Matrix elements

Matrix elements are especially made use of in the older JETSET-originated implementation of the process $\mathrm{e}^+\mathrm{e}^-\to \gamma^* / \mathrm{Z}^0\to \mathrm{q}\overline{\mathrm{q}}$.

For initial-state QED radiation, a first order (un-exponentiated) description has been adopted. This means that events are subdivided into two classes, those where a photon is radiated above some minimum energy, and those without such a photon. In the latter class, the soft and virtual corrections have been lumped together to give a total event rate that is correct up to one loop. This approach worked fine at PETRA/PEP energies, but does not do so well for the $\mathrm{Z}^0$ line shape, i.e. in regions where the cross section is rapidly varying and high precision is strived for.

For final-state QCD radiation, several options are available. The default is the parton-shower one (see below), but the matrix-elements options are also frequently used. In the definition of 3- or 4-jet events, a cut is introduced whereby it is required that any two partons have an invariant mass bigger than some fraction of the c.m. energy. 3-jet events which do not fulfil this requirement are lumped with the 2-jet ones. The first-order matrix-element option, which only contains 3- and 2-jet events therefore involves no ambiguities. In second order, where also 4-jets have to be considered, a main issue is what to do with 4-jet events that fail the cuts. Depending on the choice of recombination scheme, whereby the two nearby partons are joined into one, different 3-jet events are produced. Therefore the second-order differential 3-jet rate has been the subject of some controversy, and the program actually contains two different implementations.

By contrast, the normal PYTHIA event generation machinery does not contain any full higher-order matrix elements, with loop contributions included. There are several cases where higher-order matrix elements are included at the Born level. Consider the case of resonance production at a hadron collider, e.g. of a $\mathrm{W}$, which is contained in the lowest-order process $\mathrm{q}\overline{\mathrm{q}}' \to \mathrm{W}$. In an inclusive description, additional jets recoiling against the $\mathrm{W}$ may be generated by parton showers. PYTHIA also contains the two first-order processes $\mathrm{q}\mathrm{g}\to \mathrm{W}\mathrm{q}'$ and $\mathrm{q}\overline{\mathrm{q}}' \to \mathrm{W}\mathrm{g}$. The cross sections for these processes are divergent when the $p_{\perp}\to 0$. In this region a correct treatment would therefore have to take into account loop corrections, which are not available in PYTHIA.

Even without having these accessible, we know approximately what the outcome should be. The virtual corrections have to cancel the $p_{\perp}\to 0$ singularities of the real emission. The total cross section of $\mathrm{W}$ production therefore receives finite $\mathcal{O}(\alpha_{\mathrm{s}})$ corrections to the lowest-order answer. These corrections can often be neglected to first approximation, except when high precision is required. As for the shape of the $\mathrm{W}$ $p_{\perp}$ spectrum, the large cross section for low-$p_{\perp}$ emission has to be interpreted as allowing more than one emission to take place. A resummation procedure is therefore necessary to have matrix element make sense at small $p_{\perp}$. The outcome is a cross section below the naive one, with a finite behaviour in the $p_{\perp}\to 0$ limit.

Depending on the physics application, one could then use PYTHIA in one of two ways. In an inclusive description, which is dominated by the region of reasonably small $p_{\perp}$, the preferred option is lowest-order matrix elements combined with parton showers, which actually is one way of achieving the required resummation. For $\mathrm{W}$ production as background to some other process, say, only the large-$p_{\perp}$ tail might be of interest. Then the shower approach may be inefficient, since only few events will end up in the interesting region, while the matrix-element alternative allows reasonable cuts to be inserted from the beginning of the generation procedure. (One would probably still want to add showers to describe additional softer radiation, at the cost of some smearing of the original cuts.) Furthermore, and not less importantly, the matrix elements should give a more precise prediction of the high-$p_{\perp}$ event rate than the approximate shower procedure.

In the particular case considered here, that of $\mathrm{W}$ production, and a few similar processes, actually the shower has been improved by a matching to first-order matrix elements, thus giving a decent description over the whole $p_{\perp}$ range. This does not provide the first-order corrections to the total $\mathrm{W}$ production rate, however, nor the possibility to select only a high-$p_{\perp}$ tail of events.