Resonance Decays

As we noted above, the bulk of the processes above are of the $2 \to 2$ kind, with very few leading to the production of more than two final-state particles. This may be seen as a major limitation, and indeed is so at times. However, often one can come quite far with only one or two particles in the final state, since showers will add the required extra activity. The classification may also be misleading at times, since an $s$-channel resonance is considered as a single particle, even if it is assumed always to decay into two final-state particles. Thus the process $\mathrm{e}^+\mathrm{e}^-\to \mathrm{W}^+ \mathrm{W}^- \to \mathrm{q}_1 \overline{\mathrm{q}}'_1 \, \mathrm{q}_2 \overline{\mathrm{q}}'_2$ is classified as $2 \to 2$, although the decay treatment of the $\mathrm{W}$ pair includes the full $2 \to 4$ matrix elements (in the doubly resonant approximation, i.e. excluding interference with non- $\mathrm{W}\mathrm{W}$ four-fermion graphs).

Particles which admit this close connection between the hard process and the subsequent evolution are collectively called resonances in this manual. It includes all particles in mass above the $\b$ quark system, such as $\t$, $\mathrm{Z}^0$, $\mathrm{W}^{\pm}$, $\mathrm{h}^0$, supersymmetric particles, and many more. Typically their decays are given by electroweak physics, or physics beyond the Standard Model. What characterizes a (PYTHIA) resonance is that partial widths and branching ratios can be calculated dynamically, as a function of the actual mass of a particle. Therefore not only do branching ratios change between an $\mathrm{h}^0$ of nominal mass 100 GeV and one of 200 GeV, but also for a Higgs of nominal mass 200 GeV, the branching ratios would change between an actual mass of 190 GeV and 210 GeV, say. This is particularly relevant for reasonably broad resonances, and in threshold regions. For an approach like this to work, it is clearly necessary to have perturbative expressions available for all partial widths.

Decay chains can become quite lengthy, e.g. for supersymmetric processes, but follow a straight perturbative pattern. If the simulation is restricted to only some set of decays, the corresponding cross section reduction can easily be calculated. (Except in some rare cases where a nontrivial threshold behaviour could complicate matters.) It is therefore standard in PYTHIA to quote cross sections with such reductions already included. Note that the branching ratios of a particle is affected also by restrictions made in the secondary or subsequent decays. For instance, the branching ratio of $\mathrm{h}^0 \to \mathrm{W}^+ \mathrm{W}^-$, relative to $\mathrm{h}^0 \to \mathrm{Z}^0 \mathrm{Z}^0$ and other channels, is changed if the allowed $\mathrm{W}$ decays are restricted.

The decay products of resonances are typically quarks, leptons, or other resonances, e.g. $\mathrm{W}\to \mathrm{q}\overline{\mathrm{q}}'$ or $\mathrm{h}^0 \to \mathrm{W}^+ \mathrm{W}^-$. Ordinary hadrons are not produced in these decays, but only in subsequent hadronization steps. In decays to quarks, parton showers are automatically added to give a more realistic multijet structure, and one may also allow photon emission off leptons. If the decay products in turn are resonances, further decays are necessary. Often spin information is available in resonance decay matrix elements. This means that the angular orientations in the two decays of a $\mathrm{W}^+ \mathrm{W}^-$ pair are properly correlated. In other cases, the information is not available, and then resonances decay isotropically.

Of course, the above `resonance' terminology is arbitrary. A $\rho$, for instance, could also be called a resonance, but not in the above sense. The width is not perturbatively calculable, it decays to hadrons by strong interactions, and so on. From a practical point of view, the main dividing line is that the values of -- or a change in -- branching ratios cannot affect the cross section of a process. For instance, if one wanted to consider the decay $\mathrm{Z}^0 \rightarrow \c\overline{\mathrm{c}}$, with a $\mathrm{D}$ meson producing a lepton, not only would there then be the problem of different leptonic branching ratios for different $\mathrm{D}$'s (which means that fragmentation and decay treatments would no longer decouple), but also that of additional $\c\overline{\mathrm{c}}$ pair production in parton-shower evolution, at a rate that is unknown beforehand. In practice, it is therefore next to impossible to force $\mathrm{D}$ decay modes in a consistent manner.