The QCD cross section for hard processes, as a function
of the scale, is given by

The hard-scattering cross section above some given
is given by

While the introduction of several interactions per event is the natural consequence of allowing small values and hence large ones, it is not the solution of being divergent for : the average of a scattering decreases slower with than the number of interactions increases, so naïvely the total amount of scattered partonic energy becomes infinite. One cut-off is therefore obtained via the need to introduce proper multi-parton correlated parton distributions inside a hadron. This is not a part of the standard perturbative QCD formalism and is therefore not built into eq. (). In practice, even correlated parton-distribution functions seems to provide too weak a cut, i.e. one is lead to a picture with too little of the incoming energy remaining in the small-angle beam-jet region [Sjö87a].

A more credible reason for an effective cut-off is that the incoming hadrons are colour neutral objects. Therefore, when the of an exchanged gluon is made small and the transverse wavelength correspondingly large, the gluon can no longer resolve the individual colour charges, and the effective coupling is decreased. This mechanism is not in contradiction with perturbative QCD calculations, which are always performed assuming scattering of free partons (rather than partons inside hadrons), but neither does present knowledge of QCD provide an understanding of how such a decoupling mechanism would work in detail. In the simple model one makes use of a sharp cut-off at some scale , while a more smooth dampening is assumed for the complex scenario.

One key question is the energy-dependence of ; this may be relevant e.g. for comparisons of jet rates at different Tevatron/RHIC energies, and even more for any extrapolation to LHC energies. The problem actually is more pressing now than at the time of the original study [Sjö87a], since nowadays parton distributions are known to be rising more steeply at small than the flat behaviour normally assumed for small before HERA. This translates into a more dramatic energy dependence of the multiple-interactions rate for a fixed .

The larger number of partons should also increase the amount of
screening, however, as confirmed by toy simulations [Dis01].
As a simple first approximation,
is assumed
to increase in the same way as the total cross section, i.e. with some
power
[Don92] that, via reggeon
phenomenology, should relate to the behaviour of parton distributions
at small and . Thus the default in PYTHIA is