Multiple Interactions -- Old Model

In this section we present the model [Sjö87a] used in older versions of PYTHIA (and still available with the switch MSTP(81)=-1) to describe the possibility that several parton pairs undergo hard interactions in a hadron-hadron collision, and thereby contribute to the overall event activity, in particular at low $p_{\perp}$. The same model can also be used to describe the VMD $\gamma\mathrm{p}$ events, where the photon interacts like a hadron. Many basic features of this model, for instance the introduction of a $p_{\perp}$ cutoff corresponding to an inverse colour screening distance, and the options for a non-trivial transverse density structure in the incoming hadrons, carry over to the new scenario. It is therefore recommended first to read this section, even if the objective should be to learn about the new scenario.

It should from the onset be made clear that this is not an easy topic. In fact, in the full event generation process, probably no other area is as poorly understood as this one. The whole concept of multiple interactions has been very controversial, with contradictory experimental conclusions [AFS87], but a CDF study [CDF97] some years ago started to bring more general acceptance.

The multiple interactions scenario presented here [Sjö87a] was the first detailed model for this kind of physics, and is still one of the very few available. We will present two related but separate scenarios, one `simple' model and one somewhat more sophisticated. In fact, neither of them are all that simple, which may make the models look unattractive. However, the world of hadron physics is complicated, and if we err, it is most likely in being too unsophisticated. The experience gained with the model(s), in failures as well as successes, could be used as a guideline in the evolution of yet more detailed models.

Our basic philosophy will be as follows. The total rate of parton-parton interactions, as a function of the transverse momentum scale $p_{\perp}$, is assumed to be given by perturbative QCD. This is certainly true for reasonably large $p_{\perp}$ values, but here we shall also extend the perturbative parton-parton scattering framework into the low-$p_{\perp}$ region. A regularization of the divergence in the cross section for $p_{\perp}\to 0$ has to be introduced, however, which will provide us with the main free parameter of the model. Since each incoming hadron is a composite object, consisting of many partons, there should exist the possibility of several parton pairs interacting when two hadrons collide. It is not unreasonable to assume that the different pairwise interactions take place essentially independently of each other, and that therefore the number of interactions in an event is given by a Poisson distribution. This is the strategy of the `simple' scenario.

Furthermore, hadrons are not only composite but also extended objects, meaning that collisions range from very central to rather peripheral ones. Reasonably, the average number of interactions should be larger in the former than in the latter case. Whereas the assumption of a Poisson distribution should hold for each impact parameter separately, the distribution in number of interactions should be widened by the spread of impact parameters. The amount of widening depends on the assumed matter distribution inside the colliding hadrons. In the `complex' scenario, different matter distributions are therefore introduced.