In a hadron-hadron collision, the initial-state radiation algorithm reconstructs one shower initiator in each beam. This initiator only takes some fraction of the total beam energy, leaving behind a beam remnant which takes the rest. For a proton beam, a quark initiator would leave behind a diquark beam remnant, with an antitriplet colour charge. The remnant is therefore colour-connected to the hard interaction, and forms part of the same fragmenting system. It is further customary to assign a primordial transverse momentum to the shower initiator, to take into account the motion of quarks inside the original hadron, at least as required by the uncertainty principle by the proton size, probably augmented by unresolved (i.e. not simulated) soft shower activity. This primordial is selected according to some suitable distribution, and the recoil is assumed to be taken up by the beam remnant.
Often the remnant is more complicated, e.g. a gluon initiator would leave behind a proton remnant system in a colour octet state, which can conveniently be subdivided into a colour triplet quark and a colour antitriplet diquark, each of which are colour-connected to the hard interaction. The energy sharing between these two remnant objects, and their relative transverse momentum, introduces additional degrees of freedom, which are not understood from first principles.
Naïvely, one would expect an event to have only one beam remnant, and an event none. This is not always correct, e.g. a interaction in an event would leave behind the and as beam remnants, and a interaction in resolved photoproduction in an event would leave behind one and one or in each remnant. Corresponding complications occur for photoproduction in events.
There is another source of beam remnants. If parton distributions are used to resolve an electron inside an electron, some of the original energy is not used in the hard interaction, but is rather associated with initial-state photon radiation. The initial-state shower is in principle intended to trace this evolution and reconstruct the original electron before any radiation at all took place. However, because of cut-off procedures, some small amount may be left unaccounted for. Alternatively, you may have chosen to switch off initial-state radiation altogether, but still preserved the resolved electron parton distributions. In either case the remaining energy is given to a single photon of vanishing transverse momentum, which is then considered in the same spirit as `true' beam remnants.
So far we have assumed that each event only contains one hard interaction, i.e. that each incoming particle has only one parton which takes part in hard processes, and that all other constituents sail through unaffected. This is appropriate in or events, but not necessarily so in hadron-hadron collisions. Here each of the beam particles contains a multitude of partons, and so the probability for several interactions in one and the same event need not be negligible. In principle these additional interactions could arise because one single parton from one beam scatters against several different partons from the other beam, or because several partons from each beam take place in separate scatterings. Both are expected, but combinatorics should favour the latter, which is the mechanism considered in PYTHIA.
The dominant QCD cross sections are divergent for , and drop rapidly for larger . Probably the lowest-order perturbative cross sections will be regularized at small by colour coherence effects: an exchanged gluon of small has a large transverse wave function and can therefore not resolve the individual colour charges of the two incoming hadrons; it will only couple to an average colour charge that vanishes in the limit . In the program, some effective scale is therefore introduced, below which the perturbative cross section is either assumed completely vanishing or at least strongly damped. Phenomenologically, comes out to be a number of the order of 1.5-2.0 GeV, with some energy dependence.
In a typical `minimum-bias' event one therefore expects to find one or a few scatterings at scales around or a bit above , while a high- event also may have additional scatterings at the scale. The probability to have several high- scatterings in the same event is small, since the cross section drops so rapidly with .
The understanding of multiple interaction is still very primitive. PYTHIA therefore contains several different options, with a fairly advanced one as default. The options differ e.g. on the issue of the `pedestal' effect: is there an increased probability or not for additional interactions in an event which is known to contain a hard scattering, compared with one that contains no hard interactions? Other differences concern the level of detail in the generation of scatterings after the first one, and the model that describes how the scatterings are intercorrelated in colour space.