Initial- and Final-State Radiation

In every process that contains coloured and/or charged objects in the initial or final state, gluon and/or photon radiation may give large corrections to the overall topology of events. Starting from a basic $2 \to 2$ process, this kind of corrections will generate $2 \to 3$, $2 \to 4$, and so on, final-state topologies. As the available energies are increased, hard emission of this kind is increasingly important, relative to fragmentation, in determining the event structure.

Two traditional approaches exist to the modelling of perturbative corrections. One is the matrix-element method, in which Feynman diagrams are calculated, order by order. In principle, this is the correct approach, which takes into account exact kinematics, and the full interference and helicity structure. The only problem is that calculations become increasingly difficult in higher orders, in particular for the loop graphs. Only in exceptional cases have therefore more than one loop been calculated in full, and often we do not have any loop corrections at all at our disposal. On the other hand, we have indirect but strong evidence that, in fact, the emission of multiple soft gluons plays a significant rôle in building up the event structure, e.g. at LEP, and this sets a limit to the applicability of matrix elements. Since the phase space available for gluon emission increases with the available energy, the matrix-element approach becomes less relevant for the full structure of events at higher energies. However, the perturbative expansion is better behaved at higher energy scales, owing to the running of $\alpha_{\mathrm{s}}$. As a consequence, inclusive measurements, e.g. of the rate of well-separated jets, should yield more reliable results at high energies.

The second possible approach is the parton-shower one. Here an arbitrary number of branchings of one parton into two (or more) may be combined, to yield a description of multijet events, with no explicit upper limit on the number of partons involved. This is possible since the full matrix-element expressions are not used, but only approximations derived by simplifying the kinematics, and the interference and helicity structure. Parton showers are therefore expected to give a good description of the substructure of jets, but in principle the shower approach has limited predictive power for the rate of well-separated jets (i.e. the 2/3/4/5-jet composition). In practice, shower programs may be matched to first-order matrix elements to describe the hard-gluon emission region reasonably well, in particular for the $\mathrm{e}^+\mathrm{e}^-$ annihilation process. Nevertheless, the shower description is not optimal for absolute $\alpha_{\mathrm{s}}$ determinations.

Thus the two approaches are complementary in many respects, and both have found use. However, because of its simplicity and flexibility, the parton-shower option is generally the first choice, while the matrix elements one is mainly used for $\alpha_{\mathrm{s}}$ determinations, angular distribution of jets, triple-gluon vertex studies, and other specialized studies. Obviously, the ultimate goal would be to have an approach where the best aspects of the two worlds are harmoniously married. This is currently a topic of quite some study.