Matching to the hard scattering

The evolution in $Q^2$ is begun from some maximum scale $Q_{\mathrm{max}}^2$ for final-state parton showers, and is terminated at (a possibly different) $Q_{\mathrm{max}}^2$ for initial-state showers. In general $Q_{\mathrm{max}}^2$ is not known. Indeed, since the parton-shower language does not guarantee agreement with higher-order matrix-element results, neither in absolute shape nor normalization, there is no unique prescription for a `best' choice. Generically $Q_{\mathrm{max}}$ should be of the order of the hard-scattering scale, i.e. the largest virtuality should be associated with the hard scattering, and initial- and final-state parton showers should only involve virtualities smaller than that. This may be viewed just as a matter of sound book-keeping: in a $2 \to n$ graph, a $2 \to 2$ hard-scattering subgraph could be chosen several different ways, but if all the possibilities were to be generated then the cross section would be double-counted. Therefore one should define the $2 \to 2$ `hard' piece of a $2 \to n$ graph as the one that involves the largest virtuality.

Of course, the issue of double-counting depends a bit on what processes are actually generated in the program. If one considers a $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}$ final state in hadron colliders, this could come either as final-state radiation off a $\mathrm{q}\overline{\mathrm{q}}$ pair, or by a gluon splitting in a $\mathrm{q}\overline{\mathrm{q}}$ pair, or many other ways, so that the danger of double-counting is very real. On the other hand, consider the production of a low-$p_{\perp}$, low-mass Drell-Yan pair of leptons, together with two quark jets. Such a process in principle could proceed by having a $\gamma^*$ emitted off a quark leg, with a quark-quark scattering as hard interaction. However, since this process is not included in the program, there is no actual danger of (this particular) double-counting, and so the scale of evolution could be picked larger than the mass of the Drell-Yan pair, as we shall see.

For most $2 \to 2$ scattering processes in PYTHIA, the $Q^2$ scale of the hard scattering is chosen to be $Q_{\mathrm{hard}}^2 = p_{\perp}^2$ (when the final-state particles are massless, otherwise masses are added). In final-state showers, where $Q$ is associated with the mass of the branching parton, transverse momenta generated in the shower are constrained by $p_{\perp}< Q/2$. An ordering that the shower $p_{\perp}$ should be smaller than the hard-scattering $p_{\perp}$ therefore corresponds roughly to $Q_{\mathrm{max}}^2 = 4 Q_{\mathrm{hard}}^2$, which is the default assumption. The constraints are slightly different for initial-state showers, where the spacelike virtuality $Q^2$ attaches better to $p_{\perp}^2$, and therefore $Q_{\mathrm{max}}^2 = Q_{\mathrm{hard}}^2$ is a sensible default. We iterate that these limits, set by PARP(71) and PARP(67), respectively, are imagined sensible when there is a danger of doublecounting; if not, large values could well be relevant to cover a wider range of topologies, but always with some caution. (See also MSTP(68).)

The situation is rather better for the final-state showers in the decay of any colour-singlet particles, or coloured but reasonably long-lived ones, such as the $\mathrm{Z}^0$ or the $\mathrm{h}^0$, either as part of a hard $2 \to 1 \to 2$ process, or anywhere else in the final state. Then we know that $Q_{\mathrm{max}}$ has to be put equal to the particle mass. It is also possible to match the parton-shower evolution to the first-order matrix-element results.

QCD processes such as $\mathrm{q}\mathrm{g}\to \mathrm{q}\mathrm{g}$ pose a special problem when the scattering angle is small. Coherence effects (see below) may then restrict the emission further than what is just given by the $Q_{\mathrm{max}}$ scale introduced above. This is most easily viewed in the rest frame of the $2 \to 2$ hard scattering subprocess. Some colours flow from the initial to the final state. The radiation associated with such a colour flow should be restricted to a cone with opening angle given by the difference between the original and the final colour directions; there is one such cone around the incoming parton for initial state radiation and one around the outgoing parton for final state radiation. Colours that are annihilated or created in the process effectively correspond to an opening angle of 180$^{\circ}$ and therefore the emission is not constrained for these. For a gluon, which have two colours and therefore two different cones, a random choice is made between the two for the first branching. Further, coherence effects also imply azimuthal anisotropies of the emission inside the allowed cones.

Finally, we note that there can be some overlap between descriptions of the same process. Section gives two examples. One is the correspondence between the description of a single $\mathrm{W}$ or $\mathrm{Z}$ with additional jet production by showering, or the same picture obtained by using explicit matrix elements to generate at least one jet in association with the $\mathrm{W}/ \mathrm{Z}$. The other is the generation of $\mathrm{Z}^0 \b\overline{\mathrm{b}}$ final states either starting from $\b\overline{\mathrm{b}}\to \mathrm{Z}^0$, or from $\b\mathrm{g}\to \mathrm{Z}^0\b$ or from $\mathrm{g}\mathrm{g}\to \b\overline{\mathrm{b}}\mathrm{Z}^0$. As a rule of thumb, to be used with common sense, one would start from as low an order as possible for an inclusive description, where the low-$p_{\perp}$ region is likely to generate most of the cross section, whereas higher-order topologies are more relevant for studies of exclusive event samples at high $p_{\perp}$.