quark constituent masses (PARF(101) - PARF(105)), the beam remnant structure (MSTP(91) - MSTP(94) and PARP(91) - PARP(100)) and the standard string fragmentation parameters.

The cluster collapse is supposed to be a part of multiparticle production. It is not intended for exclusive production channels, and may there give quite misleading results. For instance, a $\c\overline{\mathrm{c}}$ quark pair produced in a $\gamma\gamma$ collision could well be collapsed to a single $\mathrm{J}/\psi$ if the invariant mass is small enough, even though the process $\gamma\gamma \to \mathrm{J}/\psi$ in theory is forbidden by spin-parity-charge considerations. Furthermore, properties such as strong isospin are not considered in the string fragmentation picture (only its third component, i.e. flavour conservation), neither when one nor when many particles are produced. For multiparticle states this should matter little, since the isospin then will be duly randomized, but properly it would forbid the production of several one- or two-body states that currently are generated.