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Colour Topologies

The second part of the model is to hook up the scattered partons to the beam remnants, in colours, in transverse momenta and in longitudinal momenta. The order in which these choices are made is partly intertwined, and one has to consider several special cases. In this description we only give an outline, evading many of the details. Especially the assignment of colours is physically uncertain and technically challenging.

Each of the two incoming beam hadrons can be viewed -- post facto -- as having consisted of a set of `initiator' and `remnant' partons. An initiator is the `original' parton that starts the initial-state cascade that leads up to one of the hard interactions. Together the initiators and remnants make up the original proton, so together they carry the proton net flavour content, and energy and momentum, and are in a colour singlet state. Therefore the remnant partons are constrained to carry `what is left' when the initiators are removed. The one exception is that we do not attempt to conserve longitudinal momentum (and thereby energy) exactly within each incoming proton, but only for the system as a whole. The reason here is that we have put all initiators and remnants on mass shell, whereas a more proper treatment ought to have included (moderate) space-like virtualities for them. Such virtualities would have dissipated in the hard interactions, and so not survived to the final state. The current choice therefore shortcuts a number of technical details that in the end should not matter.

In an event with $n$ multiple interactions, a corresponding set of $n$ initiator partons is defined for each of the two incoming beams. This also defines the left-behind flavour content of the remnants. At most there could be $n+3$ remnant partons for a proton with its 3 valence flavours, but there could also be none, if all valence quarks have been kicked out and no unmatched companions have been added. In the latter case a gluon is inserted to represent the beam remnant and carry leftover momentum, but else we do not introduce gluons in the beam remnant.

In the string model, the simplest representation of a baryon is the Y-shape topology, where each of the three valence quarks is connected via its string piece to the central junction (see section [*]). This junction does not carry energy or momentum of its own, but topologically it is the carrier of the baryon number. Under normal circumstances, the legs in the Y are quite short. Interactions may deform the topology, however. The simplest example would be Deeply Inelastic Scattering, where one of the valence quarks is kicked violently, so that one of the three strings of the Y will stretch out and fragment into hadrons. In this topology the other two will remain so close to each other and to the junction that they effectively act as a single unit, a diquark.

In a hadronic interaction, a valence quark is kicked out by a coloured gluon rather than a colourless photon. This colour exchange implies that the string from the junction will no longer attach to the scattered quark but rather to some other quark or gluon, but the other two quarks still effectively form a diquark. When two or three valence quarks are kicked out, the junction motion will become more complicated, and must be considered in full. Equivalently, when all of the valence quarks have large relative momenta, by definition no two of them have a small invariant mass and hence a diquark description would be inappropriate.

Multiple valence quark interactions are rare, however. The bulk of interactions are gluonic. In this case we can imagine that the gluon originates from an unresolved emission off one of the valence quarks, i.e. that its anticolour is attached to one of the quarks and its colour attached to the junction. After the gluon is kicked out, colour lines will then connect this quark and the junction to partons in the final state of the interaction. If a second gluon is kicked out, it can, in colour space, have been located either between the quark and the first gluon, or between this gluon and the junction, or between the junction and one of the other two quarks. (We do not include the possibility that this gluon together with the first one could have been radiated from the system as an overall colour singlet system, i.e. we do not (yet) address diffractive event topologies in this framework.) If all of these possibilities had equal probability, the junction would often have two or all three legs reconnected, and the baryon number could be moved quite dramatically in the event. Empirically, this does not appear to happen (?), and furthermore it could be argued that perturbative and impact-parameter arguments both allow much of the activity to be correlated in `hot spot' regions that leave much of the rest of the proton unaffected. Therefore a free suppression parameter is introduced, such that further gluons preferably connect to a string piece that has already been disturbed. In this way, gluons preferentially will be found on one of the three colour lines to the junction.

The order in which they appear along that line is more difficult to make statements about. So far in our description, no consideration has been given to the resulting momentum picture, specified after transverse and longitudinal momenta have been picked (see below). This in general implies that strings will be stretched criss-cross in the event, if the initiator gluons are just randomly ordered along the string piece on which they sit. It is unlikely that such a scenario catches all the relevant physics. More likely is that, among all the possible colour topologies for the final state, those that correspond to the smaller total string length are favoured, all other aspects being the same. Several options have been introduced that approach this issue from different directions (see MSTP(89) and MSTP(95)). One is to attach the initiator gluons preferentially in those places that order the hard-scattering systems in rapidity (the default), another to prefer the attachments that will give rise to the smaller string lengths in the final state. A third option is to rearrange the colour flow between the final-state partons themselves, again giving preference to those rearrangements which minimize the overall string length. So far no preferred scenario has been identified.

A complication is the following. Two $\mathrm{g}+ \mathrm{g}\to \mathrm{g}+ \mathrm{g}$ scatterings each on their own may have a perfectly sensible colour flow. Still, when the two initial gluons on each side of the event are attached to each other and to the rest of the remnants, the resulting colour flow may become unphysical. Specifically the colour flow may `loop back' on itself, such that a single gluon comes to form a separate colour singlet system. Such configurations are rejected and alternative colour arrangements are tried.

Another, rare, occurence is that the two junctions of the event can come to be connected to each other via two strings (graphical representation: -j=j-, where each dash corresponds to a string piece). Since we have not (yet) programmed a fragmentation scheme for such events, we simply reject them and generate a new event instead.

So far, interactions with sea quarks have not been mentioned, either a quark-antiquark pair that both scatter or a single sea scatterer with a leftover companion quark in the remnant. However, we have already argued that each such pair can be viewed as coming from the branching of some initial nonperturbative gluon. This gluon can now be attached to the beam Y topology just like the gluons considered above, and therefore does not introduce any new degrees of freedom. When then the colours are traced, it could well happen that a companion quark together with two remaining valence quarks form a separate colour singlet system. It is then likely that this system will be of low mass and collapse to a single baryon. Such possibilities are optionally allowed (see MSTP(88)), and correspondingly a companion antiquark could form a meson together with a single valence quark. As already mentioned, diquarks can be formed from two valence quarks.

next up previous contents
Next: Primordial Up: Beam Remnants (and Multiple Previous: Flavour and Correlations   Contents
Stephen Mrenna 2007-10-30