In an event with several interactions, it is convenient to impose an ordering. The logical choice is to arrange the scatterings in falling sequence of . The `first' scattering is thus the hardest one, with the `subsequent' (`second', `third', etc.) successively softer. It is important to remember that this terminology is in no way related to any picture in physical time; we do not know anything about the latter. In principle, all the scatterings that occur in an event must be correlated somehow, naïvely by momentum and flavour conservation for the partons from each incoming hadron, less naïvely by various quantum mechanical effects. When averaging over all configurations of soft partons, however, one should effectively obtain the standard QCD phenomenology for a hard scattering, e.g. in terms of parton distributions. Correlation effects, known or estimated, can be introduced in the choice of subsequent scatterings, given that the `preceding' (harder) ones are already known.
With a total cross section of hard interactions
to be distributed among
(non-diffractive, inelastic) events, the average
number of interactions per event is just the ratio
As a starting point we will assume that all hadron collisions are
equivalent (no impact parameter dependence), and that the different
parton-parton interactions take place completely independently of
each other. The number of scatterings per event is then distributed
according to a Poissonian with mean . A fit to S
collider multiplicity data [UA584] gave
which corresponds to
. For Monte Carlo
generation of these interactions it is useful to define
The probability that the hardest interaction, i.e. the one with
highest , is at , is now given by
With the help of the integral
Whereas the ordinary parton distributions should be used for the
hardest scattering, in order to reproduce standard QCD
phenomenology, the parton distributions to be used for subsequent
scatterings must depend on all preceding values and flavours
chosen. We do not know enough about the hadron wave function to
write down such joint probability distributions. To take
into account the energy `already' used in harder scatterings, a
conservative approach is to evaluate the parton distributions, not at
for the :th scattered parton from hadron, but at
the rescaled value
In a fraction of the events studied, there will be no hard scattering above when the iterative procedure in eq. () is applied. It is therefore also necessary to have a model for what happens in events with no (semi)hard interactions. The simplest possible way to produce an event is to have an exchange of a very soft gluon between the two colliding hadrons. Without (initially) affecting the momentum distribution of partons, the `hadrons' become colour octet objects rather than colour singlet ones. If only valence quarks are considered, the colour octet state of a baryon can be decomposed into a colour triplet quark and an antitriplet diquark. In a baryon-baryon collision, one would then obtain a two-string picture, with each string stretched from the quark of one baryon to the diquark of the other. A baryon-antibaryon collision would give one string between a quark and an antiquark and another one between a diquark and an antidiquark.
In a hard interaction, the number of possible string drawings are many more, and the overall situation can become quite complex when several hard scatterings are present in an event. Specifically, the string drawing now depends on the relative colour arrangement, in each hadron individually, of the partons that are about to scatter. This is a subject about which nothing is known. To make matters worse, the standard string fragmentation description would have to be extended, to handle events where two or more valence quarks have been kicked out of an incoming hadron by separate interactions. In particular, the position of the baryon number would be unclear. We therefore here assume that, following the hardest interaction, all subsequent interactions belong to one of three classes.
Since a or scattering need not remain of this character if initial- and final-state showers were to be included (e.g. it could turn into a -initiated process), radiation is only included for the hardest interaction. In practice, this is not a serious problem: except for the hardest interaction, which can be hard because of experimental trigger conditions, it is unlikely for a parton scattering to be so hard that radiation plays a significant rôle.
In events with multiple interactions, the beam remnant treatment is slightly modified. First the hard scattering is generated, with its associated initial- and final-state radiation, and next any additional multiple interactions. Only thereafter are beam remnants attached to the initiator partons of the hardest scattering, using the same machinery as before, except that the energy and momentum already taken away from the beam remnants also include that of the subsequent interactions.