Dispelling the N 3 myth for the k t jet - finder Matteo Cacciari and Gavin

At high-energy colliders, jets of hadrons are the observable counterparts of the perturbative concepts of quarks and gluons. Good procedures for identifying jets are central to experimental analyses and comparisons with theory. The kt family of successive recombination jet finders has been widely advocated because of its conceptual simplicity and flexibility and its unique ability to approximately reconstruct the partonic branching sequence in an event. Until now however, it had been believed that for an ensemble of N particles the algorithmic complexity of the kt jet finder scaled as N, a severe issue in the high multiplicity environments of LHC and heavy-ion colliders. We here show that the computationally complex part of kt jet-clustering can be reduced to two-dimensional nearest neighbour location for a dynamic set of points. Borrowing techniques developed for this extensively studied problem in computational geometry, kt jet-finding can then be performed in N lnN time. Code based on these ideas is found to run faster than all other jet finders. Partons (quarks and gluons), are the concepts that are central to discussions of the QCD aspects of high-energy collisions such as those at the Fermilab Tevatron and the future Large Hadron Collider (LHC) at CERN. Quarks and gluons, however, are not observable, and in their place one sees jets, collimated bunches of high-energy hadrons which are the result of the fragmentation and hadronisation of the original hard (high-energy) partons. Today’s limited understanding of non-perturbative QCD is such that it is not currently possible predict the exact patterns of hadrons produced. Instead one makes predictions in terms of quarks and gluons and relates these to observations in terms of hadron jets. Naively, jets are easily identified — one simply searches for bunches of collimated hadrons. However, to carry out accurate comparisons between parton-level predictions and hadron-level observations one needs a well-defined ‘jet-finding’ procedure. The jet-finder is applied both to perturbatively predicted partonic configurations and to observed hadronic configurations and one then directly compares distributions for the predicted partonic jets and the observed hadronic jets. Though partonic and hadronic jets are not equivalent, there is strong evidence (theoretical [1] and experimental [2]) that the comparison can be performed with controlled accuracy. Insofar as jet-finding is an approximate attempt to invert the quantum mechanical processes of QCD branching and hadronisation, it is not a unique procedure. Various kinds of jet-finders have been proposed, among them cone-type [1, 3] and sequential-clustering [4, 5, 6, 7] jet-finders (for alternatives, see [8, 9, 10, 11]). Cone jet-finders are the most frequently used at the Tevatron. They are based on identifying energy-flow into cones in (pseudo)rapidity η = − ln tan θ/2 and azimuth φ, together with various steps of iteration, merging and splitting of the cones to obtain the final jets. Cone jet-finders tend to be rather complex, different experiments have used different variants (some of them infrared unsafe), and it is often difficult to know exactly which jet-finder to use in theoretical comparisons. In contrast, the cluster-type jet-finders, generally based on successive pair-wise recombination of particles, have simple definitions and are all infrared safe (for reviews see [12, 13]). We shall focus here on the most widely used of them, the kt jet-finder [5], defined below. Among its physics advantages are (a) that it purposely mimics a walk backwards through the QCD branching sequence, which means that reconstructed jets naturally collect most of the particles radiated from an original hard parton, giving better particle mass measurements [14, 15] and gaps-between-jets identification [16] (of relevance to Higgs searches); and (b) it allows one to decompose a jet into constituent subjets, which is useful for identifying decay products of fast-moving heavy particles (see e.g. [17]) and various QCD studies. This has led to the widespread adoption of the kt jet-finder in the LEP (e e collisions) and HERA (ep) communities. Despite its advantages, kt clustering has so far seen only limited study [18] at the Tevatron. The reasons for this are not entirely clear. One known drawback of the kt jet finder for high-multiplicity hadron-collider environments is its apparent algorithmic slowness: to cluster N particles into jets requires O (N) operations in current implementations [19]. For a typical event at the upcoming LHC, with an expected multiplicity of N = O (2000), this translates into a clustering time of O (10 s) of CPU time on a modern O (3 GHz) processor; this is considerable given that the clustering has to be repeated for millions of events. For a typical heavy-ion event at LHC, where N = O (50000), the clustering time would grow to an unsustainable O (10 s), i.e. more than one day! The slowness of the kt jet-finder has been one of the motivating factors behind proposals for alternative jet-finders [9, 10]. Here we will show that the kt jet-finder can in fact be formulated in an algorithmically fast (N lnN) manner. A C++ implementation of this (and a related N) algorithm will be shown to run orders of magnitude faster than currently available implementations, making it feasible (and easy) to use the kt jet finder for efficiently studying high-multiplicity events. ‘Jet-algorithm’ is often used in the literature to refer to the choice of the rules for finding a jet; here instead ‘algorithm’ refers to the translation of a given set of jet-finding rules into explicit steps on a computer.