The Complexity of High-Energy Processes

To first approximation, all processes have a simple structure at the level of interactions between the fundamental objects of nature, i.e. quarks, leptons and gauge bosons. For instance, a lot can be understood about the structure of hadronic events at LEP just from the `skeleton' process $\mathrm{e}^+\mathrm{e}^-\to \mathrm{Z}^0 \to \mathrm{q}\overline{\mathrm{q}}$. Corrections to this picture can be subdivided, arbitrarily but conveniently, into three main classes.

Firstly, there are bremsstrahlung-type modifications, i.e. the emission of additional final-state particles by branchings such as $\mathrm{e}\to \mathrm{e}\gamma$ or $\mathrm{q}\to \mathrm{q}\mathrm{g}$. Because of the largeness of the strong coupling constant $\alpha_{\mathrm{s}}$, and because of the presence of the triple gluon vertex, QCD emission off quarks and gluons is especially prolific. We therefore speak about `parton showers', wherein a single initial parton may give rise to a whole bunch of partons in the final state. Also photon emission may give sizable effects in $\mathrm{e}^+\mathrm{e}^-$ and $\mathrm{e}\mathrm{p}$ processes. The bulk of the bremsstrahlung corrections are universal, i.e. do not depend on the details of the process studied, but only on one or a few key numbers, such as the momentum transfer scale of the process. Such universal corrections may be included to arbitrarily high orders, using a probabilistic language. Alternatively, exact calculations of bremsstrahlung corrections may be carried out order by order in perturbation theory, but rapidly the calculations then become prohibitively complicated and the answers correspondingly lengthy.

Secondly, we have `true' higher-order corrections, which involve a combination of loop graphs and the soft parts of the bremsstrahlung graphs above, a combination needed to cancel some divergences. In a complete description it is therefore not possible to consider bremsstrahlung separately, as assumed here. The necessary perturbative calculations are usually very difficult; only rarely have results been presented that include more than one non-`trivial' order, i.e. more than one loop. As above, answers are usually very lengthy, but some results are sufficiently simple to be generally known and used, such as the running of $\alpha_{\mathrm{s}}$, or the correction factor $1 + \alpha_{\mathrm{s}}/\pi + \cdots$ in the partial widths of $\mathrm{Z}^0 \to \mathrm{q}\overline{\mathrm{q}}$ decay channels. For high-precision studies it is imperative to take into account the results of loop calculations, but usually effects are minor for the qualitative aspects of high-energy processes.

Thirdly, quarks and gluons are confined. In the two points above, we have used a perturbative language to describe the short-distance interactions of quarks, leptons and gauge bosons. For leptons and colourless bosons this language is sufficient. However, for quarks and gluons it must be complemented with the structure of incoming hadrons, and a picture for the hadronization process, wherein the coloured partons are transformed into jets of colourless hadrons, photons and leptons. The hadronization can be further subdivided into fragmentation and decays, where the former describes the way the creation of new quark-antiquark pairs can break up a high-mass system into lower-mass ones, ultimately hadrons. (The word `fragmentation' is also sometimes used in a broader sense, but we will here use it with this specific meaning.) This process is still not yet understood from first principles, but has to be based on models. In one sense, hadronization effects are overwhelmingly large, since this is where the bulk of the multiplicity comes from. In another sense, the overall energy flow of a high-energy event is mainly determined by the perturbative processes, with only a minor additional smearing caused by the hadronization step. One may therefore pick different levels of ambition, but in general detailed studies require a detailed modelling of the hadronization process.

The simple structure that we started out with has now become considerably more complex -- instead of maybe two final-state partons we have a hundred final particles. The original physics is not gone, but the skeleton process has been dressed up and is no longer directly visible. A direct comparison between theory and experiment is therefore complicated at best, and impossible at worst.