Starting from the hard interaction, initial- and final-state radiation corrections may be added. This is normally done by making use of the parton-shower language -- only for the process does PYTHIA offer a matrix-element option (described in section ). The algorithms used to generate initial- and final-state showers are rather different, and are therefore described separately below, starting with the conceptually easier final-state one. Before that, some common elements are introduced.
As a further doubling-up, recently new transverse-momentum-ordered showers were introduced as an alternative to the older virtuality-ordered ones. The -ordering offers several advantages, on its own and especially in combination with the new, more sophisticated multiple interactions scenarios described in section . In the long run, the new algorithms may be the only ones to survive, but they are not yet sufficiently well established that the older can be removed; in addition, comparisons between different orderings are helpful for a better understanding of the powers and limitations of the shower approach [Ple05,Ska05]. While the newer routines are quite different in many respects, they still share a lot of the philosophy of the older ones. Therefore it is feasible to give a reasonably detailed presentation of the old formalisms and only provide a brief summary of the main differences introduced by the -ordering.
The main references for virtuality-ordered final-state showers are [Ben87a,Nor01] and for ditto initial-state ones [Sjö85,Miu99], while the transverse-momentum-ordered showers of both kinds are described in [Sjö04a].