Recent Progress in Jet Algorithms and Their Impact in Underlying

Recent developments in jet clustering are reviewed. We present a list of fast and infrared and collinear safe algorithms, and also describe new tools like jet areas. We show how these techniques can be applied to the study of underlying event or, more generally, of any background which can be considered distributed in a sufficiently uniform way. 1 Recent Developments in Jet Clustering The final state of a high energy hadronic collision is inherently extremely complicated. Hundreds or even thousands of particles will be recorded by detectors at the Large Hadron Collider (LHC), making the task of reconstructing the original (simpler) hard event very difficult. This large number of particles is the product of a number of branchings and decays which follow the initial production of a handful of partons. Usually only a limited number of stages of this production process can be meaningfully described in quantitative terms, for instance by perturbation theory in QCD. This is why, in order to compare theory and data, the latter must first be simplified down to the level described by the theory. Jet clustering algorithms offer precisely this possibility of creating calculable observables from many final-state particles. This is done by clustering them into jets via a well specified algorithm, which usually contains one or more parameters, the most important of them being a “radius” R which controls the extension of the jet in the rapidity-azimuth plane. One can also choose a recombination scheme, which controls how partons’ (or jets’) four-momenta are combined. The choice of a jet algorithm, its parameters and the recombination scheme is called a jet definition [1], and must be specified in full (together with the initial particles sample) in order for the process {particles} jet definition −→ {jets} (1) to be fully reproducible and the final jets to be the same. While (almost) any jet definition can produce sensible observables, not all of them will produce one which is calculable in perturbation theory. For this to be true, the jet algorithm must be infrared and collinear safe (IRC safe) [2], meaning that actions producing configurations that lead to divergences in perturbation theory, namely the emission of a very soft particle or a collinear splitting of a particle into two) must not produce any change in the jets returned by the algorithm. †Talk given at MPI@LHC’08, “Multiple Partonic Interactions at the LHC”, Perugia, Italy, October 2008. ar X iv :0 90 6. 15 98 v1 [ he pph ] 9 J un 2 00 9 Jet algorithm Type of algorithm, (distance measure) algorithmic complexity kt [5, 6] SR, dij = min(k2 ti, k 2 tj)∆R 2 ij/R 2 N lnN Cambridge/Aachen [7, 8] SR, dij = ∆R2 ij/R 2 N lnN anti-kt [10] SR, dij = min(k−2 ti , k −2 tj )∆R 2 ij/R 2 N3/2 SISCone [9] seedless iterative cone with split-merge N2 lnN Table 1: List of some of the IRC safe algorithms available in FastJet. SR stands for ‘sequential recombination’. kti is a transverse momentum, and the angular distance is given by ∆R ij = ∆y 2 ij + ∆φ 2 ij . The importance for jet algorithms to be IRC safe had been recognized as early as 1990 in the ‘Snowmass accord’ [3], together with the need for them to be easily applicable both on the theoretical and the experimental side. However, many of the implementations of jet clustering algorithms used in the following decade and a half failed to provide these characteristics: conetype algorithms were typically infrared or collinear unsafe beyond the two or three particle level (see [1] for a review), whereas recombination-type algorithms were usually considered too slow to be usable at the experimental level in hadronic collisions. This deadlock was finally broken by two papers, one in in 2005 [4], which made sequential recombination type clustering algorithms like kt [5, 6] and Cambridge/Aachen [7, 8] fast, and one in 2007, which introduced SISCone [9], a cone-type algorithm which is infrared and collinear safe. A third paper introduced, in 2008, the anti-kt algorithm [10], a fast, IRC safe recombination-type algorithm which however behaves, for many practical purposes, like a nearly-perfect cone. This set of algorithms (see Table 1), all available through the FastJet package [11], allows one to replace most of the unsafe algorithms still in use with fast and IRC safe ones, while retaining their main characteristics (for instance, the MidPoint and the ATLAS cone could be replaced by SISCone, and the CMS cone could be replaced by anti-kt). 2 Jet Areas A by-product of the speed and the infrared safety of the new algorithms (or new implementations of older algorithms) was found to be the possibility to define in a practical way the area of a jet, which measures its susceptibility to be contaminated by a uniformly distributed background of soft particles in a given event. In their most modest incarnation, jet areas can be used to visualize the outline of the jets returned by an algorithm so as to appreciate, for instance, if it returns regular (“conical”) jets or rather ragged ones. An example is given in Fig. 1. Jet areas are amenable, to some extent, to analytic treatments [12], or can be measured numerically with the tools provided by FastJet. These analyses disprove the common assumption that all cone-type algorithms have areas equal to πR2. In fact, depending on exactly which type of cone algorithm one considers, its areas can differ, even substantially so, from this naive estimate: for instance, the area of a SISCone jet made of a single hard particle immersed in a background of many soft particles is πR2/4 (this little catchment area can explain why other iterative cone algorithms with a split-merge step, like the MidPoint algorithm in use at CDF, Fig. 1: Typical jet outlines returned by four different IRC safe jet clustering algorithms. From [10]. have often been seen to fare ‘well’ in noisy environments). One can analyse next the kt and the Cambridge/Aachen algorithms, and see that their single-hard-particle areas turn out to be roughly 0.81πR2. Finally, this area for the anti-kt algorithm is instead exactly πR2. This fact, together with its regular contours shown in Fig. 1, explains why it is usually considered to behave like a ‘perfect cone’. Jet areas also allow one to use some jet algorithms as tools to measure the level of a sufficiently uniform background which accompanies harder events. This can be accomplished by following the procedure outlined in [13]: for each event, all particles are clustered into jets using either the kt or the Cambridge/Aachen algorithms, and the transverse momentum pt,j and the area Aj of each jet are calculated. One observes that a few hard jets have large values of transverse momentum divided by area, whereas most of the other, softer jets have smaller (and similar) values of this ratio. The background level ρ, transverse momentum per unit area in the rapidity-azimuth plane, is then obtained as ρ = median { pt,j Aj }