Cluster Finding

Global event measures, like sphericity or thrust, can only be used to determine the jet axes for back-to-back 2-jet events. To determine the individual jet axes in events with three or more jets, or with two (main) jets which are not back-to-back, cluster algorithms are customarily used. In these, nearby particles are grouped together into a variable number of clusters. Each cluster has a well-defined direction, given by a suitably weighted average of the constituent particle directions.

The cluster algorithms traditionally used in $\mathrm{e}^+\mathrm{e}^-$ and in $\mathrm{p}\mathrm{p}$ physics differ in several respects. The former tend to be spherically symmetric, i.e. have no preferred axis in space, and normally all particles have to be assigned to some jet. The latter pick the beam axis as preferred direction, and make use of variables related to this choice, such as rapidity and transverse momentum; additionally only a fraction of all particles are assigned to jets.

This reflects a difference in the underlying physics: in $\mathrm{p}\mathrm{p}$ collisions, the beam remnants found at low transverse momenta are not related to any hard processes, and therefore only provide an unwanted noise to many studies. (Of course, also hard processes may produce particles at low transverse momenta, but at a rate much less than that from soft or semi-hard processes.) Further, the kinematics of hard processes is, to a good approximation, factorized into the hard subprocess itself, which is boost invariant in rapidity, and parton-distribution effects, which determine the overall position of a hard scattering in rapidity. Hence rapidity, azimuthal angle and transverse momentum is a suitable coordinate frame to describe hard processes in.

In standard $\mathrm{e}^+\mathrm{e}^-$ annihilation events, on the other hand, the hard process c.m. frame tends to be almost at rest, and the event axis is just about randomly distributed in space, i.e. with no preferred rôle for the axis defined by the incoming $\mathrm{e}^{\pm}$. All particle production is initiated by and related to the hard subprocess. Some of the particles may be less easy to associate to a specific jet, but there is no compelling reason to remove any of them from consideration.

This does not mean that the separation above is always required. $2\gamma$ events in $\mathrm{e}^+\mathrm{e}^-$ may have a structure with `beam jets' and `hard scattering' jets, for which the $\mathrm{p}\mathrm{p}$ type algorithms might be well suited. Conversely, a heavy particle produced in $\mathrm{p}\mathrm{p}$ collisions could profitably be studied, in its own rest frame, with $\mathrm{e}^+\mathrm{e}^-$ techniques.

In the following, particles are only characterized by their three-momenta or, alternatively, their energy and direction of motion. No knowledge is therefore assumed of particle types, or even of mass and charge. Clearly, the more is known, the more sophisticated clustering algorithms can be used. The procedure then also becomes more detector-dependent, and therefore less suitable for general usage.

PYTHIA contains two cluster finding routines. PYCLUS is of the $\mathrm{e}^+\mathrm{e}^-$ type and PYCELL of the $\mathrm{p}\mathrm{p}$ one. Each of them allows some variations of the basic scheme.